

COUNTING in CIRCLEMATHS

by

Robert Michael Taylor

Copyright 2018 by Robert Michael Taylor

Smashwords Edition

ISBN: 978-0-908681-32-7

I

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Contents

ARITHMETIC - IT'S ALL ABOUT YOU

Meet the circles

COUNTING REGISTERS

Flea Counting

Multiplication

Catlish

BASES

Compound Number

Mr Fly and his Compound Eye

Ms Kittie and her Compound Eye

Preventing War Between Ms Kittie and Mr Flea

Human Counting

A BIG PROBLEM - BEING TOO LITTLE

Patterns

Catlish Same Thing

Summary

Bases Inside Bases Inside Bases

THE 100-CIRCLE

CODING ANSWERS

Using Tiny Circles as Code

Going Over 100

Human Counting Base

USING A 100-CIRCLE TO ADD

BOOK SUMMARY

About Circlemaths

Other Titles

Contact the Author
Arithmetic

It's All about

YOU

Most people understand

that arithmetic is

a very useful tool

which accountants, scientists

and commercial people

use.

That's true.

But we're going a bit deeper here.

Actually

Number

is

about

YOU.

It's a description

of how you look

feel and think.

So let's start

by seeing how that is.

by looking at what

number really is.

YOU

What are you?

Mind and body

Two opposites

packaged together

like

Mountain and Valley

Opposites which can't be pulled apart

Body is solid.

Flesh and blood.

Heavy. Very Real.

Mind is like a ghost.

Thought. Feelings.

Ideas.

They are opposites.

Yet they come together, naturally.

Number comes from YOU.

And so its no surprise

it works this way too.
The Birth of Number

is 0 and 1

They're special

They're not real numbers!

Let's say you look out your window

and are suprised to see

a number of cows

have walked onto your lawn.

You tell your friend

"There's a number of cows on our lawn"

"How many cows?"

they ask.

None.

Zero cows.

Does that really make sense?

Not really.

Because a number

is

a group.

How about they ask

"How many cows?"

and you reply:

One.

One cows.

Oops.

That's not quite right either, is it?

It should just be:

One cow.

Cows

is plural

meaning "many"

a group of them

more than

one.

So in one way of looking at things

zero and one

are not numbers.

They are what the other numbers are made of.

Bricks aren't made of brick.

Sheep aren't made of sheep.

They're a bit different

to all the other

numbers.

Zero and one are the parents

of all the other numbers.

How is that?

Think of the number 3.

It is ONE 3.

For after all, two three's are six.

So it's a "one"

by being "one" three.

What's inside it?

Not the three little pigs,

but three little ones.

3 = 1 + 1 + 1

On the left is **ONE** number, the 3.

On the right, three numbers, each of them **ONE** s.

And you can do that with all numbers.

So all numbers are "ones".

Do all numbers have a zero in them?

Yes!

The 3 is 3 steps up from zero.

5 is 5 steps up from zero

and so on.

The size of a number

is measured

from

zero.

Without zero

number couldn't exist!

Look how bad our rulers would be

without a zero:

So every number

inherits from 0 and 1

which are the parents.

They are special

"semi-numbers".

There is another side to this story.

There usually is to all stories...

In another way

we can say:

"Think of a number between 0 and 10"

and you could answer

"one"

or

"zero"

and that would be perfectly fine.

So although

strictly speaking

as we have just shown,

0 and 1 are

not numbers,

in normal everyday usage

they are.

So are 0 and 1 numbers?

You take your pick!

:-)

Anyway,

like the atom

which is very small

but made the atomic bomb

and makes up everything we see

0 and 1 are very small

but

they make up all the bigger numbers.

So being special we'd better become

"atomic numberists"

and look at them

more closely!
ZERO or NOUGHT

Let's start with some exercises.

Exercise 1:

Point to 2 things you can see.

Exercise 2:

Point to 1 thing you can see.

Exercise 3:

(the futile exercise)

Point to 0 things you can see.

You could do them all except the last, right?

That was a trick question!

How can you point to "no things"

Nothing.

You can't.

Which tells us something about zero.

Zero

is the nothing

of number

It doesn't exist "out there"

in the real world.

If you point to it

you've got it wrong.

It's simply

not out there!

You can't find

zero things

in the world!

So where does zero live?

It's inside you!

0 is an idea

in your mind.

It is actually your idea!

Its in you.
ONE

You come as a

double pair of opposites.

Mind and body.

So the ideas inside you

do likewise.

0 is like YOU as mind.

1 is like YOU as body.

0 vanishes away.

1 is solid.

They are both ideas.

0 attaches to nothing.

1 attaches to everything.

They are opposites.

But they naturally exist together.

When we say

"think of a number between 0 and 10"

and we answer

0

we are saying

nought is a number.

We give it an existence

(well, it exists as an idea)

we make it a reality.

Something real.

A ONE.

What is the

First One?

That would have to be

Oneself

You.

See.

You are actually

an expert in number!

That's quite right actually.

You don't know it, but you use numbers

all day long

(not just counting sheep at night!)

Let's find out how that is so

by finding out

what exactly

is

ONE?

Firstly

one can be anything.

Everything is a one.

One star, one grain of sand, one cat.

Even one football game.

What do they have in common

that makes them all the same,

that makes them

all ones?

Actually they don't have a lot in common...

That's a tough question...

Let's look at an example of ONE.

Suppose we look at

ONE

bike:

What's it got in common

with a star, a grain of sand

and a football game?

Nothing.

Except this.

YOU

can observe

the bike and

all of those things.

Actually

that's the only thing

they have in common!

Your attention.

Every time you've ever seen

any of those things

YOU"VE

been there

watching

right?

YOU

are the

ONE

common factor

linking

all those things!

So in search of

What ONE means

let's look at your

attention!

Exercise

Look at ONE word.

Notice it is made up of

lots of letters

inside it.

And there are

lots of other words

outside it.

But the moment YOU selected it

as your focus of attention

you made it

"The ONE"

Pick another word

on the same page

a bit away from it.

Focus on that.

Did you notice

how the original word

got lost

blended in to the background

and faded away

from your attention?

If not, have another look.

When YOU change

the focus of your attention

and select

another ONE word

the original word

blends and fades

from view.

End of Exercise

That's your attention at work.

Casting the net of ONEness

here, there and everywhere

it looks.

Let's get back to our bike example.

Here is a prime example of a ONE.

ONE bike:

One bike is just one bike.

But also it can be exploded into many parts.

It has little ONEs

inside it.

The wheels, the seat, the bell

and so on.

Each of these

is a ONE in its own right:

Inside the ONE

we find its made of

lots of little ONEs.

You could switch your attention

to the bell

and see you have

ONE

bell.

ONE chain

ONE frame

ONE wheel

Another ONE wheel

ONE seat

and so on.

All of them are equally ONEs.

Inside the ONE bike.

But notice

they don't split off

and become

ONE's

until

someONE

namely

YOU

pays them

attention!

Your brain is so clever!

It takes all these little parts

and FUSES them together

to make

just ONE bike

you are looking at!

We forget, dismiss, ignore

the differences between the parts

and fuse them together

in our minds eye

to make

just

ONE

bike.

That's an interesting idea.

Ignoring, dismissing, forgetting

is actually

part of the process

by which your mind

focuses and pays attention

in order to select

a

ONE.

That's making

nothing out of something.

NOUGHT.

There's another side to this story too.

(There usually is, to most stories)

The bike could be sitting next to other bikes.

Behind it might be a mountain, some clouds in the sky

some grass...

All sorts of things.

But when we point to the bike

as the example of ONE

we don't include them.

We EXCLUDE them.

This time our mind doesn't

FUSE them TOGETHER

in the ONE

rather

it EXCLUDES the background.

Ignores it.

Says its unimportant to the bike.

But the background

is

ALWAYS THERE!

Even though we ignore it,

even though it may change,

it doesn't go away.

It is ALWAYS there.

So it is like the zero,

the silent partner of the ONE thing you see.

So an

EXCLUDED BACKGROUND

is actually

part of the story of

ONE.

Excluding the stuff around the bike,

ignoring it, dismissing it,

relegating it to the background

is how our

focus of attention

works!

We make them vanish from our attention.

It's a bit like

making them go to

ZERO

Once again

we have two opposite ideas

Paying attention to something

and

Ignoring, dismissing, relegating

something else to the background

which seem to be

part and parcel

of the same process

we call paying attention

to ONE thing.

Isn't it strange how our minds work?

We can see other examples of

seeing something

because something else

else is switched off or hidden:

A movie.

We can't see the movie

until the lights are turned off.

Then we see it.

Cutting off the light

brings the movie to our attention.

The moon and the stars.

Did you know the stars are up in the sky

all day long?

It's just that we can't see them

because the sun gives off too much light.

When the sun sets

and darkness falls

we can see

the moon and the stars.

A dream.

When we close our eyes

and ignore any sounds

we fall asleep

and then

we can see our dreams!

The sides of binoculars

cut off the light from the sun

helping you see

what you are focused on.

There are lots of similar examples.

I'll bet you could think of some.

I hope you are starting to see a link

between 0 and 1

seeing and "ignoring"

and how you perceive the world

with the help of

your mind and your eyes.
The IDEAL ONE

Let's put it all together

to draw a picture

of

ONE.

Basically

ONE

fuses everything together

inside it

It doesn't see

"little parts"

inside itself.

(no little "ones")

At the same time

ONE

excludes, cuts off and ignores

the background.

It doesn't recognise

anything else

outside and alongside it.

(no twos or threes next to it)

Nothing

inside it

Nothing

outside it.

Just ONE

The PURE IDEA of ONE.

It's just like we've drawn it.

Simple.

Shocking News!

The Ideal One

is

TWO!!

Why?

Our basic idea as drawn

has

an INSIDE

and

an OUTSIDE

That's TWO

And that's no good!

We were meant to be getting the pure idea

of just ONE!

Let's fix it!

Let's get rid of one of them!

In the above cartoon, we are shrinking the INSIDE

until its just a

vanishing dot!

Then it winks out of existence!

Ha Ha!

Good riddance!

BUT

The end result

is

NOTHING

A blank sheet of paper!

This is not ONE

It is

the **tiny** zero.

Our ONE turned into nothing (zero).

Let's try another tack...

This time,

let's get rid of that pesky background!

In the above cartoon we shrink the background...

The INSIDE of the ONE expands!

It fills the whole page!

The end result

is

NOTHING

A blank sheet of black paper

with NOTHING on it.

This is the BIG zero.

It is like the zero which is the emptiness of outer space,

infinite and empty forever and ever.

A big

NOTHING

Zero.

In two opposite ways

(tiny zero and Big zero)

One turns back to Zero

when we try to stop it being

TWO.

All we have really shown

is that 0

becomes 1

becomes 2

0, 1, 2...

That's some of the magic

of how ideas work

deep in your mind.

They flow and change.

Opposite ideas are bound together

and turn into one another.

It's strange.

But that's just the way

nature works.
Meet the Circles

0-circle

ONE

SELF

In Circlemaths

the zero

represents

the observer.

YOU.

The ultimate zero

is not the symbol

0

it's the circle itself.

It's probably not an accident

that a zero

is drawn in the form

of a circle.

A circle on its own

is called

0-circle.

It's the first circle.

0-circle represents

only an outer

circle of awareness

but

nothing inside it

to be aware of.

This represents you when you are fast asleep.

"Dead to the world"

No dreams.

Nothing is seen.

Nothing is heard.

Nothing is thought.

You are not even aware of yourself.

But that apparent nothingness

is aware nevertheless.

Make a noise and see what happens!

You instantly spring awake!

This is the most basic level

of awareness that exists.

Inheritance

You inherit certain features

from your parents.

Perhaps you have the same colored eyes

or a similar face or height.

Numbers also inherit

from their parents.

0-circle

is the parent of all

number circles.

Every number inherits

this circle.

Every number henceforth appears

inside a circle.

Another way of saying it is that

the basic level of awareness

present in deepest sleep

is also present

throughout your entire life.
1-circle

Quick Exercise

Without looking in a mirror

are you aware of yourself now?

As a thinking person?

That means you are

aware you are aware.

Self-awareness.

We could represent that

mathematically

using a 1-circle

like this:

If the circle

is your awareness

and what is inside the circle

is what you are aware of

then in this case

you are aware

of yourself

and nothing else.

Congratulations.

You've just woken up.

This circle represents

you having just awoken

without yet opening your eyes

or bothering to listen to noises

around you.

You aren't taking in anything

from the outside world yet.

You are just

aware of yourself.

Just lying in bed awake

but with your eyes closed.

The zero inside the circle

represents YOU.

The circle itself

is inherited from 0-circle.

It is the awareness

that is always present.

Inheritance

If 0-circle is the mother

then 1-circle is the father

of all circles.

They are the parents.

All circles

inherit what they have.

All circles from here on

(so excluding only the 0-circle)

have a zero inside them

at the very top.

0-circle doesn't have a

zero inside it

because it

IS

the true

zero.
2-circle

This circle has both

a 0 and a 1

inside it.

It represents

YOU

when you have woken up

and are now looking

at something.

The always present awareness

of the circle

has two things

inside its awareness:

The 0 is you.

(you are aware of yourself)

The 1 is something you are looking at.

The 1 is a solid thing you can see.

The 0 is you as thought, ideas, not solid at all.

The surrounding circle is the deepest level

of awareness which is always present.

Circles only know what is inside them.

2-circle knows itself (as the 0)

and it knows what 1 is.

But it doesn't understand

the idea of TWO yet.

There is no "2" inside 2-circle!

Mind you

2-circle has

TWO

"semi-numbers"

as shown below:

and it has

TWO

journeys

you can make

around the circle

in

TWO

directions

as shown below:

It doesn't have

a TWO inside it

because

it IS TWO.

It represents what TWO

really means.

2-circle

represents

the IDEA of TWO.

It's On the Tip of My Tongue

Have you ever had the experience

of knowing what a word means

but not being able

to quite remember it?

We say

"its on the tip of my tongue".

We understand what we want to say

but we can't find the word.

This is what 2-circle is

for the idea of 2.

It IS the very idea of 2.

But inside it

there is no 2.

That is like

understanding the idea of 2

without knowing the word "two"

or recognising the symbol "2"

2-circle

is the understanding and meaning

of what TWO is

but its

on the tip of my tongue!

Inheritance

2-circle has a 0 up top

and a 1

down below

in the middle.

0 is even

1 is odd

and

1 is half of 2.

From now on

every second number

will also take this form.

For example

here are the next two even circles,

4-circle and 6-circle:

The 4-circle

has a 2

at the bottom

cutting it in half

and 2 is half of the 4-circle.

Likewise the 6-circle

has a 3 at the bottom

cutting it in half

and 3 is half of the 8-circle.

Every second number is even.

0, 2, 4, 6...

Not so for odd numbered circles.

Every second number is odd.

1, 3, 5, 7...

The 3-circle has nothing

at the half way mark,

and neither does the 5-circle.

Nor does any odd circle

ever have

a number

dead center at the bottom.

There will always be

a gap there.

This inheritance of TWOness

breaks

all number

into

Odd and Even

forms.
The Circle

Counting Line

Let's look at all the circles

as they step out into a counting line:

3-circle

is what we

mean

by the symbol

"3"

2-circle

is what we mean

by the symbol

"2"

and so on.

Here are some more circles in that line:

Let's count up to 10-circle...

We will find when it comes to calculating

that 9 and 10-circle

are especially valuable.

What's Missing?

In a 4-circle

4

is missing.

The biggest number in

4-circle

is a 3.

We name the circle

after the number

that is missing.

Because that's the number

it IS.

The definition of "cat"

in the dictionary

shouldn't use the word

"cat"

"A cat is a type of cat"

is a poor definition.

For the same reason

the 4-circle is the meaning behind

the symbol

4

so it shouldn't appear

inside it.

Circles are always named

after the number

that is

missing.

The Missing Person

Captain Checkit

was asked to go aboard

a submarine

and phone back

how many people

were in it.

He looked

and saw

2

people in front of him.

He called back "2 people!"

His commanders wrote down

3

on board.

Captain Checkit

had forgotten to count himself!

This is a picture of it:

The 0

represents Captain Checkit

He is invisible

to his own eyes.

He only sees 1 and 2

so he made a mistake.

Sent on to another submarine

he sees just

1

other person

and relays back

"1 person on board"

His commanders dutifully write down

2

people on board.

He's done it again!

This is the picture:

Captain Checkit is the 0

He only sees the 1

but really

together they make

2

One day he is asked to go on board

an empty submarine.

He looks and looks

but can't see anyone there!

"No-one on board Sir!

he reports.

0

Of course,

he made the same mistake.

His commanders write down

1

person on board.

Captain Checkit himself.

Drawn:

Captain Checkit is the 0

There is no-one else in the room

(in the circle that is)

But this is a 1-circle

because it has

ONE

zero

in it.

Last case!

Captain Checkit is asked to check

how many people are on board

once again.

He decides

NOT to go into the room.

That way he won't make his previous mistake!

"If I don't go into the room,

I won't have to worry about

forgetting to count myself"

he thinks.

So he stands outside

and knocks

raher loudly.

No-one answers.

He concludes

"No-one in the room"

and reports back

0

This is the scenario:

This time he got it right!

However,

there is another interesting possibility.

What if there were hundreds of people

in that room

but they all decided

to not answer the door

and not tell him they were there.

Could he tell?

No!

He never went into the room!

So in fact

there could have been any number

of people in that room!

At least, potentially anyway...

He can't be sure...

And that is a lot like zero.

It is the idea of number.

The potential idea

behind _any_ number.
Number

as Groups

We've seen that 0 and 1

are not really numbers.

They form a special group

all of their own.

All the other circles

represent numbers.

A number is more than one.

It is a group.

Did you know you

see things in groups

all the time?

We see in numbers.

Certain groups

are recognised by a part of your brain

almost instantaneously.

Exercise

How many red dots are in each square below?

Do this as quickly as possible.

You look at a square

and either

you instantly know

how many are in it  
(like "1")

or you don't.

If you don't

DO NOT COUNT

just move to the next square.

It should all be over in 4 seconds tops!

What was the highest number

you could instantly recognise?

It probably wasn't very high.

I could instantly recognise only up to 4.

Whatever the number was

its usually pretty small.

But you recognise it

INSTANTLY!

If the dots

are put in certain shapes

you can recognise them instantly

by their shape.

This means your instant recognition

appears to be higher.

Try these.

Do the same test.

Glance at each square.

You either know instantly

how many

or move on to the next square:

How did you go this time?

Numbers in patterns

are easier to spot.

These patterns are familiar

as they are off dice.

Playing dice games

is a good way

to help build your

number abilities.

Dots that don't come in familiar patterns

are harder to instantly recognise.

Nevertheless

something in your brain

is capable of

instant number recognition

even without

any patterns.

Let's try counting a large number of dots.

You probably can't

instantly recognise

how many dots are in the pictures below.

So count them.

When you do,

observe HOW you count them!

Here's how I counted them:

I found they broke into groups

I could

instantly recognise.

I recognised the 5 group

because it was off the dice.

Same with the 4 group

which was a square.

The two 2's and the two 1's

I instantly recognised.

Then I just added up those parts.

5 +4 + 2 + 2 + 1 + 1 = 15

That was combining

instant recognition

with adding.

Slower than just

instantly recognising 15,

but much quicker

than counting the dots

one-by-one.

Count the dots in the squares below.

You won't instantly recognise them:

Did you find the 8, 9 and 10 were easier?

The 8 was two lots of 4 (in square pattern)

The 9 was three lots of 3 (triangles)

The 10 was a 6 (off a dice) plus a 4 (square pattern).

Did you notice the other numbers also added to

8, 9 and 10?

But see the difference.

They weren't in easy patterns.

So counting them was much slower!

You may have recognised patterns while counting them.

If you didn't,

try again,

counting them in two's

or threes

at a time.

I hope all this has convinced you

that you instantly recognise some numbers

and that when you look at things

your brain breaks them into

tiny instantly recognisable

chunks.

That's what you do.

That's how you look.

If you want to take another look

at how you spot patterns,

try counting these:

The answers

should be 12, 15, 13 and 14.

Count them yourself.

Watch HOW you count them.

I've read that

some people can instantly recognise

more than a hundred dots.

It can only be taught to babies

and only if they are under about 3 years old.

A chap named Glen Doman

figured out how to teach them.

Great work Mr Doman!

But me,

I'm stuck with 3!

Most people are like me.

The important thing

is not how high you can go

but to note

there are two types:

numbers you instantly recognise

(like 0, 1, 2)

and numbers you have to count

(like 9, 8, 7)

Even the children taught by Glen Doman

have an upper limit.

Once they hit a few thousand dots

they're counting too!
Why do we have

Instant Number Recognition?

Over a hundred years ago

some hunters were trying to capture some crows

that were eating their grain in the barn.

A hunter would go into the barn

and wait for the crows.

But the crows were too smart!

They remembered he was in there!

So the hunter decided to trick the crows.

He brought along a friend,

and they both went into the barn together.

Then one of them left,

hoping to trick the crows into thinking

they had both left.

But the crows were too smart!

They could count!

They knew there was another hunter in there!

So the next day they tried again.

This time three hunters went into the barn.

And two walked out.

Again hoping to trick the crows!

But once again,

the crows proved to be too smart!

They could definitely count up to three!

They knew a hunter was still inside the barn!

The following day

four hunters went into the barn

three came out,

leaving one hunter behind.

The crows flew into the barn

thinking eveyone had gone

and were caught.

The crows could count up to about four.

Then they lost count.

You can see how that helped them survive.

All animals have this instant number recognition.

Different animals can count

up to different numbers.
COUNTING REGISTERS

Computers have tiny memory banks

called

registers.

They only have a few of them.

The registers can do arithmetic.

Very very quickly!

In fact, the registers

are the secret of the computer's speed.

When we do arithmetic

and the sum is quite large

we write bits of it down

on paper

to help us remember

as we go along.

Computers do the same.

If a sum is too large

it has to store bits of the answer

not on paper

but in memory.

This slows the computer down.

It has to store them,

then when it wants to use them

it has to retrieve them.

Back and forth

back and forth

all very time consuming

for the poor old computer.

So the computer has registers.

They don't have to store anything

in memory.

They are a tiny chunk of memory

built directly into the chip.

A chip is what makes the computer run.

It's the control center.

The registers can do tiny bits of arithmetic

at fantastic speed!

We have registers too.

As do crows, ducks, penguins and gorillas.

Our registers

are the circles.

The circle-registers

help us understand

and instantly recognise

numbers.

As it turns out

it is likely a human has 7 to 8 registers.

Psychologists have made speed tests

and this seems to be the case.

We can recall

about 7 to 8 things

in our short term mermory.

After that

if we have to remember another thing,

we forget one of the others!

The crow had about 3 or 4.

A flea might only have 3!

A cat may have 6.

Maybe an elephant has 11 or 12!!!
Flea Counting

Here is a flea's biggest circle-register:

It's a flea-circle!

I mean a 3-circle!

It's a flea's 3-circle register

for instant number recognition.

Question

Can the flea count?

Answer

Yes it can.

But like the crow,

it quickly loses count.

Let's see how counting works

using a single 3-circle register

for a flea.

The flea can successfully count up to 2

as shown below:

He starts at 0

and moves on to 1

then on to 2.

Yay!

He's counting!

But when it tries to count past 2

trouble looms!

When

he reaches 2

his next step

takes him right back

to the start

to zero!

He loses count!

He can't count to 3

because

the 3-circle register

doesn't have a

3

Poor flea!

We could draw it this way:

One Too Many

The flea

can't count to 3

His 3-circle register

doesn't have a 3 in it.

He loses count after 2

He really counts things this way:

One, Two, MANY!

Adding that extra 1

onto 2

is

"one too many"

for him!

One leg to jump on...

Two legs to jump on...

and when he spots a third leg

he loses count

and says instead:

MANY legs to jump on!

Have you heard of

infinity?

Infinity is meant to be

a number so big

you can't even conceive it.

For Mr Flea,

that's easy.

Infinity for him

is

3

Because that's the number

so big

he can't even conceive it

in his little

3-circle register world

which doesn't have

a

3

By the way

starting at 0 on 3-circle

and counting out 3 steps

brings you full circle back to 0

So in a funny way

if 3 is infinity

for Mr Flea,

then so is zero!

Perhaps we should call it

infini-flea

instead!

Ha ha!

There is evidence

that primative humans

counted

One, Two, Many

as well.
Flea Arithmetic

The atom is tiny

but by studying the tiniest of things

scientists have created

iPads, the GPS, and computers!

A baby is thought to be quite simple

but you can give a baby

to French parents

or Greek parents

or Japanese parents

and that baby

will learn to speak

French or Greek or Japanese

fluently

within a few years.

The baby can learn

any language in the world

without using a dictionary

in just a few years.

Actually

the baby is a genius

in disguise!

So is Mr Flea!

Never overlook the simple!

We'll be sure not to overlook

the simple arithmetic

of the Flea!

Let's look more closely

at how he counts...

We've already shown

Mr Flea would count

to 3 and lose count like this:

To turn a number into

flea-talk

just count it out

around the

flea-circle

For example,

here's how to turn 7

from English into Flealish:

Exercise 1

Count these numbers out

around the

3-circle

to convert these numbers

to Mr Flea's language:

1_1. What is 4 in Fleaish?

1_2. What is 8 in Flealish?

1_3. What is 10 in Flealish?

1_4. What is 15 in Flealish?

1_5. What does 17 come to in Flealish?

1_6. What is 23 in Flealish?

1_7. What does 28 come to in Flealish?

Answers to Exercise 1:

Answer to 1_1. What is 4 in Flealish?

4 counts out to 1

Answer to 1_2. What is 8 in Flealish?

8 counts out to 2

1_3. What is 10 in Flealish?

10 counts out to 1

Answer to 1_4. What is 15 in Flealish?

15 counts out to 0

Answer to 1_5. What does 17 come to in Flealish?

The answer is 2

Answer to 1_6. What is 23 in Flealish?

It counts out to 2

Answer to 7. What does 28 come to in Flealish?

It comes to 1

One good thing about Mr Flea

is that he keeps things simple!

I think I like his small numbers!

Exercise 2

2_1. Turn 3, 6 and 9 into "Flealish"

2_2. Turn 4, 7 and 10 into Flealish

2_3. Turn 5, 8 and 11 into Flealish

2_4. Turn 31 into Flealish

2_5. Turn 62 into Flealish

Answers Exercise 2

**Answer 2_1** : Turn 3, 6 and 9 into "Flealish"

3 is once around the circle and back to 0

6 is twice around the circle and back to 0

and 9 is three times around the circle and back to 0

Easy!

**Answer 2_2** : Turn 4, 7 and 10 into Flealish

If 3 comes to 0 then 4 is one more which makes it 1

If 6 comes to 0 then 7 is one more which makes it 1

if 9 comes to 0 then 10 is one more which makes it 1

**Answer 2_3:** Turn 2, 5 and 8 into Flealish

If 3 comes to 0 then 2 is the number

just before 0

which is 2

if 6 comes to 0 then 5 is the number

just before 0

which is 2

if 9 comes to 0 then 8 is the number

just before 0

which is 2

**Answer 2_4:** Turn 31 into Flealish

Every time we count 3

we go back to 0.

Because 30 is 10 threes

it is 10 times around the circle

and back to 0

So 31 will just be 1 more on from there:

Answer: 31 is 1 in Flealish

**Answer 2_5:** Turn 62 into Flealish

60 is 30 + 30

and just above we showed 30 comes to 0

so 60 is 0 + 0 = 0

That means 62 must come to 2

around the flea-circle

To help you further

we could write out the flea's

circlar counting line.

He counts like this:

Every time he should get to 3

he loses count instead!

Compare our counting line

with Mr Flea's circular counting line:

If the flea wanted to count

how many pointy bits

a triangle has

he would say

0

when we would say

3

Poor Mr Flea!

Suppose we ask

Mr flea!

How many corners

does a piece of paper have?

"Let me see..."

says the flea

He starts at 0: then counts 1 2 0 1

That's

1

he says!

(check it's "correct" in the picture above

by seeing what flea-number is under 4)

Poor flea!

Not a very good counting line.

He has the same problem the crows had!

To help you answer the next questions

(although you could count them out

around the circle if you prefer)

here is a longer list

comparing our counting line

to Mr Flea's:

So let's see

how astounding Mr Flea's

simple

3-circle-register

"calculator"

can be!

It might astound you

but the flea's simple 3-circle-register

can do any arithmetic in the world

so long as the answer

comes to

0, 1 or 2

As long as the final answer

is 0, 1 or 2

he will get it right!

Here's an example:

Use Fleamath to find:

5 + 8 - 11 = ?

First convert the numbers to Flealish:

5 = 2

8 = 2

11 = 2

The flea sees:

5 + 8 - 11

as

2 + 2 - 2

looks easier through his eyes

doesn't it!

I think you can easily see

that 2 + 2 - 2 must be 2

The answer is 2.

Just check that:

With a pencil and a 3-circle

Start at 0

count out 2 (to get to 2)

count out another 2

(you'll go right round and on to 1)

weird stuff eh?

then count one lot of 2 BACKWARDS

to confirm the answer

is 2 as shown below:

5 + 8 - 11 = 2

is correct.

NOTE

5 + 8 - 11 came to 2 + 2 - 2

which you could count out

around the circle

as we did above

which is a totally

WEIRD WAY

to do it

OR

you could just see

in 2 + 2 - 2 that

+2 and -2

cancel each other

leaving 2.

which is

the EASY way.

IMPORTANT

Do each sum

BOTH

ways!

First,

do it the easy way.

Then

do it the

WEIRD way

Shortcuts are good

but so is weird

in this case.

Doing it by counting it out

around the circles

will familiarize you

with some very

very

very

important

principles.

You'll absorb them

automatically

without even knowing it!

So don't skip the weird way!

And don't skip the easy way either!

Try them both!

Which do you prefer?

We'll sometimes show the weird way

sometimes the easy way.

Sometimes

both ways are needed

together.

So it's important

to know them

BOTH!

Here are some sums for you to try.

Use the English/Flealish comparison list to help you.

The answers always come to 0, 1 or 2:

Exercise 3

3_1) What is 11 - 9?

3_2) What is 21 - 19

3_3) What is 7 + 6 - 10 - 2

3_4) What is 16 + 15 + 14 - 18 - 12 - 14

3_5) What is 23 + 29 - 18 - 17 - 16

Answers to Exercise 3

Answer to 3_1. What is 11 - 9?

Look up the conversion:

11 is 2

9 is 0

Mr Flea sees

11 - 9

as

2 - 0

which comes to 2.

Answer: 11 - 9 = 2

Ans 3_2. What is 21 - 19

Look up the conversion:

21 is 0

19 is 1

Mr Flea sees

21 - 19

as

0 - 1

Put your pencil on 0

at top of the 3-circle.

Go backwards 1

You arrive at 2

0 - 1 = 2

in Flea-circle

Answer: 21 - 19 = 2

Let me show you how 0 - 1 = 2:

Easy eh?

Ans 3_3. What is 7 + 6 - 10 - 2

Look up the conversions:

7 is 1

6 is 0

10 is 1

and 2 is 2

Mr Flea sees

7 + 6 - 10 - 2

as

1 + 0 - 1 - 2

Easy way: 1 - 1 cancel leaving 0 - 2 = 1

as illustrated below:

Weird way:

Start on the 1

Do nothing to take off 0

Move BACK 1 to take away 1

(this will bring you to 0 again)

Move BACK 2 more steps

which brings you to 1

The answer is 1

7 + 6 - 10 - 2 = 1

Ans 3_4. What is 16 + 15 + 14 - 18 - 12 - 14

Convert the numbers to Flealish:

16 = 1

15 = 0

14 = 2

18 = 0

12 = 0

14 = 2

Mr Flea sees

16 + 15 + 14 - 18 - 12 - 14

as

1 + 0 + 2 - 0 \- 0 - 2

Easy way:

+2 -2 cancel leaving 1 and a whole lot of 0's!

It's 1

Weird way:

Start on 1

Do nothing to add 0.

Go forward 2 more.

This brings you to 0 again!

Do nothing to take away 0.

Do nothing again to take away the other 0.

Go backwards 2 steps to arrive on 1

Answer: 16 + 15 + 14 - 18 - 12 - 14 = 1

Ans 3_5. What is 23 + 29 - 18 - 17 - 16

Convert the numbers to Flealish:

23 = 2

29 = 2

18 = 0

17 = 2

16 = 1

Mr Flea sees

23 + 29 - 18 - 17 - 16

as

2 + 2 - 0 - 2 \- 1

Start on 2

Go forwards 2 to arrive on 1

Do nothing to subtract 0 (stay on 1)

Go BACK 2 steps to arrive on 2

Go BACK 1 more step to arrive at 1

The anwer is 1

23 + 29 - 18 - 17 - 16 = 1

MULTIPLICATION

To do the next group of sums

you need to have

just a little

information

about

multiplication.

Not a lot...

A flea's bite sized amount

will do...

It's very simple.

The multiplication

"2 x 3"

is shorthand

for an addition.

It means:

"two lots of three"

which means

3 + 3

whereas

"3 x 2"

would mean:

"three lots of two"

or

2 + 2 + 2

Just to make sure you have got it,

in the exercise below

match the sum to the

correct picture:

Exercise 4

Match the sum to the correct picture:

4_1) 3 x 2

4_2) 4 x 2

4_3) 4 x 5

4_4) 5 x 3

4_5) 3 x 5

4_6) 2 x 3

Answers to Exercise 4

4_1) 3 x 2 = 2 + 2 + 2

4_2) 4 x 2 = 2 + 2 + 2 + 2

4_3) 4 x 5 = 5 + 5 + 5 + 5

4_4) 5 x 3 = 3 + 3 + 3 + 3 + 3

4_5) 3 x 5 = 5 + 5 + 5

4_6) 2 x 3 = 3 + 3

So all multiplications

are just shortcuts for additions

and you can turn any multiplication

back into the addition it came from.

Notice how the pictures

take on the shape

of the numbers?

Here's a fun one

See if you can figure it out

3 x 3 x 3

or,

even better...

It's quite fun

drawing multiplication

patterns

Why don't you

make up

some of your own?

See what pictures

you can make?

But of course,

all these multiplications

are

WAY TOO BIG

for Mr Flea!

No three for Mr Flea!

So let's look

at some

SMALLER

multiplications

involving

little "numbers"

like 0 and 1 for a start...

The ZERO Times Table

Remember

Mr Flea can't count to 3!

Here are all the

0 x Tables

for Mr Flea:

0 x 0 = 0

0 x 1 = 0

0 x 2 = 0

spot the pattern?

All the answers are 0!

The ONE Times Table

Here are all the

1 x Tables

for Mr Flea:

1 x 0 = 0

1 x 1 = 1

1 x 2 = 2

spot the pattern?

All the answers

are the same

as what the 1

is multiplying!

So 0 and 1

times tables

are easy!

They always work that way.

No matter which circle register you are using!

See! I told you 0 and 1 were special!

The TWO Times Table

The BIGGEST number

in Mr Flea's 3-circle register

is 2

Let's look at the 2 x table.

There is a surprise there...

That's right, that's wrong!

The last one is _wrong_!

Mr Flea

can't count to three,

so how can the answer

be

4?

It can't!

Let's see how Mr Flea

really sees

2x2

by looking at his

3-circle-register:

2x2

is shorthand for

Two lots of 2

and as you can plainly see

for Mr Flea

two lots of 2

add to

1

So he writes

2 x 2 = 1

and is correct

in his tiny world.

Now you know

ALL

the times tables answers

in Mr Flea's world

let's try doing some

multiplications!

EXAMPLE

What is (4 x 5) - (6 x 3) - 1

I've written the multiplications

in brackets.

Think of brackets

like a pair of bulldozers.

They crush the numbers within them

until they leave

only

one

number.

Look below

to see them

crushing

2+2

into

just a single

4

When they are finished

we can simply throw them away!

Heh heh...

Tossing aside bulldozers!

what fun!

We'd better repeat our

conversion codes

for convenience here:

What is (4 x 5) - (6 x 3) - 1

Convert the numbers to Flealish:

4 = 1

5 = 2

6 = 0

3 = 0

1 = 1

Mr Flea sees

(4 x 5) - (6 x 3) - 1

as

(1 x 2) - (0 x 0) - 1

1 x 2 = one lot of 2 = 2

0 x 0 = no lots of 0 = 0

so this sum becomes

(2) - (0) - 1

let's throw away the brackets!!!

it's now

2 - 0 - 1

Start at 2

Do nothing to subtract 0.

Go BACK 1

This brings you to 1

Answer: (4 x 5) - (6 x 3) - 1 = 1

clever little flea!

EXERCISE 5

Here are some

multiplications for you to try:

5_1) What is (5 x 3) - (7 x 2)

5_2) What is (7 x 5) - (2 x 16) - 1

5_3) What is (8 x 22) - (9 x 20) + 6

5_4) What is (6 x 18) - (5 x 20) - 7

ANSWERS

Ans 5_1) What is (5 x 3) - (7 x 2)

Convert the numbers to Flealish

5 = 2

3 = 0

7 = 1

and

2 remains 2

Mr Flea sees

(5 x 3) - (7 x 2)

as

(2 x 0) - (1 x 2)

2 x 0 = 0

1 x 2 = 2

so this becomes

(0) - (2)

throw away the brackets!

it's just

0 - 2

we'll draw that for you:

Start at 0 on the 3-circle.

Go BACK 2 steps

This puts you on 1

The answer is 1.

(5 x 3) - (7x2) = 1

Ans 5_2. What is (7 x 5) - (2 x 16) - 1

Convert the numbers first:

7 = 1

5 = 2

2 remains 2

16 = 1

1 remains 1

Mr Flea sees

(7 x 5) - (2 x 16) - 1

as

(1 x 2) - (2 x 1) - 1

That comes to

2 - 2 - 1

because

1 x 2 = 2

and

2 x 1 = 2

To find 2 - 2 \- 1:

Easy way:

2 and -2 cancel leaving -1 = 2

Weird way:

Start at 2

Go BACK 2 steps to return to 0

Go BACK 1 more step.

You arrive at 2.

The answer is 2.

(7 x 5) - (2 x 16) - 1 = 2

Ans 5_3. What is (8 x 22) - (9 x 20) + 6

Convert the numbers first:

8 is 2

22 is 1

9 is 0

20 is 2

6 is 0

Mr Flea sees

(8 x 22) - (9 x 20) + 6

as

(2 x 1) - (0 x 2) + 0

That comes to

2 - 0 + 0

because

2 x 1 = 2

0 x 2 = 0

Start at 2

Do nothing to subtract 0 (stay on 2)

Do nothing to add 0 (stay on 2)

The answer is 2

(easy eh?)

(8 x 22) - (9 x 20) + 6 = 2

Ans 5_4. What is (6 x 18) - (5 x 20) - 7

Convert the numbers first:

6 = 0

18 = 0

5 = 2

20 = 2

and

7 becomes 1

Mr Flea sees

(6 x 18) - (5 x 20) - 7

as

(0 x 0) - (2x2) - 1

That comes to

0 - 1 - 1

because

0 x 0 = 0

and

2 x 2 = 1

2x2 = 1

is the flea's toughest sum!

So here's a picture.

2 x 2 means two lots of 2

added together.

Drawn below they come to 1

in the flea-brain:

So now we have to work out

0 - 1 - 1

to get the answer!

Put your pencil on 0 to start.

From zero take off 1

by moving BACK 1 step

to land on 2.

Finally, take off another 1

by moving BACK 1 more step

to land on 1.

The answer is 1.

(6 x 18) - (5 x 20) - 7 = 1

Here's a few more.

You can actually make up your own

using a calculator.

Just make sure the final answer

is

0, 1 or 2

and any sum will work!

The flea only gets tiny answers

but he gets them perfectly correct

every time!

EXERCISE 6

6_1) 19 + 17 \- 13 -18 -15 +29 - 16 - 1

6_2) 15 + 16 \+ 17 - 18 - 14 -13 - 2

6_3) 25 + 14 \- 19 - 28 + 17 - 9

6_4) 29 + 25 \+ 17 - 22 - 26 - 24 +3

6_5) 17 + 17 \+ 17 + 23 - 18 - 18 - 18 - 19

6_6] 36 - 29 \+ 15 - 17 + 28 - 19 + 25 - 29 - 8

ANSWERS to EXERCISE 6

6_1) 19 + 17 - 13 -18 -15 +29 - 16 + 1

becomes

1 + 2 - 1 - 0 \- 0 + 2 - 1 + 1 = 1

6_2) 15 + 16 + 17 - 18 - 14 -13 - 2

becomes

0 + 1 + 2 - 0 \- 2 - 1 - 2 = 1

6_3) 25 + 14 - 19 - 28 + 17 - 9

becomes

1 + 2 - 1 - 1 \+ 2 - 0 = 0

6_4) 29 + 25 + 17 - 22 - 26 - 24 +3

becomes

2 + 1 + 2 - 1 \- 2 - 0 + 0 = 2

6_5) 17 + 17 + 17 + 23 - 18 - 18 - 18 - 19

becomes

2 + 2 + 2 + 2 \- 0 - 0 - 0 - 1 = 1

to check that:

2 + 2 = 1 and 1 + 2 = 0

0 + 2 = 2

forget the 0's

2 - 1 comes to 1

6_6] 36 - 29 + 15 - 17 + 28 - 19 + 25 - 29 - 8

becomes

0 - 2 + 0 - 2 \+ 1 - 1 + 1 - 2 - 2 = 2

checking:

0 - 2 = 1

forget the 0

1 - 2 = 2 and 2 + 1 = 0

0 - 1 = 2 and 2 + 1 = 0

0 - 2 = 1 and finally 1 - 2 = 2

Finally, to really make sure you've got it,

a mixture of them all!

EXERCISE 7

7_1] (17 x 19) - 17 - 18 + (16 x 18)

7_2] 28 - (25 x 16) - 13 + (18 x 22) -10

7_3] (29 x 29) + 15 + 17 - (28 x 28) - (27 x 3) - 8

7_4] 22 - (15 x 23) - 17 + (14 x 28) - (5 x 10)

7_5] (21 x 13) - (14 x 19) + 17 + 22 - (11 x 16) + (5 x 27) - 3

ANSWERS to EXERCISE 7

7_1] (17 x 19) - 17 - 18 - (16 x 18)

becomes

(2 x 1) - 2 \- 0 - (1 x 0)

bulldozers to

2 - 2 - 0 - 0 = 0

7_2] 28 - (25 x 16) - 13 + (18 x 22) -10

becomes

1 - (1 x 1) \- 1 + (0 x 1) - 1

which is

1 - 1 - 1 + 0 - 1 = 1

Easy way:

1 - 1 cancel and forget the 0

leaving -1 -1 = -2 which is 1

7_3] (29 x 29) + 15 + 17 - (28 x 28) - (27 x 3) - 8

becomes

(2 x 2) + 0 \+ 2 - (1 x 1) - (2 x 0) - 2

which is

1 + 0 + 2 - 1 - 0 - 2 = 0

Easy way:

1 - 1 cancel

+2 -2 cancel

forget the 0's

nothing left = 0

7_4] 22 - (15 x 23) - 17 + (14 x 28) - (5 x 10)

becomes

1 - (0 x 2) \- 2 + (2 x 1) - (2 x 1)

which is

1 - 0 - 2 + 2 - 2 = 2

Easy way:

-2 and +2 cancel

forget the 0

1 - 2 = 2

5] (21 x 13) - (14 x 19) + 17 + 22 - (11 x 16) + (5 x 27) - 3

becomes

(0 x 1) - (2 x 1) + 2 + 1 - (2 x 1) + (2 x 0) - 0

which is

0 - 2 + 2 + 1 - 2 + 0 - 0 = 2

Easy way:

-2 and +2 cancel

leaving 1 - 2 = 2

whew!

It's pretty amazing

what a flea-brain

with a tiny

3-circle-register

can do,

isn't it?
Catlish

The cat is of course

many times smarter than a flea.

We don't really know,

but let's say

the cat has a 4-circle-register

to instantly recognise

numbers with.

If the flea can do arithmetic,

so can the cat,

only better!

If we speak English,

the cat speaks Catlish.

If we do arithmetic

the cat does catithmetic.

heh heh...

The cat uses her 4-circle-register

in exactly the same way

the flea does.

Lucky you.

You already know how it works!

I'll still show you a few examples

very briefly just to make sure you've got it...

Unlike the flea,

Ms Kitty Kat

can

count to

three

in her 4-circle-register:

As you can see above,

she _can_ count to three!

But as you probably guessed,

she _can't_ count to 4!

Yep.

Same problem

as Mr Crow and Mr Flea

Ms Kitty

loses count

at

4

and starts back at

zero

again.

Which means she also

has a circle counting line

which goes:

But notice she has a 3 in it!

That's a big improvement!

As with Mr Flea

you can convert

from

English

to

Catlish

by

counting around her

4-circle-register.

Ms Kitty

is showing off below.

She is so proud she can count to 3,

that she's adding

three 3's in a row

to get to 1:

We know three 3's are 9

and if you count up her

three 3's you will see

that counting to 9

is the same

as counting to 1

in her

4-circle-register.

Some say a cat has "9 lives".

She thinks she only has 1.

Just as with Mr Flea

you can convert from

English

to

Catlish

either by counting around the circle

(as above)

or by writing out the line

as below

(take your pick!)

In this case you can see

6 = 2

and

9 = 1

in Catlish.

So Ms Kitty Kat

would see

9 - 6 = ?

as

1 - 2 = ?

To do that

(see above picture)

Start at 1

then

move BACK 2 steps

to land on...

you guessed it...

it had to be...

the THREE!

That cat is really showing off!

9 - 6 = 3

(Poor Mr Flea!)

It's not time for a catNAP

it's time for some

catMATH

Either draw a 4-circle

to count around

or

use this list

to help you do the following

cat-calulations (should that be "catulations"?)

(by the way, we stopped the above table at 39

but you could count it out to 1939 or larger

if you wanted!)

Exercise 8

8_1) 35 - 34

8_2) 25 - 23

8_3) 37 - 34

8_4) 32 - 24 \- 5

8_5) 27 + 29 \- 35 - 19

8_6) 37 + 28 \+ 30 - 37 - 18 - 39

Answers to Exercise 8

8_1) 35 - 34

converts to

3 - 2 = 1

8_2) 25 - 23

converts to

1 - 3 = 2

8_3) 37 - 34

converts to

1 - 2 = 3

8_4) 32 - 24 - 5

converts to

0 - 0 - 1 = 3

8_5) 27 + 29 - 35 - 19

converts to

3 + 1 - 3 - 3 = 2

Easy way:

3 - 3 cancel leaving 1 - 3 = 2

8_6) 37 + 28 + 30 - 37 - 18 - 39

converts to

1 + 0 + 2 - 1 - 2 - 3 = 1

Easy way:

1 - 1 cancel

so does 2 - 2

leaving 0 - 3 = 1

Multiplication

Ms Kitty has 0, 1, 2 and 3

to deal with.

Let's look at her

idea of the

times tables.

The 0 x Table

All answers are 0

nothing x anything is still nothing.

The 1 x Table

1 x something doesn't change it.

1 x 3 = 3

for example.

The 2 x Table

The TWO times table

looks different

to Mr Flea's:

Mr Flea says

2 x 2 = 1

Ms Kitty says

No! 2 x 2 = 0

Rubbish! says Mr Flea!

Even my son knows 2 x 2 is one!

Who is right and who is wrong?

(Have you ever heard people arguing

and each is convinced they alone have the truth

and they think the other is totally wrong?)

Acually,

sometimes they can BOTH be right!

That is certainly the case here.

According to Einstein

people in rockets moving at different speeds

will have clocks that tick at different speeds

and rulers of different lengths

but

each will think their one is correct

and the other one

is wrong.

In fact both their rulers and clocks

are correct

for the speed they are travelling!

Apparently going fast

shortens rulers (by a tiny tiny bit)

and slows down time (by a tiny tiny bit again!)

The same thing is happening here.

Remember we said

whenever we see something

there is always

a background

which we ignore

but which is always there.

The background is part of the picture.

It changes how we see the picture.

The backgrounds in this case

are the

3-circle and the 4-circle.

2 x 2 = 1

is correct

in Mr Flea's world

while

2 x 2 = 0

is correct

in Ms Kitty's world.

So

we must always

state the background!

2 x 2 = 0

is not entirely true.

What is better is:

2 x 2 = 0

in

Ms Kitty's 4-circle-register

Sometimes

it's possible to write it this way

so they don't get mixed up:

(2 x 2 = 0)4

and

(2 x 2 = 1)3

The little 4 and 3

outside the brackets

tell us which circle-register

the answer belongs to.

Brackets have more than one use!

We won't use them just now

because

we know

we are working

in the 4-circle-register

of Ms Kitty Kat.

Let's prove Ms Kitty's answer

is equally correct:

And of course

here is 2x3 = 2

which Mr Flea

can't even imagine exists:

There is more he doesn't know.

The entire three times table!

Here it is:

3 x 0 = 0

3 x 1 = 3

3 x 2 = 2

3 x 3 = 1

Notice the first two

follow the rule

for 0 and 1

so let's look at:

3 x 2 = 2

and

3 x 3 = 1

And speak of coincidences!

Look at that!

3 x 2 = 2

and

2 x 3 = 2

The turn around's

come to the same answer!

I wonder if that works for other turn-arounds?

Anyway, now we can do

multiplications in Ms Kitty's world!

As long as any sum comes to an answer of

0, 1, 2 or 3

Ms Kitty can do it

and get the same answer

our calculator would!

Here's some to try:

EXERCISE 9

To convert the smaller numbers

from English to Catlish

count around Ms Kitty's

4-circle-register

but

it might be easier

to use this chart

to convert the bigger numbers

like 39!

9_a] (5 x 6) \- (9 x 3)

9_b] (33 x 22) - (25 x 29)

9_c] (39 x 37) - (35 x 25) - (26 x 21) - 21

9_d] (16 x 18) - (13 x 17) + (14 x 14) - (16 x 17) + 9

9_e] (36 x 38) + (30 x 26) - (33 x 35) - (39 x 26) + 23

ANSWERS to EXERCISE 9

Ans to 9_a] (5 x 6) - (9 x 3)

converts to

(1 x 2) - (1 x 3)

which is

2 - 3 = 3

check that:

Start on 2

go BACK 3 around the 4-circle

you arrive at 3

Ans to 9_b] (33 x 22) - (25 x 29)

converts to

(1 x 2) - (1 x 1)

which is

2 - 1 = 1

Ans 9_c] (39 x 37) - (35 x 25) - (26 x 21) - 21

converts to

(3 x 1) - (3 x 1) - (2 x 1) - 1

which is

3 - 3 - 2 - 1 = 1

check that:

3 -3 cancel

-2 - 1 = - 3

Start on 0

count BACK 3 steps to 1

Ans 9_d] (16 x 18) - (13 x 17) + (14 x 14) - (16 x 17) + 9

converts to

(0 x 2) - (1 x 1) + (2 x 2) - (0 x 1) + 1

which is

0 - 1 + 0 - 0 \+ 1 = 0

check:

forget the 0's

-1 + 1 cancel to 0

Ans 9_e] (36 x 38) + (30 x 26) - (33 x 35) - (39 x 26) + 23

converts to

(0 x 2) + (2 x 2) - (1 x 3) - (3 x 2) + 3

which is

0 + 0 - 3 - 2 \+ 3 = 2

check:

forget the 0's

-3 +3 cancel

leaving -2

Put pen on 0 to start

Go BACK 2 steps

and see that

0 - 2 = 2

in the 4-circle-register
BASES

While it's pretty amazing

what can be done with these tiny register brains

you'll notice

all the answers have to come out

small

or we can't do them.

After all,

Mr Flea

can't count to three.

He can only recognise

0, 1 and 2

as possible answers.

So if we give him a sum

like

6 - 1

where the answer is 5,

he can't do it!

For Mr Flea

6 - 1

becomes

0 - 1 = 2

So he writes:

6 - 1 = 2

which is

the same answer

as

3 - 1 = 2

oh oh...

His answer is correct in a 3-circle-register

world without 5's

But maybe we want to be able to get

bigger answers than that!
Compound Number

First of all

let's look at

an ordinary clock.

What is the hour

_after_ 11 o'clock?

That's right!

It's 12 o'clock!

That makes sense.

We move from 11 to 12.

What is the number of the hour

_before_ 1 o'clock?

It is 12!

Do we normally count down:

3, 2, 1, 12?

No. We do not.

That does not make sense.

The 12 on the clock

is "objectively"

(that means "the way it actually is")

12

but it also doubles

as a

zero.

If you look at what is actually there

you see a 12

but you understand

it can fill in for a

0

when you recognise

12 as the number before 1

(which should be 0)

With circlemaths it is the

exact reverse!

What number comes

_before_ 1 in a 3-circle?

It is the zero!

That makes sense!

What number comes

_after_ 2 in a 3-circle?

It should be 3

but its 0!

We don't normally count

0, 1, 2, 0!

The 0 on the 3-circle

is "objectively"

(the way it acually is to everyone)

a

0

but it also doubles

as a

3

If you look

you see a 0

but you understand

it can also mean a

3

As you know by now

by the time we count

UP

to 0

around a 3-circle

we have really counted

up to

3

We went

0, 1, 2, and a bit... nearly 3...

and

when we reach the 0

exactly

that's when we would be

exactly

on

3

Except its not.

We're on 0.

So the 0

in a 3-circle

is also

the idea of 3

It's two things in one.

We could draw it

this way perhaps

(see below):

The zero

has a tiny "3" inside it.

Then we would see

0 is before 1

(correct)

and

3 follows 2

(correct)

The cost

is that

the top position

now has _two_ numbers:

0 and 3!

In fact

we don't usually draw the 3

inside the zero.

The 3 inside the 0

exists as

an idea

only.

Look at the strange balance in that:

The 3 is a "solid" number

but it isn't drawn

as it's only an idea at this stage,

while

the zero is drawn

which makes it "real"

but zero doesn't exist

by definition!

Let's compare the circles

to some coins:

You can see the link

between objects

and the number

of places

in each circle.

1-circle

has 1 place

2-circle has 2

3-circle has 3

and so on...

However

the first place

in each circle,

which is the top place,

is always given

to something

which doesn't

really

exist.

Zero:

As a result

the circle doesn't contain

the number

that would name it.

For example

the "3"

gets kicked out

of the 3-circle below:

Mr Flea's

3-circle

has

3 places (see below)

but one of them

is taken by

zero (see below)

There is no more room.

The "3"

gets kicked out:

This means

3-circle

has no 3

and cannot name itself!

And this happens

to all the circles.

The entire 3-circle

IS

the idea

of

3

It's got 3 dots

and 3 arrows

and we reach

the count of 3

when we go around it

once

exactly

...and return back to

nought.

Nought means

"making _naught_ of the idea of 3"

Zero exactly

coincides with

the IDEA

of three

...but when we

stop and look

we see

the reality of

zero.

This is

the idea world

versus

the "real" world!

Number belongs in both worlds!

So that's why Mr Flea

can't count to three.

Because there is no 3.

It keeps vanishing to nought!

That's why he calculates:

5 - 2 = 0

because he sees

no 3

for him

3 is 0

Scott Flansburg

is a prodegy calculator.

World class.

He has a video

(https://www.youtube.com/watch?v=hesKQ_y1P7k)

where he says

we must always start from zero

and he counts as shown below:

He says

there is no such thing as 10

when we count our fingers.

9 is the biggest number.

But that means he counts

this as three fingers:

(see below)

Most people would say

that Scott is wrong and that

the correct image of 3 fingers

is that shown below:

However

he is not the world's fastest calculator

for nothing!

Scott is trying to show us

the register

inside our heads

that actually does the counting!

Not the fingers!

He is entirely

correct!

The 10-circle-register

we use is shown below:

It starts from zreo.

It has 10 places.

It has 10 journeys between each place.

Its biggest number is 9.

It has no "10" in it.

Exactly as Scott describes it.

Circle registers

work that way.

Mr Flea is stuck

because he always loses

the count

at 3 exactly:

Ms Kittie does the same thing.

And so do we.

In our mind

we throw away the number

that counts the circle.

Every circle throws away

the number which names it

and returns to zero instead.

Can we do something about it

or are Mr Flea and Ms Kittie

(and us!)

doomed to never count

above 3 or 4?

Is there a way

we could somehow

CATCH

the

3

that was

thrown away

by Mr Flea?

The answer is yes!

The answer lies in the common fly...
Mr Fly

and his Compound Eye

If you look at the eye of a fly

closely with a magnifying glass

you'll see

Mr Fly doesn't have two eyes

he has

hundreds of eyes.

Each of his eyes is actually made up

of hundreds of tiny eyes!

It looks a bit like this (see below)

Each tiny eye is in a slightly different spot

and looks at a slightly different angle

and sees a slightly different world.

Each eye sends its slightly different image

to the tiny brain of Mr Fly

and that tiny

but incredible

brain

puts all those images together

and creates a

single

3-D

image

of the world

enabling Mr Fly

to navigate while flying

and avoiding objects,

even objects that are trying to catch him!

It's almost impossible

to catch a fly

while its flying.

It can see you.

It can dodge you!

Mr Flea and Ms Kittie

can break free

of tiny numbers

by using what's called

compound number

It works a bit like

the compound eye

of Mr Fly.

Except we are talking here

about "seeing"

in the brain.

As in

"I see what you mean"

Understanding.

In this case

understanding

bigger numbers

than 3!

Firstly though, some people say

"there is no magic in the world"

but actually it is all around you.

It is just hidden

behind the ordinary.

Here is some magic

in drawing

a simple

circle...

First we put our pen

on the paper

and make a single dot.

The dot

is a tiny

circle

The beginning

and the end

are

together.

We start to draw our circle

and as we go

we create

a line

with

a beginning

and

an end...

It gets closer and closer

to becoming a circle...

Finally

The beginning

and

The End

become ONE

and

vanish.

Lift our pen off the paper:

The circle is born

with

an inside

and

an outside

but

no beginning nor end.

That's magic!

Somehow the beginning and the end

turned into

an inside and an outside.

A circle has an outside.

And we will use it to create

what's called a

"compound number".

Outside Mr Flea's

3-circle-register

we place

another 3-circle-register.

When we go around

the entire circle

and return to

0

the count of

3

and the zero

become

ONE

(one lot of 3 at the

exact count of 0)

and

vanish.

We are left

with

0

But outside that circle

the other circle

picks up the count

of the

ONE

lot of 3

that vanished.

It does this

by counting

on to the

ONE.

It's shown below:

10

means

**ONCE** around the circle

and

back to the **NOUGHT**.

The new circle

keeps count

of how many times

the original 3-circle

loops around.

Each time the original circle

loops around once

the new circle

gains

ONE

So each of its

ONE's

really represents

an entire world journey,

an entire circle's worth

which is what

we would call

a

3

Has this solved Mr Flea's

and Ms Kittie's problem

of not being able to count very high?

Yes it has!

Let's just check out Mr Flea's

new ability to create numbers

bigger than 2:

What we call 3

would be

once around 3-circle

and

back to zero

or

10

What we would call 4

would be

once around 3-circle

and

on one more to 1

making it

11

What we would call 5

would be

once around 3-circle

and

on one more to 2

making it

12

What we would call 6

would be

TWICE around 3-circle

and

back to zero

making it

20

Can you figure out

what 7 and 8 would be?

Here they are:

What we would call 7

would be

TWICE around 3-circle

and

on one more to 1

making it

21

What we would call 8

would be

TWICE around 3-circle

and

on one more to 2

making it

22

This is Mr Flea's

new improved

much bigger

counting line:

He can even do

his own arithmetic

on it.

We'll see how shortly.
Every Number

can be a

Compound Number

The numbers from 10 on

are called

compound numbers

because each number

is a compound

coming from

two

different circles.

Noitce you have to

lift your pen

off the paper

to write

a compound number.

There are

three

stages of number.

1

0 and 1

(the foundation)

2

All the single digit numbers

which includes 0 and 1

(0, 1 and 2 for Mr Flea

and 0, 1, 2 and 3 for Ms Kittie

and

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

for you and me!)

3

and all the compound numbers

form stage 3

It turns out

we can write

any single digit number

as

a compound number

if we want.

Here's how:

We can treat the single digit

0

as a compound

00

The diagram above

shows that the new circle

has seen

NO

whole circles have been made

and we are still on

zero.

So it reads

00

The single digit

1

is the compound

01

The above diagram

shows that the new circle

has seen

NO

whole circles have been made

however

the original circle

has moved on

to the 1.

It reads

01

Similarly

2

becomes

02

And Mr Fleas counting line

now reads as shown below:

and is composed

entirely of

double digit

compound numbers!

So all three stages

end up in one!

Flea - Compound-eye-Fly

Arithmetic

Here is some arithmetic

Mr Flea can now do

giving answers

in "Compound-Fleanglish"

that are bigger than 2:

Some of his sums

look familiar

like the one above

and...

Some don't!

It doesn't look right to us

that

12 + 2 = 21

we would expect

12 + 2 = 14

but Mr Flea

has no 3

and certainly

no 4

It turns out

his answer is actually correct

in his tiny world.

Count

the dots to see

that both sums make sense

to Mr Flea.

Remember

every time Mr Flea

sees "3"

they vanish to 0

and count as a 1

in the new

circle.

EXERCISE 10 (a)

Can you help Mr Flea

with these sums?

Use the dots

to fill in the sums

and answers

below

and

draw your way

to the answers!

(treat every number

as a compound number,

so 1 becomes 01 for example):

Draw the dots

to get the answers

to these sums:

1 + 2 = 10

2 + 2 = 11

11 + 2 = 20

Answers to Exercise 10 (a)

and the answers to

the other sums look like this:

EXERCISE 10 (b)

Use the dots

to fill in the sums

and the answers

to these subtractions:

Draw the dots yourself

to get the answers

to these sums:

12 - 02 = 10

10 - 2 = 01

21 - 12 = 02

and these multiplications:

2 x 2 = 11

2 x 1 = 02

Answers to Exercise 10 (b)

and the answers to the other subtractions

look like this:

and here are the answers

to the multiplications for exercise 10(b):

Ms Kittie

and her Compound Eye

How does Ms Kitty fare?

EXERCISE 11

See if you can do it!

Try to write out

Ms Kittie's

compound counting line

starting from

00

and going

all the way up to

33

ANSWER to EXERCISE 11

She starts

turning single digits

into compounds

using two

4-circle-registers

starting at

00

She can then count

00 01 02 03

but she can't write

04

as she has

no

4

Instead

she jumps to

10

as shown below:

The meaning

of

10

for Ms Kitty

is

ONCE around the circle

and

back to the zero

exactly as it was

for Mr Flea

except her circle

is bigger.

Then she counts on one more

to reach

11

and she can

continue to count

this way to

13

00 01 02 03 10 11 12 13

before she runs out

of numbers

in her

first circle

Then she jumps

to

20

(see below)

and continues

all the way to

23

20 21 22 23

what's next?

Yes.

She jumps

to

30

and so on

all the way

to

33

Here is her

counting line

from

00 to 33

Ms Kittie can do

lots of arithmetic

with numbers

bigger than 3

now!

Yay!

Go Kittie Kat!

Exercise 12

Help Ms Kittie

with these sums.

Use the dots

to fill in the sums

and answers

below.

Draw your way

to the answers!

Do the same with

the sums below.

We've left it for you

to circle the

groups of 4

the way you'd prefer

in some of them:

ANSWERS to EXERCISE 12

and the second lot of answers

look like this:

Exercise 13

Draw the dots

to get the answers

to these sums:

23 + 10 = 33

3 + 23 = 32

13 + 13 = 32

Answers to Exercise 13

Exercise 14 (a)

Use the dots

to fill in the sums

and the answers

to these subtractions.

Again we're letting

you circle the groups

of 4

the way you like

in some of them:

Exercise 14 (b)

Draw the dots yourself

to get the answers

to these sums:

32 - 11 = 21

22 - 13 = 03

20 - 2 = 12

and

these tables

Exercise 14 (c)

3 x 2 = 12

3 x 3 = 21

2 x 2 = 10

10 x 2 = 20

TIP for 10 x 2

don't forget

10 is "4"

in

Catlish!

Answers to Exercise 14(a)

Answers to Exercise 14(b)

32 - 11 = 21

22 - 13 = 03

20 - 2 = 12

look like this:

Answers to Exercise 14(c)

The answers to the multiplications

3 x 2 = 12

3 x 3 = 21

2 x 2 = 10

10 x 2 = 20

look like this:

and this:

For the last one

10 x 2

doesn't mean

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2

which is

10 two's

for us

Ms Kittie's version of 10

comes from a 4-circle

so

10 x 2

means

2 + 2 + 2 + 2

for her.

Tricky eh?!!
Preventing War

Between Ms Kittie and Mr Flea

Ms Kittie and Mr Flea

have totally different

world views:

As you can see

Mr Flea thinks $12

means what we would call

$5

while

Ms Kittie thinks $12

means what we would call

$6

They are going to argue!

To fix this

we state what

"base"

we are in.

We could write

($12) Flea

($12) Cat

or

even better

($12) 3

($12) 4

If they both call it

$12

they will disagree.

But if they call it

($12) 3

or

($12) 4

they will know

they

mean

different amounts

and will be able to

understand one another

and agree

on the price!

Here is one way

to translate

between

Catlish

and

Flealish:

Translatorcise 1

Mr Flea and Ms Kittie

met

Doggly

who has

a very smart

5-circle-register

brain

Could you translate

the piles of coins

shown below

into

Flealish, Catlish

and Doglish

and

regular English

which we all use?

Answers

1. (12)3 = (11)4 = (10)5 = 5

2. (2)3 = (2)4 = (2)5 = 2

3. (22)3 = (20)4 = (13)5 = 8

4. (11)3 = (10)4 = (04)5 4

and

Translatorcise 2

Draw dot pictures

to help you

translate

these numbers

from

Catlish

into

Doglish and English:

a

(22)4

b

(12)4

c

(30)4

d

(21)4

e

(13)4

f

(20)4

Answers 2

a

(22)4 = (20)5 = 10

b

(12)4 = (11)5 = 6

c

(30)4 = (22)5 = 12

d

(21)4 = (14)5 = 9

e

(13)4 = (12)5 = 7

f

(20)4 = (13)5 = 8

Translatorcise 3

Draw dot pictures

to translate these

calculations

from

Catlish

to

Doglish

and

plain old English:

(a) (11 + 11 = 22)4

(b) (12 + 12 = 30)4

(c) (23 - 2 = 21)4

(d) (12 - 3 = 3)4

(e) (21 - 2 = 13)4

Answers 3_(a)

We've drawn the pictures

in some of them

for you

Answer 3_(a)

(11 + 11 = 22)4

becomes

(10 + 10 = 20)5

or

5 + 5 = 10

(just count the dots ordinarily

to get 5+5 =10)

Answer 3_(b)

(12 + 12 = 30)4

becomes

(11 + 11 = 22)5

or

6 + 6 = 12

Answer 3_(c)

(23 - 02 = 21)4

becomes

(21 - 02 = 14)5

or

11 - 2 = 9

Answer 3_(d)

(12 - 03 = 03)4

becomes

(11 - 03 = 03)5

or

6 - 3 = 3

Answer 3_(e)

(21 - 2 = 13)4

becomes

(14 - 02 = 12)5

or

9 - 2 = 7
Human Counting

When we count

we use the single digits

0 1 2 3 4 5 6 7 8 9

Let's make that into a circle

and look at it

Shown above

are those ten single digit numbers

arranged in a 10-circle

with of course

no

10

in it!

Take a look at the dots below.

We have circled

the whole tens in it

to turn them into

10-circle-register

compound

numbers:

Do you notice anything

unusual?

The numbers

inside the brackets

are identical

to our ordinary

count!

(24)10 = 24

(16)10 = 16

and in fact

that happens

for all

numbers.

Test it for yourself!

Take a few moments

to write out

the counting line

of that

10-circle

and you will find

a surprise!

Here it is:

The surprise

is that

it is our

ordinary

counting line!

Our ordinary counting line

is based on

a

10-circle-register

Because its

based

on ten

we call it

**base** ten.

Fine.

Then

why don't we

write all our sums like this:

(2 + 2 = 4)10

Well,

we could,

but look what it looks like:

($1.50)10

($1.50)10

($3.00)10

Which do you prefer?

That

or this:

$1.50

$1.50

$3.00

I know which I prefer!

_Without_ the little

10

marker

thanks very much!

It's cleaner, simpler!

The idea is

that because we

always

use the base 10

then there is no need

to say it

each and every time

we do a sum!

Because most books

we read are exclusively

in English

we never need

to tell people

"this book is written in English"

They understand that!

In the same way

everyone understands

that the basis

for our numbering

is

base ten

So the

unspoken rule

is

IF

a number

doesn't state

what base it's in

then its

assumed

to be

base ten

A Peculiar

Naming Twist

Do the French speak French?

No.

They do not.

If you ask them

they will tell you

they speak

"Francaise"

"Francaise"

is French for French!

Do the Germans speak German?

No.

They don't!

They speak

Deuche!

Deuche

is the German word

for German!

Have you noticed

that all the other circles

throw out

the very number

that would name them.

3-circle

has no 3

so it can't call itself

3-circle!

4-circle has no 4.

But actually

EVERY

counting compound system

has a

10

Remember Mr Flea

counts

00, 01, 02, 10

and

Ms Kittie

counts

00, 01, 02, 03, 10

and

Doggley

counts

00, 01, 02, 03, 04, 10

They ALL have a 10!

So they could

ALL

claim

to be

base 10!

Likewise

we

say Mr Flea

counts in

base three

but

if we asked

him

he would say

"what's a 3?

Sounds to me

like a number

that couldn't ever be!"

He doesn't even know

what a

3

looks like!

He would say

"I count

in

base 10"

Because the number

above 2

for him

is

10

Ms Kittie

would disagree!

she would say

"you work in base 3

It is

I

who work

in

base 10!"

We are doing the same

as Mr Flea

and Ms Kittie

and no doubt

Doggley...

We claim

it is

WE

who work

in Base 10!

Not anyone else!

It's all a matter of

point of view.

Until dogs

cats and fleas

can talk

we'll be OK

with claiming that!
A BIG PROBLEM

\- being too little -

Can we really

count the grains of sand

on a beach

with our

compound numbers?

Well,

to be honest...

er...

no

We run out of numbers...

Ooops...

You may have

noticed this

and wondered about it

earlier.

Let's go back

to Mr Flea's

compound

counting line

to see the problem.

It goes:

00 01 02 10 11 12 20 21 22

but what comes

after

22

?

Let's take a look.

Below we see Mr Flea

has just arrived

at 22

Now we want

to add on

1

more

We can't

get 23

there is no 3!

What happens

first

is that the

original 3-circle-register

(the one on the right)

clocks on

1

more

to arrive

back at nought:

It loses count!

But that's ok!

It

tosses out

ONE

whole circle's

worth,

what WE

would call a

"3"

as a

ONE

The other circle

will

catch the count

as a

1

and it will clock on

1 more time

and

catch the count!

But can it?

It tries to.

It moves on

1

more

and arrives...

ouch !!

back at nought

itself!

IT

loses count

also!

(see below)

They have

BOTH

lost count!

The answer

revolves

back to

00

!!!

Wow!

That means

the compound

counting line

really goes like this:

It starts where I've colored it blue

counts to

22

then goes back to

00

where I've colored it red

as you can see,

then

carries on

to

22

again

then

back to

00

in the blue again

and so on!

Round

and

round!

It's going around a circle!

A much bigger circle than 3-circle

but a circle

just the same!

Remember

the original

3-circle line

which went

0 1 2 0 1 2 0 1 2...

This is similar.

Just a bit more

"runway"

before it

loops!

Let's look at it!

What circle

is it looping

around?

To find out

just wrap

the numbers on its

counting line

into a circle

That's

00 01 02 10 11 12 20 21 22

They make the circle

drawn below:

If you look

you can see

it has something

that looks a bit like

a

3-circle

hidden

inside it!

I've highlighted

the 00, 10 and 20

which fit in the same places

that

0, 1 and 2

fit in

an ordinary

3-circle

It has

9

places

so we might think

its a

9-circle

And indeed it is!

BUT

it's not written

in human language!

It's written

in

Flealish!

So what would

Mr Flea

call that circle?

Unfortunately

he's lost count

and can't name it.

Which means

he's stuck again!

Can we help him?

To resolve

a problem,

you must

look

at the problem.

Let's go back

to where the problem

began...

A

ONE

in the form of

ONE

"3"

was lost

thrown out

and caught

by the second circle

But now

that circle too

has succumbed

and returned

to zero.

But wait!

That means

IT

lost count

IT

lost

ONE

"3"

as well!

What if we

had

ANOTHER

circle

to catch

it's count???

Like this:

Then

it would read

100

That's the solution!

And that's what

Mr Flea

would name

that unusual

"9"-circle

we found!

It's

the

100-circle

of

Mr Flea!

Have we really

solved the

"too small"

counting

problem?

What comes

after

the number

222

?

The same thing will happen!

The third new circle

will lose count

We will end up

with a bigger

counting line

which will go

from

000

all the way

up to

222

but then

it will loop

into a circle

Here is the circle

it loops into

We haven't drawn

all the numbers

in it

because

its too big!

For example

between

000 and 010

there are

000 001 002 010

and between

010 and 020

there are

010 011 012 020

This circle

is certainly

Big

...but it is still

limited!

But this time

we know

what

to do!

We'll add

another 3-circle

to

catch the count!

Then we'll see

that that

BIG

circle

was really

the

1000-circle

for

Mr Flea!

Clearly

just like Mr Fly

with his

compound eye

we could have

hundreds

of circles we add

to keep catching the count!

It could go on

for as long as we chose!

Perhaps

Mr Flea

can

finally

count the grains of sand

on a beach

and the stars

in the sky

using just his

tiny 3-circle

register!
Patterns

Look at the circle sizes

and names

for Mr Flea:

10-circle

100-circle

1000-circle

which count

0 to 2

00 to 22

000 to 222

care to guess the next?

If you guessed

10,000-circle

capable of

counting

from

0000 to 2222

you would be

100%

correct!

Here's another pattern:

10-circle is the start.

Don't forget

for Mr Flea

10

means

what we call

3

Mr Flea's

100-circle

has 3

3-circles

hidden in it.

Woops!

Speaking in Flealish

I mean

Mr Flea's

100-circle

has

10

10-circles

hidden

within it!

You can see them

if we cover up

the first number

in each number

in the circle:

For us,

we would say

the 100-circle

is really only just a 9-circle

and it has

3 lots of 3-circle

in it

but

Mr Flea would say

the 100-circle

has 10 lots of 10-circle

in it

and he would write

100 = 10 lots of 10

100 = 10 x 10

What about the 1000-circle?

First

we can see

a 3-circle

hidden inside it:

It's a 3-circle

written in hundreds.

If we shade

the first digit

then magically

we will find

3

100-circles

inside

the 1000-circle.

Ooops!

I've done it again!

Must remember to speak

Flealish

when in Flealand!

I'll say that again:

magically

we will find

10

100-circles

inside

the 1000-circle!

Don't forget

we couldn't fit all the numbers in!

So

00 10 20

stands for

00 01 02 10 11 12 20 21 22

which is

an entire

100-circle!

So 1000

is

10 lots of 100

which means its

10 x 100

It's a times table

1000 = 10 x 100

And

in the picture below

I've

zoomed in

on one of those

100-circle chunks

inside

the 1000-circle

and

you can see

IT'S

made up of

"3"

"3"-circles

I mean

in Flealish

make that:

IT'S

made up of

10

10-circles

Each

100-circle

has

10 lots of 10-circle

inside it

so

100 = 10 x 10

which means

as we wrote

1000 = 10 x 100

then also

1000 = 10 x 10 x 10

all in Flealish

of course

where Mr Flea's

10

is our

3

That gives us this pattern:

10 = 10

100 = 100

100 = 10 x 10

1000 = 1000

1000 = 10 x 100

1000 = 100 x 10

1000 = 10 x 10 x 10

which gives us

another great pattern!

10 = 10

100 = 10 x 10

1000 = 10 x 10 x 10

guess what

10,000 is?

Yes!

10,000 = 10 x 10 x 10 x 10

We could keep going

up to a million:

10 = 10

100 = 10 x 10

1000 = 10 x 10 x 10

10,000 = 10 x 10 x 10 x 10

100,000 = 10 x 10 x 10 x 10 x 10

1,000,000 = 10 x 10 x 10 x 10 x 10 x 10

Whew!

Return to Simplicity

When we travel

around a circle

the numbers get

bigger and bigger

then

suddenly

drop to

zero

When things get

more and more

complex

suddenly

they often

become

simple

To simplify

everything above

just count the zeros

on both sides

of the equal sign:

10 = 10

1 zero on both sides

100 = 100

2 zeros on both sides

100 = 10 x 10

2 zeros on both sides

1000 = 1000

3 zeros on both sides

1000 = 10 x 100

3 zeros on both sides

1000 = 100 x 10

3 zeros on both sides

1000 = 10 x 10 x 10

3 zeros on both sides

and so on...

So we can guess

that

1,000,000

which has 6 zeros

will also equal

1000 x 1000

which has 6 zeros

or it could also be

10 x 10 x 10 x 10 x 10 x 10

or

100 x 10,000

all of which

have 6

zeros!

What multiplies

by

10,000

to make

1,000,000?

1,000,000 = 10,000 x 100

6 zeros on both sides

What multiplies

by 10,000

to make

1,000,000,000?

Try it yourself!

Did you get

1,000,000,000 = 10,000 x 100,000

(9 zeros on both sides)

or even

10,000 x 100 x 1000

or

10,000 x 10 x 10 x 10 x 10 x 10

or any others

so there are

9 zeros on each side?

All are correct!
The Caterpiller

Have you ever watched

a caterpillar

as it walks?

It moves like this:

It forms a curve

then straightens out

then forms a curve

then straightens out...

Mr Flea's

counting

moves

like a caterpillar too!

We went from

10-circle

to

short straight line

0 1 2

then we used compound number

and got

a longer straight line

but

it turned out it was actually

a 100-circle

so we added more compound digits

to make an even longer

straight line

but

that turned out to be

a

1000-circle

and so on

we went

from

circle

to

straight line

to circle again!

Just like the caterpiller!

Mystery Question

So what's at the end?

If the numbers keep

getting bigger

this way

will we end up with

a circle

or

a line?

Equally Mysterious Answer

We can't really say.

Actually

if we had

an indefinitely great circle

could you tell the difference

between that

and an infinite

straight line?

If you drew a line

all the way around

the world

would a little bit of it

look straight

or round?

It would look straight!

Perhaps we should

blend the two ideas

and call it

a

lincle

or a

cirline

!

Because

actually

no-one

can tell the difference!

An indefinitely great

counting circle

is

indistinguishable

from

the infinite

straight counting line

When it reaches this stage

we call it

a base.

So

whereas admittedly

all bases

could be called

"base 10"

for us

from our human

point of view

we say:

Mr Flea

uses

base 3

Ms Kittie

uses

base 4

Doggley

uses

base 5

and

we

use

base 10

For the astute

Actually,

all that means

is that while we can name

the smaller bases

our ten circle

has nothing

that can name

itself.

So having no name

we are forced to call it

what every base

could equally be called,

"base 10"

Every base

including ours

has a 10

so it is the common

default name

which could apply

to anything.

In other words

"Base 10"

means

"we just don't have

a single digit name

for it!"

In other other words

its the base

we

use!
Catlish

Same Thing

Ms Kittie

has the same problem

and the same

solution!

She gets to count

all the way

from

00

up to

33

But

alas at 33

if she adds one more

the same thing happens!

She returns to

00

This means that

she loops

around

a circle

with these numbers:

How many places

does this circle have?

Answer: 16

It is a 16-circle

for us!

But Ms Kittie

can apply

the same solution,

use another

compound circle

and she would then discover

that

for her

(in Catlish)

the circle is a

100-circle

again

(see below)

So she jumps

from 10-circle

to 100-circle

and the next step

would be

1000-circle

just like Mr Flea!

Of course

her idea

of a 10-circle

a 100-circle

and a 1000-circle

is much bigger

than Mr Flea's

but it works the same way.

You can still see

"4

4-circles"

or I should say

10

10-circles

(speaking Catlish now)

inside

her 100-circle

as shown below

And all the rest is the same.

10 = 10

100 = 100

100 = 10 x 10

1000 = 1000

1000 = 10 x 100

1000 = 100 x 10

1000 = 10 x 10 x 10 x 10

and so on

and her numbers

move like a caterpillar

too,

endlessly

straightening out

folding back into a circle

looping

straightening out

folding back into a circle

looping...

What About Us?

The same thing happens

for Doggley

and all the other

bases

including

our base ten!

We start with

10-circle

which gives us

the counting line

0 1 2 3 4 5 6 7 8 9

and then

jump to

compound number

giving us

00

all the way

up to

99

Here's

the straight line

we get

(sorry, it wouldn't fit across the page)

with two

10-circle-registers:

And that

folds up into

our

100-circle

Note:

there are far too many numbers

for them all to be shown!

You just need

to remember

that there are numbers

between the tens!

For example

between

50 and 60

there should really

be

**50** 51 52 53 54 55 56 57 58 59 **60**

Likewise we can

draw the 1000-circle

of base ten

by only showing

the hundreds:

So what's the difference

between

1

01

and

001

?

Not much.

1 is 1 in 10-circle

01 is still 1

but in addition it tells us

that we are in

a 100-circle

and

001 is still just 1

but it tells us that we are in

a 1000-circle.

It's just a bit of

additional information

which does turn

out to be useful

in certain situations.
Summary

Let's look at the number

0 1 2 3 4

written in "humanglish"

(base 10):

This digit

is looked after

by the first 10-circle

It counts

ones

(sometimes called "units")

only.

Here it sees

4

ones

which is 4

Look again:

The second digit

is looked after

by the second 10-circle

It counts

whole tens

only

Here it sees

3

whole tens

which is worth

3 x 10 = 30

Together

they form a

100-circle.

Its biggest number

is

99

Currently

it is set on

34

in 100-circle

Look again:

The third digit in

is looked after

by the third 10-circle.

It counts

whole hundreds

only.

Here it sees

2

whole hundreds

which is worth

2 x 100

which is the same as

2 x 10 x 10 = 200

Together

all three form a

1000-circle.

Its biggest number

is

999

Currently

it is set on

234

in 1000-circle

Look again:

The fourth digit in

is looked after

by the fourth 10-circle.

It counts

whole thousands

only.

Here it sees

1

whole thousand

which is worth

1 x 1000

which is the same as

1 x 10 x 10 x 10 = 1000

Together

they form a

10,000-circle.

Its biggest number

is

9999

Currently

it is set on

1234

in 10,000-circle

Lastly

This fifth digit in

(or the first digit "out"!)

is looked after

by the fifth 10-circle.

It counts

whole ten-thousands

only.

Here it sees

0

whole ten-thousands

which is worth

0 x 10,000

or

"no lots of ten-thousand"

which is the same as

0 x 10 x 10 x 10 x 10 = 00000

or nothing.

It doesn't change

the count.

Together

they form a

100,000-circle

Its biggest number

is

99,999

Currently

it is set on

01234

in 100,000-circle

And do you know what is funny?

That's all true

no matter

who sees it!

Its an

absolute truth,

not a

relative truth.

The ten of Mr Flea

is worth "3"

the ten of Ms Kittie

is worth "4"

these are

relative truths.

But

the fact that

10 x 10 = 100

no matter

who it is for

is

an absolute truth.

It's true for

Mr Flea, Ms Kittie, Doggley

and for

you and me!

EXERCISE 15

for Humans

Assume we are

using the human

base 10 counting circle

15_(a)

What is the

5

in 14,256

worth?

Answer to 15_(a):

It is worth

5 x 10 = 50

15_(b)

What circle

does the

56

of

14256

belong to?

Answer to 15_(b):

It belongs

to the

100-circle

whose biggest number

is 99

15_(c)

What is the

2

in 14256

worth?

Answer to 15_(c):

It is worth

2 x 100 = 200

15_(d)

What circle

does the

256

of

14256

belong to?

Answer to 15_(d):

It belongs

to the

1000-circle

whose biggest number

is 999

15_(e)

If we were told

the number

14,256

was written

by

Ms Octopus

who uses

8-circle-registers

would all of what was said above

still be true?

Answer to 15_(e):

Yes.

But each of her

10's

would only be worth

8

from our

point of view.

15_(f)

What would

her version

of the number

20

be worth

to us?

Answer to 15_(f):

For her

20

would be

2 x 10 = 20

but

for us

her "20"

would only be

2 x 8 = 16

15_(g)

If we were told

the number

14,256

was written

by

Mr Flea

who uses

3-circle-registers

would all of what was said above

still be true?

Answer to 15_(g):

No!

Because

Mr Flea cannot see

anything above 3!

So 4, 5, and 6

would make no sense to him at all!

He believes

there are

no such numbers!
Bases Inside Bases Inside Bases

In the real world

outside us,

things are outside things.

By that I mean

that a cup and a brick

can't be in the same place

at the same time.

If we try and squeeze

two things together

at the same time

like driving two cars

into one another,

they explode.

They exclude one another.

Things are outside things.

Inside

in the mind

the very reverse holds!

In the mind

ideas are inside ideas!

I'll show you what I mean.

Firstly

doesn't a 4-circle

contain a 3?

But

3

is the idea

of Mr Flea's

entire circle!

So

the 3-circle

of Mr Flea

is wrapped up inside

the 4-circle

of Ms Kittie

in the form of

a

"3"

Similarly

it turns out

the 100-circles

counting lines

all fold up

inside one another!

Take a close look

at our

base ten

100-circle

counting line:

Can you spot

Mr Flea's

100-circle

counting line

inside it?

How about

Ms Kittie's counting line?

Mr Flea's

100-circle

counting line

is inside

Ms Kittie's

100-circle counting line

which is inside

Doggley's

100-circle counting line

and so on

all the way up

to the fact

that they are

all

inside our

100-circle counting line!

Imagine

a giant

who has a

bigger counting line

than even us.

Then our

100-circle

counting line

and all the others

along with it

would sit

inside the giant's

100-circle

counting line!

And on it would go

forever

or until

we ran out

of giants!

So by writing out

our ordinary

counting line

to 100

as shown above

you also get

for free

ALL

the 100-circle

counting lines

of

ALL

the bases

below it!

Pretty good value eh!

EXERCISE 16

Write out

our 100-circle

counting line.

Use it

to quickly count

all the numbers

from zero

to 100

in bases

two, four, six, eight and ten.

ANSWERS to EXERCISE 16:

Base 2

0, 1, 10, 11, 100

Base 4

0, 1, 2, 3, 10, 11, 12, 20... up to 32, 33, 100

Base 6

0, 1, 2, 3, 4, 5, 10, 11, 12, 13... up to 53, 54, 55, 100

Base 8

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13... up to 75, 76, 77, 100

Base 10

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11... up to 97, 98, 99, 100
The 100-circle

It turns out

that the 100-circle

can be very useful

for arithmetic

So let's look

at the various 100-circles!

Starting with

the very smallest

the

100-circle

of

Ms Microbe

who uses

two

2-circle

registers

to create

her

compound

100-circle:

Her 100-circle

counts

00 01 10 11

the next number

is

100

There it is!

The quickest way

in the world

to count

to 100!

Give someone a race

to count to 100

and then count

00 01 10 11 100!

You win!

The base

Ms Microbe uses

by the way

is called

"Binary"

and it runs

our computers!

What patterns do we see in it?

00 is the start

11 is the end

They are both

double numbers

like 22, 33, 44, 55 and so on.

They form a diagonal:

The 01 and 10

are the reverse of

one another

Going down a column

from 00 to 10

adds 10

as does going

from 01 to 11

so

Hopping

up or down a column

to the next line

either subtracts or adds

10

(depending on which way

you are going)

Moving right

from 00

takes you to 01

and adds

1

Moving right

from 01

takes you to

10

and adds

1

Going the other way

from 10 to 01

or from 01 to 00

subtracts

1

each time

So

moving

one place

to the

left or right

either subtracts or adds

1

The 00

is

11

apart

from the

11

on the diagonal

The biggest number

on the top row

is

01

If we add

01 to 01

we get

to 10

which forms

the other diagonal

with 01

shown above.

Folded into circle form

her 100-circle

looks

like this:

Look at the lines

joining the numbers

in the circle

00 and 00

11 and 01

10 and 10

11 needs 01

to go once around the circle

10 needs another 10

to go once around the circle

00 needs 100

to go once around the circle

and 100 would land back on 00

so 100 is the same as

00

in that circle

We can say

00 (as zero)

is the **complement** of

00 (as nought which is worth 100)

in that circle.

A

complement

completes

the circle

So for example,

01

is the complement of

10

because together

they add

to

one whole circle

Because

this is the first base

ever

we can expect

the other bases

will inherit

these characteristics!

Let's look

at the 100-circle

of

Mr Flea and see

if this is so:

Look down the diagonal

and see

00 11 and 22

are double numbers

exactly as for Ms Microbe:

Notice

01 and 10

are reverses of one another

02 and 20

are reverses of one another

12 and 21

are reverses of one another

Just as they were

for Ms Microbe

Moving

from 00 to 10

adds

10

moving

from 10 to 20

adds 10

moving

from 01 to 11

adds 10

and so on

Moving down

a column

adds 10

Moving up

a column

subtracts 10

Exactly as for Ms Microbe.

Moving

to the right

from 00 to 01 to 02

adds

1

each step.

The step from 02 to 10

adds

1

also.

Moving

to the left

from 22 to 21 to 20 to 11 etc.

subtracts

1

off each time

Just as for Ms Microbe.

Finally

the biggest number

on the top row

is

02

If we add

02 to 02

we get

to 11

and if we then add

another 02 to 11

we get

to 20

which forms

the other diagonal

with 02

shown here:

Everything

replicates

what happened

for Ms Microbe.

Mr Flea's 100-circle

inherited

all of Ms Microbe's

patterns!

EXERCISE 17

Test Ms Kittie's

100-circle

(shown below)

It might be useful for you

to copy this on paper

17_(a)

Start at 00 and run

down the column.

What does it add on

each time?

Repeat that

starting at 01, 02 and 03

Does it do the same?

Answer to 17_(a)

It adds 10 on as you go down a column

from whatever number

you start from.

17_(b)

Start at 30

and run your finger

up the column.

How much does it subtract

each time?

Repeat that

starting at 31, 32 and 33

Answer to 17_(b)

It takes away 10 as you go up a column

from whatever number

you start from.

17_(c)

Start at 00

Add on 1

add on another 1

add on another 1

add on yet another 1

What numbers did you get as you went?

Which way did you travel?

Start at 33

take off 1

take off another 1

take off another 1

take off yet another 1

what numbers did you get?

Which way did you travel?

Answer to 17_(c)

Adding from 00

you get

00 01 02 03 10

and you moved

to the right each time

Taking away from 33

you get

33 32 31 30 22

and you moved

to the left each time

17_(d)

Run your finger

along the

\

diagonal

What numbers do you get?

How far apart

are the numbers

on this diagonal?

Run your finger

along the

/

diagonal

What numbers do you get?

How far apart

are the numbers

on this diagonal?

Answer to 17_(e)

You get

00 11 22 33

and the numbers

are exactly spaced

11 apart from one another.

On the other diagonal

you get

03 12 21 30

and the numbers

are spaced exactly

03

apart

which you can test

by adding

03 to 03

to get 12

then add 03 to 12

to get 21

and finally add

03 to 21

to get 30
Coding Answers

Recall how we could do

quite a bit of arithmetic

using tiny circles?

We can use those tiny circles

to code answers

for our

compound

counting lines

too!

To show you

let's use

Ms Kittie's

compound

Base 4

Take a look at her 100-circle

The trick to coding

is to pick

tiny circles

one below

and

one above

hers.

Ok

That's

Mr Flea's

3-circle

and

Doggerly's

5-circle

The tiny circles

form a counting line

remember

which just goes

0 1 2 3 4 0 1 2 3 4 0 1 2 3...

and

0 1 2 0 1 2 0 1 2...

forever

I've colored them

green and red

Green for the bigger circle

Red for the smaller one

just to make things easier

to spot

what's what!

Hey,

while we're about it

let's make

Ms Kittie's base 4

00 01 02 03 10 11...

blue!

Here they are

all together:

Make a copy of this

on paper!

Let's take a look

for patterns!

EXERCISE 18

In the diagram below

Find and run your finger along

all the red 3-circle 0's

(there are 6 of them!)

Find and run your finger along

all the red 3-circle 1's

How many 1's are there?

Find and run your finger along

all the red 3-circle 2's

How many 2's are there?

Answers to Exercise 18

There are 6 x 0's

(read that as "six zeros")

and

5 x 1's

5 x 2's

(i.e. five one's and five two's)

They all form diagaonals

EXERCISE 19

In the diagram below

Find and run your finger along

all the green 5-circle 0's

How many of them did you find?

Find and run your finger along

all the green 5-circle 1's

How many 1's are there?

Find and run your finger along

all the green 5-circle 4's

How many 4's are there?

Find and run your finger along

all the green 5-circle 2's

Did you find all three of the 2's?

Find and run your finger along

all the green 5-circle 3's

How many 3's are there?

Answers to Exercise 19

There are 4 x 0's

and

3 x 1's

3 x 2's

3 x 3's

3 x 4's

They all form the other diagaonals

The Red Patterns

That long diagonal

splits the square

into two

parts

I've called one part

EASY

and the other part

HARD

Here's why

Easy Part

It turns out

you can

add

Ms Kittie's blue digits

to get the red 3-circle answer

00 = 0 + 0 = 0

00 = red 0

01 = 0 + 1 = 1

01 = red 1

02 = 0 + 2 = 2

02 = red 2

10 = 1 + 0 = 1

10 = red 1

11 = 1 + 1 = 2

11 = red 2

20 = 2 + 0 = 2

20 = red 2

Isn't that easy!

All answers add

below 3.

The Diagonal Itself

In a 3-circle

3 = 0

So anything

adding to 3

is

0

Running down Ms Kittie's

diagonal numbers

03, 12, 21, 30

and looking at the

3-circle

below them

note:

03 = 0 + 3 = 0 + 0 = 0

03 = red 0

12 = 1 + 2 = 3 = 0

12 = red 0

21 = 2 + 1 = 3 = 0

21 = red 0

30 = 3 + 0 = 3 = 0

30 = red 0

All answers

add to 3

exactly!

Hard part

All these numbers

add _over_ 3!

13 = 1 + 3 = 1 + 0 = 1

13 = red 1

22 we'll come back to

31 = 3 + 1 = 0 + 1 = 1

31 = red 1

23 = 2 + 3 = 2 + 0 = 2

23 = red 2

32 = 3 + 2 = 0 + 2 = 2

32 = red 2

33 = 3 + 3 = 0 + 0 = 0

33 = red 0

and we add

22 = 2 + 2 in 3-circle like this:

22 = 2 + 2 = 1 in 3-circle

22 = red 1

It's easy to see what they add to

in the easy area

it's just a tad more hidden

to see what they add to

when they add

to 3 or more

This will help you code

most of

Ms Kittie's numbers

without looking them up!

For example

11 will be red

1 + 1 = 2

in 3-circle

Perhaps

this might help you

also:

So use whichever

is easiest for you.

Work them out

(as shown above by adding)

or

look them up.

I prefer to work them out

because I'm too lazy

to look them up!
The Green Patterns

The second long diagonal

splits the square

into two

parts

EASY and HARD

once again

Here's why

This time

subtract

Ms Kittie's blue digits

to get the green 5-circle answer

The diagonal

00 = "0 from 0" = 0

00 = green 0

11 = "1 from 1" = 0

11 = green 0

22 = "2 from 2" = 0

22 = green 0

33 = "3 from 3" = 0

33 = green 0

all these numbers

are what I call level or flat...

They are the same size

as each other.

Easy Part

01 = "0 from 1" = 1

01 = green 1

12 = "1 from 2" = 1

12 = green 1

23 = "2 from 3" = 1

23 = green 1

02 = "0 from 2" = 2

02 = green 2

13 = "1 from 3" = 2

13 = green 2

03 = "0 from 3" = 3

03 = green 3

All these numbers

are what I call

"uphill numbers"

uphill numbers

go smaller to bigger

as we read from left to right.

Hard Part

Remember

this is a 5-circle

where

-1 = 4

-2 = 3

-3 = 2

(see below)

The numbers

in the hard part

are

10, 21, 32

20, 31, 30

They are all

downhill

numbers.

Downhill numbers

go bigger to smaller

as we read from left right

Lets subtract them:

In 5-circle

10 = "1 from 0" = -1

and -1 = 4

so 10 = green 4

(or -1)

In 5-circle

21 = "2 from 1" = -1

and -1 = 4

so 21 = green 4

(or -1)

In 5-circle

32 = "3 from 2" = -1

and -1 = 4

so 32 = green 4

(or -1)

In 5-circle

20 = "2 from 0" = -2

and -2 = 3

so 20 = green 3

(or -2)

In 5-circle

31 = "3 from 1" = -2

and -2 = 3

so 31 = green 3

(or -2)

30 = "3 from 0" = -3

and -3 = 2

so 30 = green 2

(or -3)

Perhaps

this might help you

remember the pattern:

Using Tiny Circles As Code

We can now use

the 100-circle of Ms Kittie

and the circles

on either side

(3-circle and 5-circle)

to do some arithmetic!

Example 1

Add

02 + 02

in Ms Kittie's base 4

Solution

Look up 02

Its code is green 2, red 2

The sum becomes

The green 5-circle

numbers

are added

2 + 2 = 4

and

the red 3-circle

numbers

2 + 2

are added

as well

but

in 3-circle

2 + 2 is not 4

it is 1

as we have met before

(shown below)

That gives us

a new code

green 4 and red 1

Look it up

to decode it.

It is 10

as shown below

So

(2 + 2 = 10)4

We'll draw that

to check it

Yes!

It's correct!

Example 2

Let's try

12 + 13

Solution

Once again

look up

12 and 13

12

is green 1, red 0

(Look it up

or see

"1 from 2" is 1

and

1 + 2 = 3 = 0

in

3-circle)

13

is green 2, red 1

(Look it up

or see

"1 from 3" is 2

and

1 + 3 = 1 + 0 = 1

in

3-circle)

So

12 + 13

is

Instead of adding

12 + 13

we add

1 + 2 = 3

in green 5-circle

and

0 + 1 = 1

in red 3-circle

which is

much easier!

(see above)

The answer

in code

is

green 3, red 1

Look it up

It decodes

to

31

(12 + 13 = 31)4

Check with drawings:

Its like

writing secret code!

It's kinda fun!

EXERCISE 20

TRY SOME YOURSELF!

20_(a)

11 + 12

20_(b)

13 + 13

20_(c)

12 + 20

20_(d)

22 + 03

Solutions

Answer to 20_(a)

11 + 12

Look up the chart

to code

(or be lazy and just

work it out as shown below)

Code for

11 is green 0, red 2

(1 from 1 is 0) and (1 + 1 is 2)

Code for

12 is green 1, red 0

(1 from 2 is 1) and (1 + 2 = 3 = 0)

Add greens

0 + 1 = 1

Add reds

2 + 0 = 2

New code

is

green 1, red 2

Decode:

in coded form

Answer

(11 + 12 = 23)4

Check by drawing

Answer to 20_(b)

13 + 13

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

13 is green 2, red 1

(1 from 3 is 2) and (1 + 3 = 1 + 0 = 1)

The other 13

is the same.

Add greens

2 + 2 = 4

Add reds

1 + 1 = 2

New code

is

green 4, red 2

Decode:

Answer

(13 + 13 = 32)4

Check by drawing:

Answer to 20_(c)

12 + 20

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

12 is green 1, red 0

(1 from 2 is 1) and (1 + 2 = 3 = 0)

Code for

20 is green 3, red 2

(2 from 0 is -2 which is 3 in 5-circle)

and

(2 + 0 = 2 in 3-circle)

Add greens

1 + 3 = 4

Add reds

0 + 2 = 2

New code

is

green 4, red 2

Decode:

Answer

(12 + 20 = 32)4

Check by drawing:

Answer to 20_(d)

22 + 03

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

22 is green 0, red 1

(2 from 2 is 0)

and

( 2 + 2 = 1 when counted in 3-circle)

Code for

03 is green 3, red 0

(0 from 3 is 3) and (0 + 3 = 3 = 0)

Add greens

0 + 3 = 3

Add reds

1 + 0 = 1

New code

is

green 3, red 1

Decode:

Answer

(22 + 03 = 31)4

Check by drawing:

Going Over 100

Don't forget

our coding uses

100-circle

It's biggest number

is

33

The number above that

is actually

100

How can we

decode an answer

that goes

above

100

?

It's actually quite easy!

First of all,

let's look at

100

In Ms Kittie's base 4

100 = 10 x 10

(shown in picture above)

For us

her

10 x 10

is

4 x 4

or 4 lots of 4

and

her hundred

is

16

For her

in her base

20 means 2 lots of 10

so

20 + 20 = "4" lots of 10

(she would say 10 lots of 10)

which is

100

If 20 + 20 is 100

then

22 + 23

must be a bit

over 100.

Knowing that

will help us do

our next sum.

EXAMPLE OVER 100

22 + 23

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

22 is green 0, red 1

(2 from 0 is 0)

and

(2 + 2 = 2 + 1 \+ 1 = 3 + 1 = 0 + 1 = 1)

Code for

23 is green 1, red 2

(2 from 3 is 1) and (2 + 3 = 2 + 0 = 2)

Add greens

0 + 1 = 1

Add reds

1 + 2 = 3 = 0

New code

is

green 1, red 0

Decode:

The answer

seems to be 12

but that is too small!

(22 + 23 = 12)4

is clearly wrong!

We know

20 + 20 = 100

so

22 + 23

must equal

one-hundred-and-something

We need a

1

for

"100"

in front

of the answer!

So let's just

stick one on!

But

112

is close

but also wrong.

To get the answer

simply

take the

1

(for "100")

out of

the 12

leaving

11

Then the final answer is

**1** 11

To get answers

in the hundreds

(1)

guess how many hundred

(2)

pull them out of the decoded answer

(3)

join them up

Check it using dots

as shown below:

No _te:_

I've shown you

how to do it

but not

why it works

here.

EXERCISE 21

Try some Sums Yourself

Adding Over 100

21_(a)

31 + 33

21_(b)

20 + 22 + 13

21_(c)

31 + 13 + 32 + 12

21_(d)

23 + 23 + 11 + 22

Solutions

Answer to 21_(a)

31 + 33

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

31 is green 3, red 1

(3 from 1 is -2 which is 3 in 5-circle)

and

( 3 + 1 = 0 + 1 = 1 in 3-circle)

Code for

33 is green 0, red 0

(3 from 3 is 0) and (3 + 3 = 0 + 0 = 0)

Add greens

3 + 0 = 3

Add reds

1 + 0 = 1

New code

is

green 3, red 1

Decode:

The apparent answer

31 + 33 = 31

is too small.

In the picture below

both 31 and 32

are just a bit under

100 each

So the answer

should be

over 100

and a little bit under 200:

That "<" sign

means

"less than"

Final step

Guess 1-hundred-and-something

Pull the

1

out of the

31

leaving

30

Answer

130

Just right!

(31 + 33 = 130)4

Answer to 21_(b)

20 + 22 + 13

Look up the chart

to code them

(or get lazy and just

work it out as shown below)

Code for

20 is green 3, red 2

(2 from 0 is -2 which is 3 in 5-circle)

and

( 2 + 0 = 2 in 3-circle)

Code for

22 is green 0, red 1

(2 from 2 is 0 in 5-circle)

and

(2 + 2 = 2 + 1 \+ 1 = 3 + 1 = 0 + 1 = 1)

Code for

13 is green 2, red 1

(1 from 3 is 2) and (1 + 3 = 1 + 0 = 1)

Add greens

3 + 0 + 2 = 3 + 2 = 0

(remember that in 5-circle, 5 = 0)

Add reds

2 + 1 + 1 = 3 + 1 = 0 + 1 = 1

New code

is

green 0, red 1

Decode:

The apparent answer

20 + 22 + 13 = 22

is too small

As shown above

20 is half Ms Kittie's idea of 100

So 20 + 22

is, roughly speaking,

about 100

(just a bit more perhaps)

and we still have

to add the 13

So the answer

should be

roughly speaking

1-hundred-and-something

That's close enough!

We just want

a rough estimate!

Final step

Guess 1-hundred-and-something

Pull the

1

out of the

22

leaving

21

Answer

121

(20 + 22 + 13 = 121)4

Answer to 21_(c)

31 + 13 + 32 + 12

Look up the chart

to code them

(or get lazy and

work them out as shown below)

Code for

31 is green 3, red 1

(3 from 1 is -2 which is 3 in 5-circle)

and

( 3 + 1 = 0 + 1 = 1 in 3-circle)

Code for

13 is green 2, red 1

(1 from 2 is 1) and (1 + 3 = 1 + 0 = 1)

Code for

32 is green 4, red 2

(3 from 2 is -1 which is 4 in 5-circle)

and

(3 + 2 = 0 + 2 = 2)

Code for

12 is green 1, red 0

(1 from 2 is 1) and (1 + 2 = 3 = 0)

Add greens

3 + 2 + 4 + 1 = 5 + 5 = 0 + 0 = 0

(recalling 5 = 0 in 5-circle)

Add reds

1 + 1 + 2 + 0 = 1 + 3 + 0 = 1 + 0 + 0 = 1

New code

is

green 0, red 1

Decode:

The apparent answer

31 + 13 + 32 + 12 = 22

is too small

The above picture shows

31 +13

are _roughly_

30 + 10 = 100

(a bit more)

and

32 + 12

are _roughly_

30 + 10 = 100

(again, a bit more)

So

our answer

should

roughly

be

a bit more

than 200

2-hundred-and-something

Final step

Guess 2-hundred-and-something

Pull the

2

out of the

22

leaving

20

Answer:

220

(31 + 13 + 32 \+ 12 = 220)4

Answer to 21_(d)

23 + 23 + 11 + 22

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

23 is green 1, red 2

(2 from 3 is 1) and (2 + 3 = 2 + 0 = 2)

The second 23

has exactly the same code.

Code for

11 is green 0, red 2

(1 from 1 is 0) and (1 + 1 = 2)

Code for

22 is green 0, red 1

(2 from 2 is 0)

and

(2 + 2 = 2 + 1 \+ 1 = 3 + 1 = 0 + 1 = 1)

Add greens

1 + 1 + 0 + 0 = 2

Add reds

2 + 2 + 2 + 1 =

2 + 1 + 1 + 2 + 1

equals

3 + 3 + 1 = 0 + 0 + 1 = 1

(we broke the second 2 into 1 + 1 to make it easier)

New code

is

green 2, red 1

Decode:

The apparent answer

23 + 23 + 11 + 22 = 13

is too small

As shown above

23 + 23

are _roughly_

20 + 20 = 100

(and a big bit more)

and

11 + 22

are _roughly_

10 + 20 = 30

which is

just a little bit

under 100

A bit over 100

a bit under 100

evens out

to

roughly

200

That's about

200

in all

as a rough guess

to our answer!

A rough guess

is all we need!

Final step

Guess about 2 hundred

Pull the

2

out of the

13

leaving

11

Answer

211

(23 + 23 + 11 \+ 22 = 211)4

Question:

What would have happened

if we had

incorrectly

guessed it began

with

1

hundred?

Well,

we would have said:

Guess 1 hundred and something

(which should be nearly 200)

Pull the

1

out of the

13

leaving

12

And the final answer

would have been

112

but we expected

our answer

to be

nearly 200

and

112

is closer to

100

so it's

still too small!

The correct answer must be

211

which is

closer to 200

than 112
BIG TIP

Before trying

another sum

here's a

big tip!

In 5-circle

4

is also

-1

(backwards 1)

Look what happens

if we start at 0

then add 4

and 4 more

and 4 more...

Our answers

go

3, 2, 1, 0, 4...

We go

backwards 1

each time

we

add 4

The answers

are the same as

counting backwards

1 at a time like this:

Instead of adding

4

you can

take away

1

in a 5-circle!

The same trick

works

in a 3-circle

with its

biggest number

being the

2

In 3-circle

2 = -1

That's why we can say:

2 + 2 = 2 - 1 = 1

Remembering

adding 4

is the same as

subtracting 1

in 5-circle

look at the next example!

EXAMPLE

21 + 21 + 21

Look up the chart

to code them

(or get lazy and just work them

out as shown below)

Code for

21 is green 4, red 0

(2 from 1 is -1 which is 4 in 5-circle)

and

(2 + 1 = 3 = 0 in 3-circle)

All the other 21's

also code to

green 4, red 0

Add greens

4 + 4 + 4 = 4 -1 - 1 = 2

(easy when we know 4 = -1)

Add reds

0 + 0 + 0 = 0

New code

is

green 2, red 0

Decode:

Once again the apparent answer

21 + 21 + 21 = 30

is too small

See above,

21 + 21

is roughly

20 + 20 = 100

(a bit more)

leaving the 21 left over

on its own

So the final answer

must

roughly

be

100 and a bit

1-hundred-and-something

Final step

Guess 1-hundred-and-something

Pull the

1

out of the

30

leaving

NOT

29

but

23

Remember

you are looking through

Ms Kitties

Base 4

eyes!

(see below)

As shown above

30 - 1 = 23

(not 29!)

in base 4

The final answer is

123

(21 + 21 + 21 = 123)4

Two things happened there

(1)

4 + 4 + 4

became

4 - 1 - 1

in 5-circle

(2)

30 - 1

was

23

(not 29)

which you read off

from

Ms Kittie's

100-circle

Those 4's

are easier than they look!

EXERCISE 22

Try Some with Green 4's

22_(a)

31 + 32 + 13 + 21

22_(b)

21 + 10 + 32

22_(c)

20 + 21 + 20 + 21

Solutions

Answer to 22_(a)

31 \+ 32 + 13 + 21

Look up the chart

to code them

(or get lazy and just

work it out as shown below)

Code for

31 is green 3, red 1

(3 from 1 is -2 which is 3 in 5-circle)

and

(3 + 1 = 0 + 1 = 1)

Code for

32 is green 4, red 2

(3 from 2 is -1 which is 4 in 5-circle)

and

(3 + 2 = 0 + 2 = 2 in 3-circle)

Code for

13 is green 2, red 1

(1 from 3 is 2) and (1 + 3 = 1 + 0 = 1)

Code for

21 is green 4, red 0

(2 from 1 is -1 which is 4 in 5-circle)

and

(2 + 1 = 3 = 0 in 3-circle)

Add greens

3 + 4 + 2 + 4 = 3 - 1 + 2 - 1 = 2 + 1 = 3

(easy when we know 4 = -1 in 5-circle)

Add reds

1 + 2 + 1 + 0 = 1 + 3 + 0 = 1 + 0 + 0 = 1

New code

is

green 3, red 1

Decode:

The apparent answer

31 + 32 + 13 + 21 = 31

is too small

As shown above

31 is _roughly_ 100 on its own

32 is also _roughly_ 100

Together

they're a bit under 200

Add the bits left over

to them

and you must be

into the 200's.

giving

a final answer

of

roughly

2-hundred-and-something

Final step

Guess 2-hundred-and-something

Pull the

2

out of the

31

leaving

23

(not 29!)

Answer

223

(31 + 32 + 13 \+ 21 = 223)4

Answer to 22_(b)

21 \+ 10 + 32

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

21 is green 4, red 0

(2 from 1 is -1 which is 4 in 5-circle)

and

(2 + 1 = 3 = 0 in 3-circle)

Code for

10 is green 4, red 1

(1 from 0 is -1 which is 4 in 5-circle)

and

(1 + 0 = 1 in 3-circle)

Code for

32 is green 4, red 2

(3 from 2 is -1 which is 4 in 5-circle)

and

(3 + 2 = 0 + 2 = 2 in 3-circle)

Add greens

4 + 4 + 4 = 4 - 1 - 1 = 2

(easy when we know 4 = -1 in 5-circle)

Add reds

0 + 1 + 2 = 0 + 3 = 0

New code

is

green 2, red 0

Decode:

The apparent answer

22 + 10 + 32 = 30

is too small

As shown above

10 + 32

is _roughly_

10 + 30 = 100

(a bit more)

which

means we have

100 and the 21 and a little bit

or

1-hundred-and-something

as our answer

roughly!

Final step

Guess 1-hundred-and-something

Pull the

1

out of the

30

leaving

23

(not 29!)

Answer

123

(22 + 10 + 32 = 123)4

Answer to 22_(c)

20 + 21 + 20 + 21

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

20 is green 3, red 2

(2 from 0 is -2 which is 3 in 5-circle)

and

(2 + 0 = 2 in 3-circle)

Code for

21 is green 4, red 0

(2 from 1 is -1 which is 4 in 5-circle)

and

(2 + 1 = 0 in 3-circle)

Code for the other

20 is green 3, red 2

(same as just above)

Code for the other

21 is green 4, red 0

(same as just above)

Add greens

3 + 4 + 3 + 4 = 3 - 1 + 3 - 1 = 2 + 2 = 4

(easy when we know 4 = -1 in 5-circle)

Add reds

2 + 0 + 2 + 0 = 2 + 0 - 1 + 0 = 1

(easy when we know 2 = -1 in 3-circle)

New code

is

green 4, red 1

Decode:

The apparent answer

20 + 21 + 20 + 21 = 10

is too small

As you can see above

20 + 21

is _roughly_

20 + 20 = 100

(and a bit)

and

so is the other

20 + 21

so

the final answer

is

roughly

200 and a few bits left over

or

2-hundred-and-something

Final step

Guess 2-hundred-and-something

Pull the

2

out of the

10

leaving

02

(not 08!)

Answer

202

(20 + 21 + 20 \+ 21 = 202)4

Subtractions

Subtracting works in much the same way.

Just subtract instead of adding.

EXERCISE 23

Try Some

23_(a)

32 \+ 21 + 33 - 12

Answer to 23_(a)

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

32 is green 4, red 2

(3 from 2 is -1 which is 4 in 5-circle)

and

(3 + 2 = 0 + 2 = 2 in 3-circle)

Code for

21 is green 4, red 0

(2 from 1 is -1 which is 4 in 5-circle)

and

(2 + 1 = 3 = 0 in 3-circle)

Code for

33 is green 0, red 0

(3 from 3 is -0) and (3 + 3 = 0 + 0 = 0)

Code for

12 is green 1, red 0

(1 from 2 is 1) and (1 + 2 = 3 = 0)

Add and Subtract greens

4 + 4 + 0 - 1 = 4 - 1 + 0 - 1 = 2

(4 = -1 in 5-circle and we are also taking away 1 anyway)

Add reds

2 + 0 + 0 - 0 = 2

New code

is

green 2, red 2

Decode:

The apparent answer

32 + 21 + 33 - 12 = 02

is too small

As you can see above

32 is roughly 100

( a bit under)

and

33 is roughly 100 also

( a bit under)

Together

they are

a bit under

200

If we add the

21 and 12

on top of that

we should be

roughly

a bit

over

200

we guess:

2-hundred-and-something

Final step

Guess 2-hundred-and-something

Pull the

2

out of the

02

leaving

00

Answer

200

(32 + 21 + 33 \- 12 = 200)4

23_(b)

23 + 13 - 31 + 22

Answer to 23_(b)

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

23 is green 1, red 2

(2 from 3 is 1) and (2 + 3 = 2 + 0 = 2)

Code for

13 is green 2, red 1

(1 from 3 is 2) and (1 + 3 = 1 + 0 = 1)

Code for

31 is green 3, red 1

(3 from 1 is -2 which is 3 in 5-circle)

and

(3 + 1 = 0 + 1 = 1 in 3-circle)

Code for

22 is green 0, red 1

(2 from 2 is 0)

and

(2 + 2 = 2 - 1 = 1 in 3-circle)

(easy when we know 2 = -1 in 3-circle)

Add and subtract greens

1 + 2 - 3 + 0 = 3 - 3 + - = 0

Add and subtract reds

2 + 1 - 1 + 1 = 3 = 0

New code

is

green 0, red 0

Decode:

The apparent answer

could be

(23 + 13 - 31 \+ 22 = 00)4

or

(23 + 13 - 31 \+ 22 = 33)4

It's easy to see 00 is wrong and 33 is just about right!

Check it with an estimate:

As you can see above

23 + 13

is about

20 + 10

which is

+30

(with some extra left over)

so if we take away

31

which is roughly

\- 30

we get

just the extra bit left over.

The answer will be

22 + some extra

roughly

So the answer

will be closer to 100

roughly.

Our answer was 33

That's close to 100.

So our answer

is absolutely

correct

just as it stands!

23_(c)

33 + 21 + 13 - 23

Answer to 23_(c)

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

33 is green 0, red 0

(3 from 3 is 0) and (3 + 3 = 0 + 0 = 0)

Code for

21 is green 4, red 0

(2 from 1 is -1 which is 4 in 5-circle)

and

(2 + 1 = 3 = 0 in 3-circle)

Code for

13 is green 2, red 1

(1 from 3 is 2) and (1 + 3 = 1 + 0 = 1)

Code for

23 is green 1, red 2

(2 from 3 is 1) and (2 + 3 = 2 + 0 = 2)

Add and Subtract greens

0 + 4 + 2 - 1 = 4 + 1 = -1 + 1 = 0

(easy when we know 4 = -1 in 5-circle)

Add reds

0 + 0 + 1 - 2 = 1 - 2 = -1 = 2

(easy when we know -1 = 2 in 3-circle)

(or draw it as below)

New code

is

green 0, red 2

Decode:

The apparent answer

33 + 21 + 13 - 23 = 11

is too small

As shown above

33

is roughly

100

and

21 - 23

is roughly

20 - 20 = 0

They cancel

each other out!

leaving

roughly 100

and a bit

(the 13)

or

1-hundred-and-something

Final step

Guess 1-hundred-and-something

Pull the

1

out of the

11

leaving

10

Answer

110

(33 + 21 + 13 \- 23 = 10)4
Human Counting Base

Does this work

for us humans?

You bet it does!

This works for all

EVEN

bases.

You can even try it

with base 6

and base 8

if you like

exploring!

(ODD bases are similar,

but a bit odd, a bit different)

Here is the

100-circle

for us

humans:

What are the circles

above and below

ten?

Think

9, 10, 11

We use

9 and 11

circles

instead of

3 and 5

circles

Our 100-circle

is so big

it's difficult

to fit on the page

with the circle

numbers

above and below!

If you want to try it

draw the 100-circle

on paper

with plenty of room

above and below

them for your

red and green numbers

Then

above

(in a green pen)

clock out the endless circle

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3...

which is

the 11-circle

and below

clock out the endless circle

0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3...

which is

the 9-circle

Here's a close up

of the start:

and a close up

of the end:

We will show fragments

for you in this sum

In fact

you can use

these same methods

to multiply numbers

into their thousands

in our base

but

at this stage

we'll just show it works!

Example

Human Base Ten

89 + 89

Look up the chart

to code them

(or get lazy and just

work them out as shown below)

Code for

89 is green 1, red 8

(8 from 9 is 1 in 11-circle)

and

(8 + 9 = 8 + 0 = 8 in 9-circle)

(in 9-circle the 9 = 0)

The other 89

is the same.

Add greens

89 + 89

becomes

1 + 1 = 2

in 11-circle

Add reds

89 + 89

becomes

8 + 8 = 8 - 1 = 7

(easy when you know 8 = -1 in 9-circle)

New code

is

green 2, red 7

Decode:

The apparent answer

89 +89 = 79

is too small

Each 89

is just a bit

under 100

So

together

they are under

200

but over

100.

Together they are

1-hundred-and-something.

Final step

Guess 1-hundred-and-something

Pull the

1

out of the

79

leaving

78

Answer

178

(89 + 89 = 178)10

In different bases

we use

different numbers

but

exactly the same

methods!

This is why

Scott Flansburg

(a whizz bang maths calculator)

says

"Don't memorize answers"

Learn the methods instead!

If you take the time

to write out

the 100-circle

of our base ten

with the

9 and 11 circles

above and below

you could

do more additions

and subtractions.

But we'll leave that

for another book!
Using a 100-circle to Add and Subtract

Directly

In this section

all answers

will be

in Doggerly's

base 5

Here is the 100-circle

of base 5:

Suppose we want to find

42 - 21

Start at 42

Go up two rows

That's 42 - 20 is 22

Now take off 1 by going left

1 step to 21

Answer:

42 - 21 = 21

We could do that this way too:

and say

4 - 2 = 2

2 - 1 = 1

Answer: 21

Subtractions gets harder

when one of the numbers

goes below zero

For example

41 \- 23

We could write out

but we don't know

what 1 - 3 is

Here's how we can do it

in 100-circle:

Start at 41

Go up 2 to remove 20

Now count left

3 steps

20, 14, 13

The answer is:

41 - 23 = 13

in this base.

You can add the same way.

For example

11 \+ 21

Starting at 11

go down 2 rows to 31

(to add 20)

then go right

1 step to 32

(to add the 1)

Answer:

11 + 21 = 32

You could have done that

this way also

based on knowing

1 + 2 = 3

and

1 + 1 = 2

But additions get hard

when one of the digits

adds

to 10 or more.

For example

13 \+ 23

We know 1 + 2 = 3

but what is 3 + 3

in this base?

Here's how we can add it anyway

using the 100-circle:

Start at 13

Move down 2 rows to 33

(to add on 20)

now go right

3 steps to 41

(to add on the 3)

Answer:

13 + 23 = 41

EXERCISE 24

Use Doggerly's

100-circle

to do these calculations

in base 5:

24_(a)

41 - 12 = ?

24_(b)

33 - 12 = ?

24_(c)

32 - 24 = ?

24_(d)

31 - 22 = ?

24_(e)

34 - 13 = ?

24_(f)

21 - 13 = ?

Answers to Exercise 24

(all in Doggerly's base 5)

Answer to 24_(a)

41 - 12 = 24

Answer to 24_(b)

33 - 12 = 21

Answer to 24_(c)

32 - 24 = 03

Answer to 24_(d)

31 - 22 = 04

Answer to 24_(e)

34 - 13 = 21

Answer to 24_(f)

21 - 13 = 03

Exercise 25

25_(a)

11 + 13 = ?

25_(b)

22 + 13 = ?

25_(c)

03 + 21 = ?

25_(d)

14 + 14 = ?

25_(e)

13 + 23 = ?

25_(f)

23 + 03 = ?

Answers to Exercise 25

Answer to 25_(a)

11 + 13 = 24

Answer to 25_(b)

22 + 13 = 40

Answer to 25_(c)

03 + 21 = 24

Answer to 25_(d)

14 + 14 = 33

Answer to 25_(e)

13 + 23 = 41

Answer to 25_(f)

23 + 03 = 31

Try some

in our human base ten

Trace out the answers with your fingers

on the diagram below:

EXERCISE 26

26_(a)

93 - 22 = ?

26_(b)

75 - 37 = ?

26_(c)

57 - 32 = ?

26_(d)

51 - 34 = ?

26_(e)

65 - 32 = ?

26_(f)

31 - 15 = ?

Answers to Exercise 26 (Human answers)

Answer to 26_(a)

93 - 22 = 71

Answer to 26_(b)

75 - 37 = 38

Answer to 26_(c)

57 - 32 = 25

Answer to 26_(d)

51 - 34 = 17

Answer to 26_(e)

65 - 32 = 33

Answer to 26_(f)

31 - 15 = 16

Exercise 27 (Human Numbers)

27_(a)

32 + 34 = ?

27_(b)

46 + 45 = ?

27_(c)

68 + 24 = ?

27_(d)

38 + 36 = ?

27_(e)

13 + 33 = ?

27_(f)

18 + 56 = ?

Answers to Exercise 27 (Human Answers)

Answer to 27_(a)

32 + 34 = 66

Answer to 27_(b)

46 + 45 = 91

Answer to 27_(c)

68 + 24 = 92

Answer to 27_(d)

38 + 36 = 74

Answer to 27_(e)

13 + 33 = 46

Answer to 27_(f)

18 + 56 = 74

Note

we add or subtract

the tens

FIRST

then the ones

That's the opposite of what most people do

but it makes a lot of sense

That way you have an idea

of how big the answer is straight away

and

you are saying the answer

immediately

reading left to right

At this stage

do yourself a favour.

Watch this You Tube video

by Scott Flansburg

<https://www.youtube.com/watch?v=hesKQ_y1P7k>

and

visit his website

<http://www.scottflansburg.com/>

and

download

his free app.

Have fun!

Note:

Scott reverses

the 11-circle

from what I've shown

for example

he says

12 is

1 - 2 = -1

and

21 is

2 - 1 = 1

whereas

I would say

12 = 1 from 2 = 1

and

21 = 2 from 1 = -1

the other way

around!

I'm sure he has his reasons,

so just be aware of it.

It's like

driving on the left

in New Zealand

and

driving on the right

in America.

Otherwise it should all

be crystal clear!

Scott is a great educator

and a brilliant calculator.

I think you'll enjoy his videos as much as I did!

I highly recommend them!
BOOK SUMMARY

Wow!

You have come

a loooooong way!

You understand

where number comes from

You understand

bases

You are calculating

in a multibase

not just

one base!

Well done!

You have a

fabulous foundation

for understanding

arithmetic!

It's like knowing

several languages

instead of

just one!

You have also

acquired skills

which will help you

unlock the secrets

of fast

mental arithmetic

but that will have to wait

for another book!

About Circlemaths

Circlemaths was the discovery of my father,

Dr Stephen Watson Taylor

a medical GP in New Zealand

Having worked for years on philosophy

he said he had found

"the key that opened every door"

He meant the pattern

behind all patterns.

When we were young

we figured that if that was so

he would have no trouble

translating his philosophical ideas

into mathematics

which

to everyone's surprise

he did.

The result you have just seen.

There is much more to it but this makes a good start!
Other Titles

Visual Circlemaths Book 1

A Parent Teacher Guide to

Speed Subtraction and

Long Division

by

Robert Michael Taylor

You May Also Like:

BIRTH

by Dr StephenWatson Taylor

(Dr Taylor's philosophy applied to Natural Birth.)

To see these free titles

visit my Smashwords Profile Page
Contact the Author

Visit my webpage at:

www.Circlemaths.com

for more free stuff

maths games

and philosophy

Favorite me on Smashwords

Robert Michael Taylor Author Page

or email me at:

circlemaths@gmail.com
