 
All right, before we
begin our lecture,
I want to make sure that
you are prepared to take
good notes through
all these lectures.
So make sure that, on your
notes, before you begin,
you write down the heading.
So in this case, it will be
Module 1, Lecture 1.1-- Decimal
Number System.
You'll have a date on the side.
The For My Eyes
Only column is here,
as we talked about in
Module 0, so that if you
have any thoughts, any questions
that you want to ask later,
you can write that here.
If you want to do any
scratch work for you answers,
you can do it here as well.
All right, we asked you
to do a counting project
at the end of the
Module 0 lecture.
But we want to take inventory
to see if you actually did it.
If you have not
attempted it yet,
please pause the video here,
and just give it a try.
Try something.
Even if you feel like
nothing is coming,
spend at least five or 10
minutes and see what happens.
You might be very surprised.
All right, if you were
trying and you're stuck,
just stop, focus, be quiet.
Bring your attention
to your breath.
Everything under observation
changes, just remember.
So focus.
Breathing in, become
aware of your in breath.
Breathing out, become
aware of your out breath.
Do that a few times,
until you feel your brain
is a little more
clearer to think.
And then continue attempting
if you're having trouble.
If you're not having
trouble, that's great.
If you're having trouble,
don't self-judge and make
assumptions about your ability.
This is not going to
serve any purpose.
But think about
all the positives.
Remember, it's OK to not be able
to do every single thing right
away.
What is most important
is the attempt part.
If you're attempting it,
then that is really good.
All right, so now we have
attempted the project,
and we're going to share
some common experiences
that many of you had.
I'm going to talk about what
some of you did in class.
We looked at groupings, or
bundlings, of some sort.
Some of you used numbers.
Some of you used letters.
And when I say numbers, you made
up some symbols for numbers.
And that's great.
But each one of you had
some sort of bundling.
And the smallest
number in your bundle
is called the base of
your number system.
Making the groups consistent
across larger quantities
of objects-- that's something
we saw that many of you did.
But we also had
many difficulties
in making our number system.
This is what I saw in
some of your notes--
some people wrote that they had
trouble thinking for even more
than two minutes, it was
extremely frustrating.
And that's OK.
Going through this
process tells you
what doing mathematics is like.
Some of you tried
using letters, but then
didn't know what to do after
the letters were exhausted,
after the letter
Z. And one of you
wrote that you needed
to ask, in class,
what other ways of
doing this project
are because you were
struggling so much.
So these are just some samples
that I'm showing you in case
you want to know
what people write
in the For My Eyes Only column.
And you don't have
to have anything.
It could be all blank.
Some other concerns
that people voiced
was they that trouble extending
their system to larger numbers.
Some of you had trouble
understanding your own system,
to communicate well
with other people.
You also had trouble
understanding
what some other
people did in class.
Also, some of you just
ran out of symbols
after a certain number of
toothpicks were exhausted.
And then, the last
thing was being
able to add, or subtract,
or multiply, or divide,
using your system
that you came up with.
So as you can see, creating a
number system is not that easy.
It actually takes
quite a lot of thought.
And it is very
important to remember
that numbers, in some
form, have existed
from primitive humans 'til the
most advanced civilizations.
They all felt a need to
develop number systems
to make objective sense of the
world, starting with bartering,
or keeping track of
possessions, like their sheep.
And then of course, now, in
advanced sciences, physics,
chemistry, biology, and even
technology-- without numbers,
there would not be
many of these things.
Numbers help us facilitate
quantitative communication.
They also allow us to make
sense of how much something is--
one quantity is bigger
than another, and so on.
The oldest form of
numeration was the use
of tally marks 25,000 to 35,000
years ago, in the Stone Age.
You know, sometimes
looking at history
gives us some motivation
and inspires us to learn,
because it humanizes
some of the mathematics,
and it's not just a
dry subject anymore.
So tally marks are
the oldest form
of numeration, which is
just shown right here.
Did you see that?
So you go, one, two,
three, four, and five.
For a fifth tally mark, you
just have a horizontal bar
going across.
Even to this day, we use this
tally mark system-- which
is quite interesting, isn't it?
People have found
bones and carvings
that show the use
of tally marks.
So let's look at some of
the past civilizations,
and what kinds of number
systems they used.
The Babylonians, 3000 to
2000 BC-- 5,000 years ago,
almost-- they used base 60
number system, which we still
use to this day when we're
looking at the clocks.
We have 60 seconds
is one minute,
60 minutes is one
hour, and so on.
The way they wrote their
numbers were upside down
triangles for ones, sideways
symbols like this for tens.
And so 12,074 was
written in this manner.
And in our system, this is
what might make sense to you.
The Aztecs and the Mayans
used base 20 system
in the fourth century, AD.
And all three of
these civilizations
did very complex
astronomical computations.
Our own decimal system
has been credited
cataloging to Bhaskaracharya,
in the 11th century, AD.
And that's the first
time that people
are seen to use the
number zero, which
was referred to as shunya, to
represent what it means now,
to us, and not just as a
positional placeholder.
This digits we use are 1,
2, 3, 4, 5, 6, 7, 8, 9.
And then, after 9, we will start
grouping in groups of tens.
So each of these digits
represents how many objects
you have.
And one more than
9 is represented
as one group of 10
and zero singletons.
 
The next number is
written as one group
of tens and one singleton.
The one after that is one
group of 10 and two singletons.
So here we have 11.
And this is referred to as 12.
Then we go 13, 14, and so on,
once you know how to count.
If you go to 99, the
one number after 99
is going to be written as 1, 0,
0-- zero singletons, zero tens,
and one group of one hundreds.
So in our decimal number system,
the position of each digit
matters.
And if you look at
the decimal point,
it separates numbers that are
bigger than 1 from numbers
that are fractional.
And we use powers of
10 to represent it.
So for example, if you have the
number 324.56, in the expanded
form, it will look like this.
And it is very important
that you understand
the values, the place values,
of each of these digits.
So for example,
this 3 is considered
to be in the hundreds place,
2 is in the tens place,
4 is in the ones place,
5 is in the tenths place,
and 6 is in the
hundredths place.
So it's important that you
understand and recognize
what each of these
place values represent.
Another way to write this would
be in the exponential notation,
where you have 10 squared
is 100, and 1 over 10
is written as 10 to the
negative 1, 1 over 100
is 10 to the negative second.
 
So with this
structure, we can see
that multiplying and dividing
by 10 becomes very easy.
For example, if you want to
multiply 324.56 by 1,000,
you're going to move the
decimal point one, two,
and three places to
the right to get you
the new number, 324,560.
If you are going to
divide by powers of 10,
you're going to move
the decimal to the left.
And how many places?
Four places.
And that's your new number.
So multiplying by
powers of 10, like 100,
10,000, 1,000, makes
the number big,
whereas dividing
by 10, 100, 1,000
is going to make
the number smaller,
in case you have
trouble remembering
which way to move the decimal.
All right, there are many
forms of decimal numbers.
Terminating
decimals, which means
they have a finite number of
digits-- so you can see they
have a certain number
of digits, and it stops.
Whereas non-terminating decimal
means the number of digits
just keeps on going.
And in that, you have two types.
With repeating pattern, you
write the repeating pattern
in the following way-- 3.41 bar.
The bar means that
the 4, 1 repeats.
So you have 41, 41,
41, 41 repeating.
Here's another example.
For a non-terminating decimal
without a repeating pattern,
the digits just keep on
going, and there does not
have to be any pattern at all.
In this particular case,
you have a pattern,
but it's not a
repeating pattern.
You can see how there's
an extra 1, then
two extra 1s, then
three extra 1s.
So it's not a repeating pattern.
All right, so in
daily life, where
are you going to use numbers?
Maybe to pay some
bills, or to buy things.
So if Julie bought
a TV for $453.68,
she could pay by credit
card or by check.
And if she wrote a check, it
would look something like this.
So you can see that
we need to know
how to work the decimal numbers,
be able to write them in words,
like this, or as numerals,
like how you see there.
So understanding decimal
numbers is the key
to understanding much more
complicated objects later.
And for example, if you take
whole numbers, like 324,
and write them in expanded
form, and then replace the 10s
with x's just for
fun, you get an object
called a polynomial of degree
2-- 3x squared plus 2x plus 4.
So I think that
this is something
to keep in mind, that
mathematics is something
that is developed from need.
And you just replace
things for fun,
to see what happens,
and then, play with it.
And playing just means how
to add, subtract, multiply,
divide, or do whatever
else you might
think of to these objects.
If you take a
decimal number system
and replace all 10s
with x's, it actually
gives rise to something
called algebraic expressions.
And so polynomials are a special
case of algebraic expressions
which we'll see later.
So here's your homework.
 
