I like the number 8.
I like the way it smells, like
2 and 4, with a hint of 3,
in a cubic sort of way.
Though that only serves
to further support
the sweet scent of 2.
There is a need, deep
within me, to express
how I feel about logarithms.
But I worry they've already
been ruined for you,
obscured by the
shuffling of numbers.
This shuffling, shuffling,
soulless zombie step,
suffering the mind's
logarithmic decay.
Institutions full of innocence
drawn into that dance of death.
And yet, if there's one thing
with true universal appeal
that all humans share, it
is not love, not desire
for food, shelter, or safety.
It's mathematical truth,
which applies equally to us
all, the normal and the
abnormal, the living
and the dead.
That so much of school is
wasted on nitpicky particulars
of elementary algebra
is incredibly tragic.
Elementary algebra is just
fancy counting, all of it.
You start by learning to count.
You were taught a
sequence of syllables.
And you parroted it back for the
reward of some authority figure
telling you, "good job,"
or sometimes, "very good."
And then, you were told
that 1 plus 1 is 2.
And parroted that
back, too, playing
a call-and-response game
of nonsense syllables
until eventually you figured
out the fundamental idea
of algebra.
There is this ordered sequence
of things, these numbers.
And they have one
true behavior, plus 1.
Each individual symbol or
syllable is just a shorthand
for some plus 1's.
2 is positive 1 plus 1.
3 is plus 1 plus 1 plus 1.
When we say 2 plus
3 equals 5, it's
not that adding 2 and 3
results in 5 or that 2 plus 3
is different but
equivalent to 5.
But that 2 plus 3 is 5.
They are different names for
the same exact thing, which
is plus 1 plus 1
plus 1 plus 1 plus 1.
That's it.
Plus 1 is all there is.
And no matter how you
group those plus 1's,
all that matters in the end
is that you don't lose any.
Oh, then, there's
reverse plus 1's,
negative numbers,
subtraction, counting down.
Those are just plus 1's
again, but going back in time.
7 minus 5 is just counting to
7, plus 1 plus 1 plus 1 plus 1
plus 1 plus 1 plus 1.
And then, going back
in time for 5 plus 1's.
You can think of it as
mutually annihilating
plus 1's and anti-1's and seeing
what's left over, if you want.
8 minus 2 is just counting to
8, and then, going back in time
2 counts.
1 minus 4 is when you find
that counting from 4 to 1
takes negative time.
You're done 3 counts
before you started.
Or the 1 annihilates with
one of the four anti-1's
and three anti-1's are left.
And if you start with
two negative numbers,
you're already back in time
and only go further back.
You've just got a
pile of anti-plus 1's
trapped in the past.
Subtraction is
egocentric counting.
7 minus 4 means you are 4.
You are the center of the
universe and compared to you,
what is 7?
Well, just count to it,
plus 1 plus 1 plus 1.
From where you're
looking, 5 is plus 1.
3 is also just plus 1 away,
but against the flow of time.
So it's an anti-1.
Minus 4 is twice as
far away back in time
as it would be if you were
looking from a 0 perspective.
To get to negative 8,
sometimes to make the harder
things simple, first, you
have to make the simple things
harder.
Addition is a process
of plus 1 plus 1 plus 1.
Multiplication is a process
of plus n plus n plus n.
For multiples of 2, instead of
counting plus 1 plus 1 plus 1,
you're counting by plus 1 plus
1, plus 1 plus 1, or plus 2,
for short.
Plus 2 plus 2 plus 2 plus 2.
Fancy counting.
Counting in a plus
2 kind of way.
Counting in a plus
5 kind of way.
Counting in a plus
10 kind of way.
What is 5 times 7?
All that means is count in a
plus 5 sort of way seven times,
which happens to be
the same as counting
in a plus 7 sort of way 5 times.
1, 2, 3, 4, 5.
This is why 8
smells like 2 and 4.
Because 8 is 4 when
you're counting in a plus
2 sort of way.
And 8 is 2 if you're counting
in a plus 4 sort of way.
Division is fancy counting.
A divided by B
asks the question,
what is A, when counting
in a plus B sort of way?
27 divided by 9 means, what is
27, in a plus 9 sort of way?
Plus 9 plus 9 plus 9, it's 3.
What is seven in a
plus 2 sort of way?
Just count, 1, 2, 3, and 1/2.
Division means counting
by plus denominators
until you get to the numerator
and seeing how long it takes.
7 divided by 1/3 means
counting by plus 1/3's
until you get to seven.
Plus 1/3 plus 1/3 plus 1/3
plus 1/3 plus 1/3, all the way
to 21.
Exponentiation-- powers--
also fancy counting.
This time, instead of
counting by plus whatever,
you count by times whatever.
Powers of 2 are just counting
in a times 2 sort of way.
First power of 2, second
power of 2. third power of 2.
The size of the times-ish
step is the bottom number.
And the number of times-ish
steps is a little exponent.
Powers of 10 count in
a times 10 sort of way.
To take 10 to the
sixth, you just
take six steps of size times
10, 10 times 10 times 10 times
10 times 10 times 10.
That's why 8 has a whiff of 3.
8 is 3 when counting
in a times 2-ish way.
Then, there's roots.
To take the fourth root
of 81, you're just saying,
hey, if 81 is 4 in a
timesy sort of way,
then how are we counting?
It's like going a
quarter of the way there,
but a timesy quarter, not an
addish quarter, times 3 times
3 times 3 times 3.
As far as times-ish
counting is concerned,
3 is a quarter of 81,
which is why 81 to the 1/4
is the same as the
fourth root of 81.
And personally, I think
it's kind of weird
that we keep around
this root notation,
when fractional powers are
so much more descriptive.
81 to this 3/4 means
you're going 3/4 of the way
to 81, in a times-ish
counting way.
A times-ish 3/4 of 81, simple.
Writing that as fourth root 81
all to the power of 3 is just,
why would you do that?
Anyway, roots or
factional powers
are the case where you know
how many steps you want to go,
and what you want to
reach when you get there.
Six times-ish steps
to get to a million
means x to the sixth
equals 1 million.
The base is the way
the exponent counts.
And the way to count to
a million in six steps is
by times 10's.
But say, you know the
size of your step,
and you know what number
you want to get to,
but you don't know how
many steps to take.
If you're counting in
a times 3 sort of way,
how many steps to 81?
Or let's write that in
a way that solves for x.
3 to the something is 81.
So in a system that counts
in a times 3 sort of way,
81 equals four steps.
When I read the
notation, I visualize it.
STCx 2 of 8.
System That Counts in
a times 2 sort of way.
That's a line of times
2's going on forever.
Then, the 8 is
somewhere in there.
And how do I get there?
In a system that counts
in a times 2 sort of way,
8 is three steps.
For a system that counts in a
times 4 sort of way, 16 is 2.
STCx 7 of 343 is 3.
STCx 2 of 1/4 is,
well, in a system
that counts in a times
2 sort of way, now,
you're going back in
time, dividing by 2.
Once, twice, that's
negative 2 steps.
The STCx 125 of 25 means
we have a system that
counts in a times 125 way.
But the number we're
trying to get to
is only a fraction
of the way to 125.
And we can divide this step into
three equal fractional steps,
which times up to 125,
each of size times 5.
Not plus 5, as if we were
dividing the normal way.
But the times 5 of dividing
in a times-ish way.
Times 5, times 5, that's 25.
Two steps along.
Just one more times
5 to 125, making 25,
2/3 of 125, in a
timesy counting way.
In a STCx 64, 128 is not
a whole multiple of 64,
but a fraction past it.
Luckily, you can divide 64
into 6 equal times 2 parts.
Times 2 times 2 times 2
times 2 times 2 times 2.
And just add one more
times 2 to get to 128.
Can you feel the 7/6-ness
of 128 in relation to 64?
Can you sense how
64 and 128 together
evoke 32, which is 5/6 of 64?
And 5/7 of 128 on this scale.
Can you smell the
sweet scent of times 2
times 2 times 2 to the fifth, 2
to the sixth, 2 to the seventh?
You can almost taste 256.
This, my good friend, is
the logarithmic scale.
This, my dear future inventor
of the mortal space robots,
is logarithms, a beast
which at times transcends
even the fanciest of counting.
And as you plus 1 your way
towards your noble goal,
I only hope that you allow
yourself the occasional moment
to stop and smell the factors.
