Hello, welcome.
I'm Jake, and this is Every Number is Special.
Wow, it's only been a week since I started this show,
and it's been one of the craziest weeks I've ever seen.
The virus is here in Chicago and it seems like we're just 
waiting to see how many people are gonna get sick,
and we're waiting to see what other sorts of
consequences that there are going to be from all this.
There's a tremendous amount of uncertainty right now
and that's really scary.
I remember when I was a kid,
sometimes when I got scared I would try to keep my 
mind occupied by counting.
So, I guess there's still a bit of that here.
I guess Every Number is Special is my attempt as an
adult to self-soothe by counting.
So, I think it's really important right now to at least 
try to think about something nice. 
For me, that's numbers and if this soothes anyone
out there, I'm really thankful for that.
Let me know if that's you, 
I'd really love to hear from you.
Okay, Episode Two
The lovely number two
All right, so if I'm gonna mention things that make
two special, I have to mention prime numbers.
So, let me just remind you,
A prime number is a natural number that is
divisible by exactly two natural numbers.
So, for example
One is not prime,
because it's only divisible by 1 natural number, itself.
And 2 is prime because it's divisible by 2 and 1 and
no other natural numbers, 
so it's divisible by 2 natural numbers.
I want to give a shout out to the concept of a bit.
A bit is a unit of information expressed as 0 or 1.
So, you have two options for what a bit could be
And somehow everything in this video is being
encoded as bits and then decoded
back as the pictures and sounds that are showing up
in front of you. So, I'm a mathematician not a computer
scientist, so I don't claim to know how
this works.
But, it is sort of mind-blowing that
 you only need two symbols to do all of this, 
to get everything you see from my desk to your screen.
I think that's *pretty* special.
I also wanted to mention the concept of a dichotomy.
A dichotomy is when you have 
two mutually exclusive cases
So, for example.
good and bad, yes and no, heads and tails,
true and false.
There's so many different dichotomies and they're
important not just in math, but in life.
So, I'm not going to say too much more about that.
Now, in category theory
and specifically in the category of sets
the two element set that contains 0 and 1 is
what's called a sub-object classifier.
So, that has something to do with the fact that
If I have a set, like this is a set that contains 5 elements, 
then subsets of this set are in correspondence with
functions from this set into a 2 element set.
So, to hopefully make that a little more clear, 
I want to pick out some subset of this set,
So, how about these 3 points?
Making that choice is the same as defining a function
from this set into the 2-element set.
So, I'm going to say that the
elements that didn't make it in are gonna get sent to 0.
And the elements that did make it in
are gonna get sent to 1.
And, if you're familiar with what I'm saying, 
you know then the function that I'm talking about is the
characteristic function of that subset.
There's one last concept I want to mention
that is related to the number 2.
And it's the concept of duality.
So, duality doesn't really refer to any one thing.
It's a phenomenon that shows up all over the place.
And it really just refers to when you've got two things,
and they're related in some way that you 
think is interesting.
So, here's an example of that.
I've got two statements and the first statement is:
Any two points determine a line.
Okay, and the second statement is:
any two lines determine a point.
These statements are related,
because I can get one from the other by
swapping point with line.
So, the first statement is true.
This is a fact about points and lines.
If I draw two points, they're going to determine a line.
There's exactly one line that goes through them.
The second statement
It might kind of look true. Here's two lines
And, they do determine this point in
their intersection.
But, it's a bit more dependent on the context.
Like, here I drew the lines in a plane, 
but if I drew them in three-dimensional space,
maybe they wouldn't intersect.
Or, if I happen to draw them parallel they
wouldn't intersect.
So.
It's these two statements that are dual to one another.
And one of them is true.
One of them is almost true,
And so, it's worth asking, are there contexts where
a statement and its dual are true?
Before I sign off, I want to remind you that I do online
math tutoring, so, if you are looking for a tutor,
please check out my schedule. 
You can find it at calendly.com/chimath
C - H - I - M - A - T - H
Also, if you want to support the show,
I want to make that easy for you, so I set up these links.
There's a Patreon link and a PayPal link.
Check 'em out if you'd like. 
And please, take care.
I'll be back for Every Number is Special, Episode 3.
Bye!
