So I mentioned in passing in another video that if you throw a dart at the real number
line your chance of hitting a rational number is exactly zero.
And there's no tricks here, the dart is guaranteed to land on the line, and there are an infinite
number of rational numbers on the number line, just waiting to be hit.
Rational numbers flooding into every crevice between two other rational numbers, every
possible ratio of one integer to another, so many numbers, but yeah, the dart won't
hit any of those.
0%.
Really, actually, 0.
Think about them written out with all their digits.
There’s some nice simple numbers with a finite number of non-zero digits before it's
infinite zeroes.
Then there’s rational numbers with repeating digits.
Every rational number has a section that repeats infinitely, sometimes it's just a repeating
zero, sometimes a single digit, and sometimes many digits.
The repeating sections may get longer and farther as you get more and more between,
but if it can be represented by a fraction, it can be represented with a repeating decimal.
And you can make up any set of repeating digits, and it will be a rational number: the repeating
section over an equal amount of 9s. .1repeating is 1/9, .11repeating is the same thing, 11/99
or 111/999, it's all 1/9th, and of course 2 times .1repeating is .2repeating or 2/9,
.3repeating is 3/9 or 1/3.
12345678901234567890 repeating is 1234567890/9999999999.
.9repeating is 9/9.
A number that repeats infinitely has only finite information.
But when you don’t repeat, there’s just so many more possibilities.
Any infinite sequence of numbers can make a real number, and if it doesn’t repeat,
the number's irrational.
You could make a random irrational number between 3 and 4 by starting with 3 and rolling
a 10-sided die an infinite number of times for the digits.
What are your chances of getting exactly pi?
What if you wanted a rational number that ended in repeating digits, like 3+1/3?
How long could you expect to keep rolling 3s, on a fair die?
Doesn't matter how lucky you are; if you literally roll infinite times, the chance of all 3s
is zero, exactly zero.
You'll eventually roll something else, and then even if you go back to threes for a while,
you'll eventually roll something else again.
Same with any rational: imagine rolling a dice and getting the same repeating sequence
over and over every time you roll it.
How long would you realistically expect that to last?
Forever and ever through infinite rolls?
No chance.
Of course, the chance of getting all 3s is the same as the chance of getting any particular
number, and if the chance of getting any particular number is exactly zero then how can you hit
any number?
It calls into question the very process of how to pick a random real number, because
you can't actually define a real number through a series of random dice rolls, and math has
problems with randomness in general, and a single infinite random numbers would be infinite
information which doesn't fit in our universe...
But before we get to transcendentals, let's talk algebraic.
Algebraic numbers can by definition be defined with algebra. (which, spoiler alert, other
numbers can't.)
So for example every rational number can be defined as some finite combination of digits,
a division sign, and another finite combination of digits, and throw a minus sign into the
first symbol's mix, but you can also use other symbols to get irrational numbers.
Some irrational numbers can be defined very simply with just a couple algebraic symbols,
like sqrt2, or just a few algebraic symbols, like Phi the so-called golden ratio which
is exactly (1+sqrt 5)/2.
Then, some numbers can only be defined as part of an unsolved equation, here there's
no algebraic way to solve for x, you can only approximate.
But we don't care if we can solve for x as long as we can fully define it, no approximations.
Which, by the way, it's totally crazy that you can make up algebraic equations that algebra
can't solve, but also entirely inevitable, and a hint that there's much cooler math lurking
beyond algebra.
And you can totally list the number of possible combinations of algebraic symbols, starting
with combinations of 1 symbol and then listing all valid combinations of 2 symbols and then
3 and so on up until an arbitrarily large but finite number of symbols.
You'll have a lot of combinations that aren't valid or that are equal to earlier ones so
they don't need to go on the list.
But since you can list all possible algebraic numbers, they match up 1 to 1 with the counting
numbers, there's exactly as many, they have cardinality aleph null.
This also matches up 1 to 1 with trying to list all numbers in decimal by putting together
all combinations of digits, which is something a lot of people mentioned in response to Cantor's
Diagonal proof.
Does this process get you all the numbers?
Well, it gets you all possible finite combinations of digits, but nothing with infinite digits,
no irrational numbers, and not most of the rationals either, since most rational numbers
end in an infinitely repeating series.
It's not a one-to-one number correspondence unless you can make a function that tells
me exactly which unique entry on the list corresponds to pi, and which to 1/3, and which
to champernowne's constant... and the answer can't be "it gets to the irrationals at infinity"
because first that's not a whole number and second that's not one to one, so, yeah, probably
not a one to one number correspondence.
It's so simple to list all possible combinations of algebraic symbols, but this list will never
get to the numbers that would take an infinite amount of symbols to define, just as listing
all combinations of digits will never get to numbers with infinite digits.
These non-algebraic numbers are the trancendental numbers, they transcend algebra.
And even if there is some infinite way to define them, this list of all combinations
still never gets there.
Maybe there's a way to approach a given transcendental number with a pattern that makes sense, like
for pi and e, but for most numbers we have no idea how to define them or whether they
can be defined at all.
You hit a random real number with a dart and we don't even know how to look at the result,
how to differentiate it from all the ones around it.
Does the dart give results in decimal?
Because no finite amount of digits will tell you what this number is.
Does the dart report back in algebraic symbols?
Because there's a 0% chance that you've hit a number that can be defined with a finite
amount of algebraic symbols.
Somehow, out of the uncountably infinite set of transcendental numbers, 100% of real numbers,
most people only know Pi and e and maybe, like, champernowne's constant, and... what
else is transcendental?
Well, maybe one reason most people don't know more transcendental numbers is that nobody
knows very many transcendental numbers, the field of mathematics is a little slow on figuring
out how to prove whether numbers are transcendental or not, there's lots of open questions in
Transcendence Theory, like, we don't even know whether Euler's constant is irrational
much less transcendental-- super embarrassing... or maybe it's rational or maybe tomorrow someone
will come up with an algebraic equation for it or maybe someone will prove that you can't
prove whether or not you can prove it's transcendental.
I mean, that kind of thing happens.
Sometimes.
Anyway, I think most people's problem with Cantor's Diagonal Proof is actually a problem
with Real Numbers, with the idea that there are numbers that cannot be defined, that have
any arbitrary combination of infinite digits, infinite information, because yeah, that's
pretty weird.
Whoever decided to call them "real" numbers didn't think that one through.
Thanks a lot, Descartes.