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• 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.

So I mentioned in passing in another video that if you throw a dart at the real number

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Transcendental Darts

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林宜悉 posted on 2020/03/30
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