Transcendental numbers powered by Cantor’s infinities

by birtanpublished on August 25, 2020

welcome to another metal a Java do to make sense of transcendental numbers is considered to be very tough and no YouTube video or other attempt at popularization I know of even gets close to a really convincing argument where

Some particular number is transcendental let alone anything beyond that today's video is my best shot at giving an introduction to transcendental numbers that is both in depth and at the same time accessible to anybody with a bit of

Common sense this introduction will be powered by the remarkably simple constructions at the core of the theory of infinite sets developed by a great mathematician Georg Cantor now the experts among you will probably familiar

With some of these constructions but I'm pretty sure that even for you there will be a couple of pleasant plot twists that you won't see coming so stick around and please at the end let me know what worked for you and what didn't okay

Quick intro to transcendental numbers and why we care about them here we go when I talk about numbers in this video in the first instance I mean the real numbers and as usual I often represent the standard set of numbers by the

Number line to start with steady integers and the rational numbers the numbers that can be expressed as fractions of integers in fact for a long time people believed that all numbers are rational and it's actually quite

Natural to do so just think about it here the numbers that are multiples of 1/2 here the ones that are multiples of one force here the ones that are multiples of one eighths and so on these numbers get as narrowly spaced as you

Wish and so it's pretty easy to see that no matter how tiny an interval I pick on the number line there will always be infinitely many rational numbers contained in it and so to start with there is really no reason to suspect

That they exist irrational numbers numbers that cannot be written as fractions but of course you probably all know that root 2 and the golden ratio are irrational numbers these two irrational x' as well as the fractions

Are examples of algebraic numbers those are the numbers that come up naturally in algebra Solutions to polynomial equations with integer coefficients for example the fraction 22 over 7 is the solution to

This linear equation air and root 2 is one of the two solutions of this quadratic equation I just randomly picked the polynomial equation at the bottom there it's got one real solution which is approximately minus one point

Zero six one and so on the algebraic numbers have a number of amazing properties for example just like the real numbers themselves and irrational numbers they form an algebraic structure called field also all numbers without

Exception that can be written as radical expressions by combining integers using plus minus times divided and roots like these are algebraic on the other hand the algebraic numbers that cannot be written as such rooting expressions that

The algebraic numbers have these properties is really quite miraculous and not obvious at all definitely was one or two separate videos I think in any case a very natural question to ask is whether there are any numbers that

Are not algebraic in other words whether they are transcendental numbers numbers that transcend the world generated by algebra ok let's warm up with a truly ingenious construction of an irrational number a number that cannot be written

As a fraction using some of gr Cantor's ideas I start with this grid here which is made up of all possible pairs of integers from here we get all the fractions like this well apart from the real fractions we also get some nonsense

Fractions with zeros of the denominators now we'll do something which at first glance may seem impossible we'll make up a list that contains every rational number exactly once country's ingenious trick that makes

This possible is to simply walk along this spiral here all right so we go and note down each rational number the first term we come across it so start with zero divided by zero that's not a number so let's move on 0 divided by 1 that's 0

Which makes zero the first rational number on our list next comes 1/1 so 1 is the second number on our list forget about 1 divided by 0 next is minus 1 here and then there's 0 but we

Already listed there also we move on to minus 1 divided by what minus 1 that's 1 which we've also got already now skipping ahead the next number we have not seen yet is minus 1/2 okay then 1/2 then 2 and so on now let's write

Down all the rational numbers in decimals you all know that some numbers have two decimal expansions one ending in a tail of zeros and the other in a tail of nines for example the number 1 can also be written as 0.99 and so on

Actually it seems that the vast majority of people watching YouTube are not familiar with the fact that 1 is equal to 0.9 and 9 and so on and that the majority of those who are aware of this basic fact are really violently opposed

To it if you don't believe me or even if you do for a real eye-opener I recommend browsing through some of the 5,000 comments on my 1 is equal to 0.99 video aimed at primary school kids anyway whenever there's a choice between two

Different decimal expansions for one of the numbers on our list we'll use the decimal expansion that terminates in zeroes now highlight this infinite diagonal of digits here all right

Using this diagonal we can straight away write down a number that is irrational so we make a copy of the diagnose for this and change all the digits in the circles here we change all the zeros to ones and all the other digits to zeroes

The new number that we get this way differs from every number on our list in one position each for example it differs from the sixth number on the list in the six digits here they're different it differs from the fifths number in the

Fifth digit and so on and this means that the new number is really different from all the numbers we've listed right but we've listed all rational numbers and therefore the new number has to be

Irrational the fact that made this neat construction possible was that the rational numbers can be listed like this infinite sets of numbers that can be listed like this are called countably infinite as part of such an infinite

List we enumerate the elements of the countably infinite set with natural numbers this also shows that in a sense they are exactly as many natural numbers as there are elements in any countably infinite set just think about this for a

Moment a listing like this shows that there are as many rational numbers as natural numbers now if you see this for the first time don't you think that this is really amazing anyway if you give me any countably infinite

Listable set then I can make up a number outside this set using cantor diagonalization which is what our nifty construction method is often called yeah and the thing is and I'll show you in a second the algebraic numbers are just

Like the rational numbers countably infinite this means paratime we can use Cantor diagonalization to construct a number that is not algebraic a number that is transcendental a very important ingredient in listing the algebraic

Numbers is the fact that a quadratic equation can have at most two solutions and a degree 21 equation like the one at the bottom there can have at most 21 real solutions and so on okay here then is a quick sketch of how you can list

The algebraic numbers we've already listed the fractions that is the solutions to all linear equations with integer coefficients a B now don't get hung up too much on the details of my method of listing the algebraic numbers

What I do here is pretty arbitrary and there are lots and lots of different ways of listing the algebraic numbers in fact it's not very hard to make up your own method if your life should ever depend on it

You know might happen okay after we've listed the rational numbers will list the quadratic ax rationals and these are the solutions of quadratic equations with three integer coefficients a B and C now just like we

Systematically inspected a 2d grid here of all integer pairs from the inside out to cycle through all fractions we can use a 3d grid of all integer triples ABC to cycle through all possible quadratic equations okay for every quadratic

Equation that we come across in this way we add those among solutions to our new list there are neither fractions and therefore already part of the first list there nor quadratic Irrational's that we've already seen in this second

Process and in this way we list all the quadratic ax rationals including root 2 and the golden ratio for example ok now we repeat this process for all Kubica rationals and so on I will compact all this all together and now this 2d grid

Contains every algebraic number exactly once so now we've got a final nifty trick we string up this grid like this and simply list the algebraic numbers as we come across them while walking along the blue path as I said this is just one

Way to list algebraic numbers they are infinitely many others Cantor diagonalization translates the one I've chosen into a transcendental number that starts out like this now pinpointing this transcendental number is not the

End of the usefulness of contrast diagonalization trick we can also use it to figure out how many transcendental numbers they are again Cantor's diagonalization trick lets us find a real number outside any countably

Infinite set of numbers what about the real numbers themselves are the real numbers also accountably infinite set well obviously not because if they were countable diagonalization would produce a real number outside the set of real

Numbers well that's obviously not possible I would be really transcendent okay anyway it's not possible to construct a real number outside the real numbers and so since this is not possible this means that the

Real numbers cannot be listed that they form an uncountably infinite set well let's have this sink in for a second uncountably infinite that's obviously a larger infinity than countably infinite right okay

So the real numbers are uncountably infinite and the algebraic numbers are countably infinite what about the transcendental numbers are they countably infinite or uncountably infinite

Well since the transcendental numbers complement the countably infinite set of algebraic numbers to the uncountably infinite set of real numbers it should be uncountably infinite itself right and here's actually chance for you to show

This in the comments just based on whatever sets of us see how you go with this anyway let me show you something stronger I want to show you that a countably infinite set of numbers like the rational numbers or the algebraic

Numbers are vanishingly small when compared to an uncountably infinite set like the real numbers the transcendental numbers or the irrational numbers here's a nice argument that illustrates is in the case of the rational numbers okay so

What I'll do is I'll label the rational numbers with the natural numbers using our list there so the rational numbers 0 gets labeled with a 1 next one gets labeled with a 2 and so on take a segment of length 1/2 it and put it on

The number line centered at the number labeled one half the remaining interval and put it on the number line centered at the number labeled 2 and so on

So what we've done is to encase the rational numbers into a set of lengths one this means and this is made rigorous in a branch of maths called measure theory that the set of rationals has lengths or measure less than one but

Then the whole construction still works if we start out with any shorter segment and chop it up and now since we can make these segments arbitrarily small the conclusion is that the rationals have lengths or measure 0 so in a sense

They're not even there although you think they're there mmm well of course the same is true for any other countably infinite set of numbers like the algebraic numbers every countably infinite set of numbers has measured 0

Hmm zero lengths that's a pretty strange conclusion isn't it but when you think about it not any stranger than some of the other things we've seen so far like that the set of algebraic numbers is no larger than the set of natural numbers

What all this also implies is that if you restrict our attention to some finite interval and pick a random number inside this interval in a paradoxical but very precise sense you have a zero chance of picking any algebraic number

Or what's saying the same you can be pretty certain to have picked a transcendental number so what about numbers like PI or e well since almost all numbers are transcendental you would expect both PI and E to be

Transcendental and that's actually true but it turns out to be super tough to prove this and is really beyond what I can do in a video like this but just for fun here's such a proof so this is four pages of super dense mathematical pain

Okay now here I just have to mention that proving in 1882 that PI's transcendental resolved the problem that already puzzled the ancient Greeks by being transcendental implies that squaring the circle is impossible so

Given a circle it is impossible to construct a square of the same area just using ideal mathematics encompass anyway since even getting close to proofs for the transcendence of Pi II or any other number we are

Obsessing over is really out of reach what's the next best thing we can for okay well the dots in the transcendental number I made up for you are the same sort of dots as in 3.14159265 three five and so on these dots only make sense in

Context on the other hand it would be great to be able to display a decimal number with a simple pattern of digits and show that this number is transcendental libras constant discovered by the great French

Mathematician Joseph Louisville just preceding countries ideas fits the bill this number is really an ocean of zeros with isolated islands of ones at one factorials two factorials three factorials etc digits since this number

Features strings of zeros of arbitrary length it clearly does not have a repeating tail and is therefore an irrational number it is one of the earliest numbers that was proven to be transcendental and in textbooks

It is usually portrayed as the transcendental number whose transcendence is easiest to prove having said that I think the way to pinpoint a specific transcendental number and prove its transcendence that I showed you in

This video is definitely much easier to explain and understand than any of the proofs of the transcendence of liberals constant that I've seen having said that the original main aim of this video was to come up with a nice YouTube bubble

Proof for the transcendence of the Leroy constant I actually managed to put together such a proof but I really think it reserves its own video some time and so this is it for today hope you enjoyed this video as usual let me know in the

Comments what did and what did not work for you you

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