All the Numbers - Numberphile

Summary

In this video, Matt Parker explores the vast world of numbers beyond just whole and rational numbers. He discusses different types of numbers including constructible, algebraic, transcendental, computable, and uncomputable numbers. Parker highlights how mathematicians categorize these numbers using Euler diagrams and shares fascinating insights about normal numbers, emphasizing how most numbers are both uncomputable and normal. Despite this, only artificially created normal numbers are known. The video concludes with a mention of Parker's new book, 'Humble Pi,' and its content related to mathematical errors.

Takeaways

• 🔢 The video discusses the categorization of numbers, starting with whole numbers and expanding to rational, algebraic, and transcendental numbers.
• 📈 The script explains that rational numbers include whole numbers as a subset and are any number that can be expressed as a ratio of two integers.
• 📐 Constructable numbers are a subset of algebraic numbers, which can be created using only a compass and straightedge, like the golden ratio (Phi).
• 📘 Algebraic numbers are solutions to algebraic equations and can include both constructable numbers and those that require higher roots or powers.
• 🌐 Transcendental numbers are not solutions to any algebraic equation and include famous constants like pi and e.
• 🤖 The concept of computable numbers is introduced, which are numbers for which an algorithm can be written to calculate their digits, such as pi and e.
• 📚 The script touches on the history of proving numbers as transcendental, with e being the first proven in 1873 and π in 1882.
• 🔍 It is mentioned that there are numbers that are known to exist but are difficult to define or understand, referred to as 'dark numbers'.
• 🎲 The video introduces the idea of normal numbers, which contain every possible digit combination with equal probability, and that Champernowne's constant is an example of a proven normal number.
• 🧩 The script highlights the vastness of numbers, indicating that most numbers are both normal and uncomputable, existing beyond our current mathematical comprehension.
• 📖 Matt Parker, a significant contributor to Numberphile, has a new book titled 'Humble Pi: A Comedy of Maths Errors', which is promoted at the end of the script.

Q & A

What are whole numbers?

Whole numbers are the set of non-negative integers starting from 0 and going up (e.g., 0, 1, 2, 3, ...). They are a subset of rational numbers.

What is the difference between rational and irrational numbers?

Rational numbers are those that can be expressed as a fraction of two integers, where the denominator is not zero. Irrational numbers, on the other hand, cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions.

What are constructable numbers?

Constructable numbers are those that can be constructed using only a straightedge and a compass. They are a subset of algebraic numbers.

What are algebraic numbers?

Algebraic numbers are the solutions to algebraic equations, which are equations of the form ax^n + bx^{n-1} + ... + k = 0 where a, b, ..., k are integers.

What are transcendental numbers?

Transcendental numbers are numbers that are not algebraic, meaning they cannot be the solution to any algebraic equation with integer coefficients. Examples include π and e.

What is the significance of e being the first number proven to be transcendental?

The proof that e is transcendental, accomplished by Charles Hermite in 1873, was a significant milestone in mathematics as it was the first time a well-known number was rigorously shown to be transcendental.

What are computable numbers?

Computable numbers are those for which there exists an algorithm or a finite process that can determine all their digits. This includes numbers like π and e, which can be approximated to any desired degree of accuracy.

What is the concept of normal numbers?

Normal numbers are those in which every possible finite sequence of digits appears with the correct limit frequency in their infinite decimal expansion. In other words, they are numbers that are uniformly distributed in the context of their digits.

What is Champernowne's constant and why is it significant?

Champernowne's constant is a number formed by concatenating all natural numbers in sequence (0.123456789101112...). It is significant because it has been proven to be a normal number and is also transcendental.

What is the difference between computable and uncomputable numbers?

Computable numbers are those for which there exists a finite, systematic method to determine their digits. Uncomputable numbers, on the other hand, cannot be computed by any algorithm or finite process, and their digits cannot be systematically determined.

What is the connection between Turing and the concept of computable numbers?

Alan Turing, in his 1936 paper on computable numbers, introduced the concept of a Turing machine, which is considered a foundational work in the field of theoretical computer science. His work on computable numbers explored the limits of computation and identified the existence of uncomputable numbers.

Outlines

00:00

🔢 Exploring the World of Numbers on Numberphile

The script begins with a discussion on the variety of numbers explored on the Numberphile channel, starting with whole numbers and moving to rational numbers, which include fractions like seventeenths and twelfths. The speaker humorously regrets not including negative numbers in their diagram, which is meant to represent different sets of numbers in an Euler diagram format. The conversation expands to include constructable numbers, algebraic numbers, and transcendental numbers, with examples like pi and e. The script touches on the historical proofs of transcendental numbers and the vastness of numbers we don't yet understand, ending with a teaser about exploring 'weird reals' beyond what Numberphile has covered.

05:03

🤖 The Turing Test of Computable Numbers

This paragraph delves into the concept of computable numbers, introduced by Alan Turing in 1936, and distinguishes them from non-computable or 'dark numbers.' The speaker explains that computable numbers like pi can be derived from a finite set of rules, whereas non-computable numbers require the knowledge of every single digit with no general formula. The script mentions Chaitin's constant, an example of a non-computable number related to the halting probability of computer programs. It also introduces the idea of 'normal numbers,' which contain every possible digit combination with equal likelihood, and Champernowne's constant, a proven normal number constructed by concatenating natural numbers.

10:05

📚 Reflections on the Limits of Mathematical Knowledge

The final paragraph reflects on the limits of our mathematical knowledge, highlighting the emptiness of the section for uncomputable normal numbers, which are currently unknown. It emphasizes that all known normal numbers are artificially created with a computable rule. The speaker also mentions that while mathematicians have made progress, the vast majority of numbers remain a mystery, with only a few examples of normal numbers known. The script concludes with a casual mention of Matt Parker's new book 'Humble Pi' and a plug for the Numberphile podcast, suggesting a light-hearted and community-driven approach to sharing mathematical knowledge.

Keywords

💡Whole Numbers

Whole numbers are the set of numbers without fractions or decimals; they include 0, 1, 2, 3, etc. In the video, whole numbers are discussed as the basic and classic category of numbers that were initially covered on Numberphile.

💡Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. The video explains that rational numbers include whole numbers and are represented as points within a Venn (Euler) diagram.

💡Algebraic Numbers

Algebraic numbers are numbers that are roots of non-zero polynomial equations with rational coefficients. The video discusses algebraic numbers in the context of being a step beyond constructable numbers and includes solutions like square roots and cube roots.

💡Transcendental Numbers

Transcendental numbers are numbers that are not solutions of any non-zero polynomial equation with rational coefficients. Examples given in the video include π and e, and these numbers are placed outside the algebraic numbers in the diagram.

💡Constructable Numbers

Constructable numbers are numbers that can be constructed using a finite number of steps involving a compass and straightedge. The video highlights the golden ratio (Phi) and square root of 2 as examples, and places these within the rational numbers.

💡Computable Numbers

Computable numbers are numbers for which there exists a finite algorithm that can generate their digits. The video mentions that numbers like π and e are computable and introduces the concept as described by Alan Turing in his 1936 paper.

💡Normal Numbers

Normal numbers are numbers in which all possible sequences of digits occur with equal frequency. The video explains that while many numbers are believed to be normal, such as π and e, this hasn't been proven. Champernowne's constant is given as an example of a normal number.

💡Uncomputable Numbers

Uncomputable numbers are numbers for which no finite algorithm can produce their digits. The video describes these numbers as being in the vast majority of the number spectrum and highlights the difficulty in understanding them. The Chaitin constant is provided as an example.

💡Venn Diagram

A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. The video refers to the use of a Venn (technically Euler) diagram to illustrate the relationships among various sets of numbers.

💡Champernowne's Constant

Champernowne's constant is a specific normal number constructed by concatenating the positive integers in order. The video highlights this number as an example of a normal number and explains how it was intentionally constructed to be normal.

Highlights

Numberphile has covered a variety of numbers including whole numbers, rational numbers, and more.

Whole numbers are a subset of rational numbers, which include ratios like seventeenths or twelfths.

The distinction between Euler diagrams and Venn diagrams is clarified, with the former used in the discussion.

The concept of 'constructable numbers' is introduced, which includes numbers that can be constructed with a compass, ruler, and pencil.

Algebraic numbers are defined as solutions to algebraic equations, while transcendental numbers do not fit this category.

Transcendental numbers such as pi and e are highlighted, with historical context on when they were proven to be transcendental.

The idea of 'dark numbers' is introduced, referring to numbers that exist but are hard to grasp or define.

Computable numbers are those for which a finite algorithm can be written to determine their digits, such as pi and e.

The Turing machine and Alan Turing's work on computable numbers is mentioned, highlighting the existence of numbers that are not computable.

Chaitin's constant is introduced as an example of an uncomputable number, related to the probability of a random program halting.

The concept of 'normal numbers' is discussed, which are numbers where every possible digit grouping is equally likely.

Champernowne's constant is presented as a proven normal number, constructed by concatenating natural numbers.

The Copeland-Erdős constant is mentioned as another example of a normal number, constructed from prime numbers.

The current inability to test if a number is normal is highlighted, with all known normal numbers being artificially generated.

A theoretical section of the diagram is noted to be empty, referring to the lack of known uncomputable normal numbers.

The assertion that most numbers are normal and most are uncomputable is made, emphasizing the vastness of the unknown in mathematics.

Matt Parker's new book 'Humble Pi' is promoted, with a mention of its content and a special offer for Numberphile viewers.

Transcripts

We're gonna do all the numbers. On Numberphile

we've done a lot of the numbers, some would say, but we haven't done all the numbers. Originally

we did the whole numbers. And these are the classics, my goodness. We've done 11, that was an early one.

We've done 3435, seventeen - right, all the whole numbers sit in here.

But then there are other types of numbers. If we go one step out, the rational numbers the ones that are ratios,

these are - you know, you get a seventeenth, you get I don't know things over twelve, you get all sorts of...

So now you've got all the rational numbers. We've gone beyond that though. And the rationals technically include the whole numbers,

they're a subset, but I'm doing this as what people call Venn diagram, which is wrong,

it's an Euler diagram. Because I'm not showing every single possible combination. - (Brady: Are negative numbers whole numbers?)

You know, I

put in a twelfth because I thought I was being hilarious and then I immediately thought, urgh I wasn't gonna put negatives on this diagram.

So I regret that, for two reasons. Opening the negative can and the expression on your face.

So I'm gonna make that a plus, there we are. In fact, this whole sheet of paper is just gonna be the

reals.

Positive re- you know, it works for negative reals, what am I saying? Have the negatives, it's fine.

But the sheet is - are all the reals and

inside here I've put whole numbers and then I've put rational numbers. If you - you can obviously get complex numbers coming up,

we're not gonna do that,

we're gonna stay down here. And I'm gonna gradually work our way out until we get a greater distance out

than Numberphile has ever gone before, right? We're going for the new Numberphile record: how far out into the weird reals

can we go? But we've done rational. Next one up the constructables. And these often aren't mentioned,

you don't have to add this in as a category, but I quite like constructables.

More importantly what people tend to go for, the next one out are the algebraic. Okay

so Simon did a fantastic video about algebraic numbers, and when you go outside,

transcendental numbers.

"I mean this number is really, really important and no one knew." - And so a lot of the number categories refer to in or out

of these different sets. So rational numbers are everything inside the blue line,

irrational numbers are everything outside the blue line.

You've got constructable numbers are anything inside the purple line, unconstructable numbers are outside this. And this light blue line out here:

algebraic numbers are everything inside there, and

transcendental numbers are everything

outside of there. And so constructable numbers are things that you can construct with a pencil and a compass and a ruler.

So Phi, you can do that, the golden ratio because you can do root 5, so you can get Phi.

You can do root 2, that lives in here,

that's kind of fun.

Algebraic numbers are the solution to an algebraic equation. If it's a square root or lower you can put it in constructable,

you can draw it. If it's higher than that

you can't. So the cube root of 2 is an algebraic number. And then outside algebraic you've got...beyond that, right?

So you've got things like pi. Pi lives out here, pi is transcendental. e is

transcendental. The natural log of 2, that's out there. Things that aren't a nice, neat solution to an algebraic equation.

And they're out there, right? This is how people tend to categorise all the numbers.

This is the fringe of, kind of, what we understand in mathematics and what we've done on Numberphile.

So e was the first number that was proven to be transcendental. And so that was proven in

1873 - so reasonably recent, given that some of these are thousands of years old. We knew e existed but we didn't know where it went.

Pi, we didn't know where that went. Pi was proven to be out here in

1882. So e to the power of pi was proven to be out here in only

1934, so that was a more recent one we managed to prove is out there. And there are loads

which we don't know. Pi to the e: we don't know. e to the e: we don't know. Pi to the pi:

we don't know, right? These are all on the cusp. We know that one of e times pi or

e

plus pi; one or more of those are transcendental,

we don't know which. Or both, right? But we know at least one of them is. Most numbers

that we know are algebraic sit around here, and we don't know if they're definitely

transcendental or if they're still algebraic. For the most part

we haven't got a clue, right. If you look at the entry for transcendental numbers on Wikipedia or MathWorld,

there's a list. Here's the only ones we know and that's it,

right. Some of our favourite big ol' numbers: Graham's number - in here. Googolplex - in here, right.

Whole numbers, doesn't matter how big it is, it's in there.

There are infinitely many whole numbers but there are also infinitely many rational numbers, same infinity.

There are infinitely many constructable numbers, infinitely many algebraic numbers;

but all countable infinity, is the smallest infinity possib-ly many of

these numbers in here. And there is one more circuit out. You know what? Should we chuck on the last one?

I can add another loop and it cleans up all of these and it puts them all in a neat bow.

This is the collection of

computable numbers.

Computable numbers? What's that- well this means we can compute them. So we can compute e,

we can compute pi, right?

It's not the solution to a nice equation, but I can write down a system by which you will get the decimal places,

right? And so we do this, we print them out on a very long bit of paper, roll 'em out on a runway,

it's hilarious, right?

"We're here on a runway because, for some reason, Brady has printed out the first 1 million digits of pi."

So this came down to Alan Turing in

1936. And everyone remembers that Turing invented the computer, with the Turing machine and that was in his paper on computable numbers.

He was looking at if you can compute all numbers, and he showed there are numbers that exist out here

but we just don't know what they are. I call them the dark numbers,

right, all these numbers that we know they exist - Turing showed us,

but they're, they're so hard to grasp. And occasionally we see them but not often.

Well hang on, why isn't pi out here? We can compute pi. Well, it takes forever,

like I could write down a

infinite series which gives you pi. And I can give you the rules for writing out the infinite series, and I could write them on

a postcard, or some finite amount of space and go: here are the rules for calculating pi.

You're going to have to do them forever, but the rules are finite. So for everything else in here

I can write a description of how to get all the digits, out here

they're only definable by writing out all the digits. And no - and that's actually most numbers. In here,

this is the countable infinity land right? There's, there's, there's infinitely many of these, but the smallest

infinitely many. Out here is a bigger infinitely many. So the vast majority,

for the strongest definition of vast majority you can come up with of numbers, are not computable. So we in this nice little

island of numbers that make sense, and then outside is this vast, vast world of all the reals

which are only definable by writing out their digits. And we've spotted a few! So there's one called the Chai-lin?

constant? I'll have to double check I got that right. Chaitin. Vaguely speaking, it's the probability,

for a certain way of writing a computer program,

if you generate a computer program at random, if it will run and come to a stop.

Right, and that probability is a naive way of describing it, and it depends on how you write a program.

So, in fact, there are lots of these constants, but we know they're all out here. The only way to get them is to

work out every single digit individually, there's no equation, no algorithm that spits it out. It's an uncomputable number. And they're so

mysterious and hard to understand at fringes of our comprehension of numbers,

but they're out there and the scary thing is most numbers are out there.

(Brady: How can we even know one of them? It seem- it feels like an unknowable unknown.)

It is insane that we even know a couple of them because when I described the numbers out here,

I'm super hand-wavy because I don't understand them, right. I've read - I like I've tried. I read the paper,

I'm like, man,

this is beyond me. Like, the maths to try and grapple with the numbers out here is insane.

But what's incredible is we have done it, right?

So if you look up uncomputable numbers, there are a few examples of ones out here. Although,

interestingly,

when you get these weird numbers you can, you can just - you can kind of make artificial ones.

Okay, so I'm now gonna kind of ruin my lovely neat diagram by putting on a whole new category.

This is the category of what are called normal numbers, which is a bit of a silly name.

It just means that every possible

sub-grouping of digits is equally likely to be in there. And a lot of people say like, for example pi. Everyone goes: your phone

number's somewhere in pi, your name is somewhere in pi, the complete works of William Shakespeare if tuned into digits are somewhere in pi.

We don't know that. We could move pi into here- - (Brady: Which we may yet do?)

We may yet do! It may be in here, everyone seems to think it is. All the digits we've checked imply

it's in here, but we not yet managed to prove that. We don't know if pi is in here.

We don't know if e is in here,

we don't know if root 2's in here. Even though for all of these, if you look through the digits,

you can find any string of digits you want. I found my name in all of them,

right, because they've all got a lot of digits that are

suitably random, and you can find sub strings in there. But we've not managed to prove any of those are normal numbers.

Would you like to see one number we have managed to prove? So this, I love this number. It's called Champernowne's constant.

Is normal. It's one of the few numbers we know is normal.

(Brady: Con-stat.) - Consta- that's an n! Look at it, it's just climbing under the a. - (Ah yeah)

And so Champernowne's constant is one of the few numbers we know is normal, and it goes zero point one two

Three - I've memorised it - four five six seven eight nine,

well, 10, one zero, 11, 12 - and it's just all

the whole numbers. 14,

15, 16 and so on. - (Brady: That's cheati-)

one eight, one nine, two zero

I've got them all,

and so on.

(Brady: That's like brute-forcing the problem)

It really is! It really is. I call this an artificial number because

Champernowne just went: can I find a number which is normal? And he came up with this procedure.

He just went, right you just put all the numbers in order, all the integers in base 10,

and you done. And it's true,

right? Because whatever you want to find - it's in there eventually, because it's just all the whole numbers listed out and then

turned into digits.

But that's all numbers are right? Whole bunch of digits, perfectly valid number, right? But - and it is computable, right.

So actually - - (Brady: It's like the least efficient way of doing it.) - It is. There's a slightly more efficient one. The Copeland-Erdős

number is the same idea but only the primes. So it's 2, 3,

5, 7, 1 1, 1 3, so on - 1 7 etc. So exactly the same idea. Slightly more efficient,

and it's also normal. So we didn't start with these numbers and go

I wonder if they're normal? These people sat down and went: I'm gonna make a normal number. How can I do that?

And they generated these. Now, Champernowne's number is transcendental. So this is it in base 10, obviously

you can do it in different bases, and it is computable. It is normal and it is transcendental.

So transcendental means it's outside algebraic, it's there.

And not the speed of light - Champernowne's constant sits in this part of the diagram.

So the only normal numbers we know for certain are artificial ones that are being constructed for that purpose.

We have never started with a number and then discovered it is normal,

we have no test. There is no process for taking a number and proving

it's normal. Like, mathematics - like hopefully one day we'll have a test,

currently we don't. Which means all the normal numbers we have are artificially generated,

we are yet to take anything from here and show that it's allowed to move over this line.

(Brady: There's an elephant on your diagram there.) - Is it over here? - (Yeah) - Well, very interesting

you should say that, Brady. This section is empty. And this is the only

properly empty section of the

diagram. Now up here we had that one number whose name I couldn't remember properly- - Chaitin.

All - this set of numbers deal with whether or not a program will halt, we've got numbers in this category, right?

So this section we've got numbers. This is

completely empty. And that's because the only normal numbers we know of are ones that we made for that purpose.

And the fact that we made them for that purpose means we have a rule for

generating them, which means they must be computable. To have an uncomputable

normal number would be incredible. But this is currently empty, but we have managed to prove one thing.

We've managed to prove that most numbers are normal, and most numbers are uncomputable.

So actually this is the biggest section. This is numbers, right?

This is a trivial blip, right, in the world, not- this is where numbers are right?

And we had none! So in the main category of numbers, where all the numbers are,

apart from a few trivial side-effects, right? We know zero of them. You know, as mathematicians

we think we're getting somewhere, but up until now we have found none of the numbers.

Hi everyone, this isn't a formal sponsorship,

it's just to let you know that Matt, who you just watched, has a new book out that I think you're like:

Humble Pi. That's what it looks like. "A comedy of maths errors".

That's a great cover.

If you'd like to be among the first

people to get your hands on it - I was actually the first person to be given one just quietly - then go to the website

in the video description, it's mathsgear, that's Matt's website. And by buying it from there not only will you get one of the first

hardcopy editions, you'll get a copy signed by Matt. As you know

Matt's been a huge friend to Numberphile over the years and just one of the small ways we can show our appreciation

is to check out his book. Now I know what you're all thinking, if this is about

mathematical errors, does it contain the parker square? Now, I don't want to give away anything,

I know people don't like spoilers, so

yes, it does! And if you can't get enough Matt Parker in your life

he is also a guest in the most recent episode of the Numberphile podcast. Have you seen the Numberphile podcast yet?

Have you listened to it? Check it out. I'll also put links down below

Chaitin.