---
title: '2.685 is (almost) everywhere - Numberphile'
source: 'https://youtube.com/watch?v=-mXaU3N9e8Y'
video_id: '-mXaU3N9e8Y'
date: 2026-08-22
duration_sec: 728
channel: 'Numberphile'
---

# 2.685 is (almost) everywhere - Numberphile

> Source: [2.685 is (almost) everywhere - Numberphile](https://youtube.com/watch?v=-mXaU3N9e8Y)

## Summary

In this Numberphile video, mathematician Ben Sparks demonstrates a surprising result involving irrational numbers and continued fractions. He shows that when you take the geometric mean of the continued fraction expansion of almost any irrational number, you always get the same constant, approximately 2.685, known as Kinchin's constant. The video explains the concepts of continued fractions and geometric means, then explores the fascinating implications and open questions surrounding this constant.

### Key Points

- **Picking an Irrational Number** [00:01] — The video begins with the host asking a guest to pick one of several irrational numbers, such as the cube root of two. The guest selects the cube root of two.
- **Defining Continued Fractions** [00:18] — The video explains that an irrational number can be represented as a continued fraction, which is an infinite expression of the form a0 + 1/(a1 + 1/(a2 + ...)). The numbers a0, a1, a2, ... are the terms of the continued fraction.
- **Defining Geometric Mean** [03:21] — The geometric mean of a set of numbers is defined as the nth root of the product of the numbers. For a list of numbers, you multiply them all together and take the nth root, where n is the count of numbers.
- **Geometric Mean of an Infinite Sequence** [04:36] — The video explores the idea of taking the geometric mean of an infinite sequence of numbers by considering the limit of the geometric means of the first n terms as n approaches infinity.
- **The Magic Trick Revealed** [05:30] — The host computes the geometric mean of the first 367 terms of the continued fraction of the cube root of two. The result is approximately 2.685, which matches the number he wrote down at the beginning of the video.
- **Kinchin's Constant** [06:38] — The surprising result is that for almost any irrational number, the geometric mean of its continued fraction terms will converge to a specific constant, approximately 2.685, known as Kinchin's constant.
- **Exceptions to the Rule** [07:18] — While 100% of real numbers yield Kinchin's constant, there are exceptions. Rational numbers, the golden ratio, and quadratic irrationals like the square root of 2 are all examples of numbers that do not converge to Kinchin's constant.
- **Open Questions** [09:56] — It is not known whether specific numbers, like pi or e, yield Kinchin's constant, as proving this for any particular number is an unsolved problem in mathematics.
- **Sponsor and Book Promotion** [10:23] — The video ends with a promotion for the sponsor, Jane Street, and a mention of Rich's new book, 'All About Numbers'.

### Conclusion

The video reveals a profound and counterintuitive result in number theory: the geometric mean of the continued fraction of almost any irrational number is always the same constant, Kinchin's constant. This constant is a deep and surprising link between different areas of mathematics, and many questions about it remain open.

## Transcript

>> Yes. Okay. Always. >> The first step is going to be I'm going to ask you to pick one of these irrational numbers. An irrational number is just a a number which when you write its decimal expansion, so you write it
its decimal expansion, so you write it out, the digits never stop and they carries on forever. So, some of these are famous numbers. You probably know Doesn't matter. We're not going to use what these numbers really mean apart
from the fact that they're irrational numbers. So, think of one. Don't tell me yet. Okay. Have you chosen one? >> Yep. >> Okay. So, I'm going to write something down here.
down here without knowing which number you've picked. Okay. So, we'll put that there and we'll come back to it and I'm not going to interfere with it. Right. you chose. >> I chose the first one there. the cube
>> Okay. Nice choice. Okay. So, we're gonna use the cube root of two as our starting use the cube root of two as our starting point. And unlike um most magic tricks, um we're going to use two mathematical tools to do something to it. See what
happens. And those two tools are firstly at its continued fraction representation. And then we're going to do something called look at its the geometric mean of that. So I'll explain
what both of those are. So the cube root of two is technical terms it's one and a of two is technical terms it's one and a bit right. So I'm going to rewrite the bit right. So I'm going to rewrite the bit as one over something. Okay let's
bit as one over something. Okay let's subtract one to give us just the bit. So that's now a number smaller than one. I'm going to write that bit as one over something. Okay. So on the calculator I can uh that
can uh that so one of the that the original bit is one over three and a new bit and then the new bit I'm to do the same thing. So we'll subtract three. So that's the new bit and then
we'll do I'm going to write that as one over something. and another bit. So subtract one.
Do one over that. And the idea is we'll keep going and keep going and keep going and keep going. Right? And by doing that we're going to get a sequence of numbers which are called the continued fraction representation of this um this number
the cube root two. And because writing all these one overs and pluses is a bit all these one overs and pluses is a bit is a bit messy, we'll just write this as to indicate that's the end of the integer part and then we just list these
numbers. The these numbers we got are one over the bits and then just look at the integer. So this carries on forever and that's actually an important part of a continued fraction. If we do it for an irrational number such as the one you
chose or any of the others you could have chosen, this will carry on forever. Right? So that's a continued fraction. So the next thing I'm going to think about is the geometric mean of some numbers. Okay. So if we had the ordinary
mean just that's that's the ordinary average. Okay. And the way we get the ordinary average of some numbers is we add them up and then we divide by the number there are. Okay. Now the geometric mean is similar except that
instead of adding we multiply. That's the idea. Okay. So we could do um let's do the first um let's do the first two of these. So the geometric mean of the
of these. So the geometric mean of the first two entries is 1 * 3. Okay. Now I don't want to divide by two because I've been multiplying. So the the corresponding thing is to take the square root. Okay. So that's the
geometric mean of the first two. And then we could take the geometric mean of the first three. So 1 * 3 * 1. And now we don't want the square root. So the root. Yeah.
>> And hopefully you can see where this is going. For the next one, we take the fourth root. I'm actually going to write it slightly differently. 1* 3 * 1 * 5. And instead of writing the root symbol, what we quite often do is write to the
power a quarter. It just means the same thing. It means the fourth root of that. And we keep going. So that's what it means to take the geometric mean of any sort of finite list of numbers. Hope hopefully that
list of numbers. Hope hopefully that makes sense. The um the question is does it make sense to do take the geometric meaning of an infinite list of numbers. Okay. And the answer is not necessarily. It might it might not work.
But what you can do and what sometimes works is if you work out the geometric three and then the first four and so on up to the first a million, the first a billion and so on. And those numbers might sort of home in on a value. they
closer to some value which it makes sense to think of as being the geometric mean of the whole infinite sequence. Okay. So what we're interested really is the geometric mean of the continued fraction the whole continued fraction
expansion. Okay. Right. And I've realized I haven't explained at all why answer is because it's part of a magic trick for now. Okay. Okay. So, um, what
sequence, the continued fraction expansion of the cube root of two, um, expansion of the cube root of two, um, but I've gone as far as 367 terms. Okay, so I've got an awful lot of terms. And what I'm going to do now is work out the
geometric mean on the computer of that long list. Okay. So if I'll type my geom long list. Okay. So if I'll type my geom mean
>> To three decimal places. >> Okay. Now have a look at the bit of >> Okay. Now have a look at the bit of paper I didn't touch from the start.
>> How did you know I was going to pick that one? It's amazing. How did I know you were going to pick the key ridd? I just thought, you know, that's the sort Well, obviously that's not the answer. I mean, a good magician doesn't reveal his
secrets, but I think we've established I'm not a good magician. So, uh, let's let's talk about the maths behind that. And the answer is whichever one you had picked, we'd get the same answer. And I think this is quite a surprising result,
which is that if you look at the geometric mean of the continued fraction of just about any irrational number you can think of, you always get the same
answer, 2.685 or thereabouts, right? Isn't that surprising? >> It is surprising. Does this is there name? >> Yes, this is Kinshin's constant. I think
it's a little little known but really interesting number um because it sort of interesting number um because it sort of underlies in some way so many other know about Kinchin's constant and what we don't know about it. What we know is
we don't know about it. What we know is that if you pick a number, a real that if you pick a number, a real number, so a decimal string at random, you have 100% probability that if you work this out, you'll get Kinch's
constant out of it. 100% probability. But when you've got infinitely many what you think it is, right? There can be exceptions, and we know plenty of exceptions. Um, so I mean, there's some obvious exceptions,
right? which is that if you happened to pick uh you know a whole number or a fraction then your you know your continued fraction would would just stop after a bit and um so you're definitely not going to get kin so you don't get
kinchin's constant for any rational number or fraction >> but the chances of picking a fraction are inconsequential when you're looking >> if you're picking at random the the chances of picking a fraction are I mean
literally zero but zero doesn't mean it's impossible it means it's somehow infinitely unlikely, right? And likewise, it's infinitely unlikely if which doesn't satisfy Kenchin's constant. And even irrational numbers,
not all of them do. So for example, um you might notice some some famous irrational numbers are not were not in my list of choices, right? So phi the golden ratio uh famously has continued fraction expansion with all ones.
you keep writing the wrong thing and so on. So of course when you work out the geometric mean of that you're not getting kinch's constant. Okay. And not getting kinch's constant. Okay. And likewise you don't get it with um so I
two. I didn't allow you to choose the square root of two because you don't get square root of two because you don't get it for any um quadratic irrational. So square roots of whole numbers don't give Kinchin's constant for basically the
same reason as this which is that the uh the the continued fraction gets loops round and also e doesn't e doesn't give it. So e is a you know a famous irrational number it doesn't give it um because because there's a sort of
underlying pattern to the continued fraction. So we know lots of numbers individually that don't give it. We know that 100% of numbers do give it. We number that does give it. >> Exactly.
>> Exactly. So those numbers, all the ones I you chose from at the start are ones where statistically people working it out. I mean, it seems to give it, but no definitely do >> because you can't go far enough down the
>> well, you can't calculate anything forever. But I mean, you might hope that would tell you you're definitely going to get Kinchin's constant out of it. to get Kinchin's constant out of it. But, um, so far as we uh so far as we
know so far, um, those pro any proof like that is really hard. No, no one's >> well, if we don't know for sure, that means we know we do know what the number is? >> Um, from an abstract argument, basically
can talk about that. >> For more on Kinchin's constant and Rich's new book, All About Numbers, check out the links below. And while you're down there, check out the link for our episode sponsor, Jane Street,
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digits of pi and you looked at the first million digits and you took the average. life. >> You have. You have? Yeah. Did you take >> No. >> Okay. I didn't I didn't add them