Almost all numbers contain the digit “3” [video] | Hacker News Reader

Almost all numbers contain the digit “3” [video]

numberphile.com

65 pointsColinWright67 comments

I really dislike these numberphile videos as they deliberately construct their arguments such that the answer makes math seem like a magic trick and that math is truly confusing, which is the opposite of helping people gain an intuition around math.

The trick here is that they start with the intuition of counting, in which each number is itself a single thing, but are actually doing a calculation based on each number being a string of single digits.

It's much less shocking if you say: Picking a random 10 digit number is the same as randomly picking 10 single digits. The more digits you pick the more likely it is that you'll get a '3' somewhere in there. So as your string of numbers increases in length the less likely it is that you're string won't contain any given number. If I said to someone "The more dice you throw the more likely it is you'll get at least one 3", I don't think I would get anyone who was surprised by that.

The unfortunate thing is that math and number theory in particular are full of genuinely fascinating observations that don't rely at all on a tricks of phrasing to be revealed, and rather than making less math literate people feel 'dumb' (as these videos tend to do), spark in interest in exploring math further.

Did we watch different videos? Because what I saw went through an entire process to explain how the math isn't a magic trick and how things that seem completely paradoxical can actually have a quite simple explanation. He even does the leg work of showing empirical results to demonstrate the basic principle of the thing before jumping into a more formal, but thoroughly accessible proof without a heavy reliance on math jargon.

To me, if he is presenting math as magic at all, he's more like Penn & Teller than David Copperfield. He uses a hook to gain your attention, then explains the trick clearly to show that there are no devils, there are no demons, it's all just science.

I mean, really, what more do you want out of the guy? He's getting people interested in math. Let's try to at least be a little more constructive in our criticisms towards him.

I teach high school math and science. I had a group of students do a mini-project around the question, "Is there anything bigger than infinity?" The core of the project was students watching this numberphile video [0], and recreating his reasoning on their own.

That project helped them reason about higher-level math in a way they never thought themselves capable. The idea of "magic" never came up, but they certainly spoke about their sense of wonder at higher-level math. I really appreciate the way he presents mathematical concepts.

[0] - https://www.youtube.com/watch?v=elvOZm0d4H0

I think the quality of numberphile videos vary quite considerably. The one you linked is a rather honest explanation of the idea of comparing the size of sets through bijections and a good explanation of Cantors diagonal argument. As a mathematician I'm quite fine with it.

The one linked in the submission is more of an obvious statement hidden by obscure use of language. Slightly silly, but not very bad.

The worst offender with "magic" is the one with the Riemann zeta function [1], which went viral a while ago. The problem here is that they get people started off on the wrong foot, confusing them with wrong arguments and hidden definitions. Now, if people really want to understand why this can be made meaningful, you first have to explain that the better portion of the video is absolutely wrong, and only then can you explain what is actually going on.

[1] http://youtu.be/w-I6XTVZXww

I disagree entirely.

You're free to elaborate, if you want.

I disagree with the notion that Numberphile makes it seem like magic.

In my personal opinion they always explain quite carefully, using neat and concise diagrams. My old maths tutor used to have a similar teaching style, which worked very well.

The post editing even adds further explanation when the editor believes certain things haven't been explained clearly.

We could note that Proofs seem magical to most until you understand how to think property about them.

I remember my professor writing an equation, doing the proof, then exclaiming "This obviously reduces to this, QED!" and the entire class staring in disbelievement. It really did seem like she was inventing math as she went, and took a long while to really get a grasp of how to think about a proof when doing one.

It's true that he probably should have mentioned the simple intuition of it, but I think the reason he went about it the way he did is because that's the way you go about proving it mathematically.

This goes against generally accepted mathematical jargon (http://en.wikipedia.org/wiki/List_of_mathematical_jargon#Des...). For countable sets like integers, "almost all" has a meaning "all except a _finite_ set". Obviously the set of numbers that do _not_ contain 3 (or any other digit) is an infinite countable subset of all integers. Frequency has nothing to do with it.

okay, but what do you call it if the frequency is diverging to zero for infinity?

"Asymptotic distribution" or "density" probably. Check out this, for an example: http://en.wikipedia.org/wiki/Prime_number_theorem

I don't think I'm going to ding what is clearly intended as a general-audience presentation for failing to precisely conform to an uncited definition on Wikipedia, under a section with the word "Informality" in its name.

That turns out not to be the case:

    > almost all
    > A shorthand term for "all except for a set of measure zero",
    > when there is a measure to speak of.

In this case there is no measure to speak of, so it is normal to talk about taking the ration in a finite initial segment and then letting the size go to infinity and seeing what happens to the ratio.

And a set of measure zero does not have to be finite, either. Almost all numbers are transcendental in the sense given above, and yet there are infinitely many non-transcendental numbers.

So I think you are mistaken, although I would be interested to hear a more complete statement of what you mean, in case I have misunderstood you.

The same excerpt, starting from "One can also speak" describes how that "almost all" terminology is applied to integers (or, by extension, to any countable set, that is already measure zero). The example statement is "almost all prime numbers are odd", because there is a finite number of primes that are even (only one number 2).

Put another way, if we stick to the definition of "almost all" as elements not having the property forming measure zero set, then you do not need the video to prove the title: all integers are already measure zero. So, for subsets of integers, "almost all" means something else in jargon.

I could not watch the video, but I found (I believe) equivalent text. And "almost all" means that as we extend the reach the share of numbers not containing 3 goes to zero. It is mildly entertaining (albeit a bit too obvious) result, but my point was that usage of "almost all" is confusing because _I_ am used to a very specific meaning of it.

It's also a little confusing because there are sets which do not have a well-defined natural density among the integers, like:

     leastpow2 x = head $ dropWhile (\k -> 2*k <= x)  $ map (2^) [0..]
     p k = k < 3 * (leastpow2 k `div` 2)
     limitless = [k | k <- [0..], k == 0 || p k]

which oscillates between attaining 1/2 and then 2/3 for its partial densities.

> In this case there is no measure to speak of, so it is normal to talk about taking the ration in a finite initial segment and then letting the size go to infinity and seeing what happens to the ratio.

Indeed it is normal to talk about it, but this condition is usually called having "density 1" (https://en.wikipedia.org/wiki/Natural_density). I think that "almost all" is not terribly common in this connection, except possibly in a community where this is the only kind of nearly-true statement that one would want to discuss.

Number Theorists use a different definition of "almost all" from the one you see in measure theory. They use the limit version shown in the video.

Wikipedia actually supports his usage on a more definitive page http://en.wikipedia.org/wiki/Almost_all. Speaking as an ex-mathematician, I started laughing as soon as I read the title, because it pretty much tells you how to demonstrate it.

By the same token, most numbers are more than 8 digits long.

In fact, most are over a billion digits long. Although I've never see a single one in the wild...

5.73x10^1000000000

There you are then. Not that impressive, is it?

3 ↑↑ 4

Where ↑ is the knuth uparrow notation, beats your example in only 4 symbols with about three thousand times as many digits

I'll see you and raise you a © .

Which in my book is c, the speed of light in a vacuum, 299792458 m/s, a mere 9 digits, but inscribed in a circle using Steinhaus–Moser notation[1] is equivalent to a number so mindbogglingly big that I'm not even going to attempt to calculate its measure.[2]

[1]: http://en.wikipedia.org/wiki/Steinhaus%E2%80%93Moser_notatio...

[2]: Not that it makes any sense to perform this operation on c, but I think that use of © in this way is much more interesting than its current usage.

I admit defeat, using standard notation I can't think of a better ratio of numeric length vs symbolic representation. Well played!

Aaronson's got a blog entry tangentially related to this competition

http://www.scottaaronson.com/writings/bignumbers.html

"In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces "Eighty-three!" The second, mightily impressed, replies "You win.""

Thank you for a nice challenge, and thanks for the interesting link :-)

As someone who is genuinely interested in AI, I'm very curious if lateral thinking like we demonstrated can ever be matched by a machine. I'm watching things unfold with great interest.

You felt the need to say "Where ↑ is the knuth uparrow notation," so I would argue that those symbols (as well as all the symbols in an article describing Knuth up arrow notation) should also be counted.

"You felt the need to say "Where ↑ is the knuth uparrow notation,""

Mere pessimism on my part about UTF-8 handling. Its standard notation. After all, its Knuth, therefore its standard.

௸

Where ௸ is my own large number notation, beats your example in only 1 symbol with about 3 ↑↑ 4 times as many digits.

Most real numbers aren't even computable, although you almost certainly haven't ever dealt with a noncomputable number. The computable numbers are countable, and the proof of that is pretty trivial.

The same reasoning applies to every digit. So almost all numbers contain your favorite decimal digit.

If I showed this video to a friend who particularly enjoys the digit "6", she would not see the connection, as the video is about "3". No matter how many times I try to explain that there is nothing special about "6", she won't hear any of it.

I am quite convinced that the same attitude is rather widespread, and could be effectively monetized by anyone both charismatic enough and unethical enough to exploit it.

At the end of the video, he does mention that it works for any digit. A more honest way to present the argument would be to start with "let x be any decimal digit..." but I suspect most non-mathematicians wouldn't be able to follow that.

Serious mathematicians will feel his title was misleading or dishonest. Considering the proportion of mathematicians to the general public, the teacher is probably not targeting mathematicians - he's a mathematician targeting the mathematically curious. With that in mind, the title is perfect.

I don't agree. I think the real power of his videos is that, if he is demonstrating anything about mathematics, it's that mathematicians aren't all pedants who can't communicate with anyone outside of their field without accuracy.

Ah, but 6 is interesting! It's the smallest number that can be expressed as the product of two different prime numbers (2 x 3).

Of course, all natural numbers are interesting: http://en.wikipedia.org/wiki/Interesting_number_paradox

This video was posted on april 1st if no one has noticed.

I hadn't noticed, but it doesn't matter. The result is correct. Are you claiming otherwise? Are you claiming that it's false?

Almost all of the news published on April 1st is garbage.

I don't think this is actually true; regular news sites and newspapers still go on producing news, and it's only a small percentage that are parodical. I would bet that less than 10% of purported news articles on April 1st are fake.

Unless you are also claiming that most news published is garbage anyway, in which case I agree.

He was making a joke.

In real life data sets, Benfords Law shows that the digit '1' occurs with much higher probability.

This one is actually true and interesting for real. If you haven't heard of this before i highly recommend reading up about it.

When measuring random things in nature, regardless of what unit and base you use to measure (inches, cm, m, feet, etc), measurements with lower first digit appear more frequent than others! This is due to things in nature usually being exponentially distributed, when you have an exponential distribution you will "move away" faster from higher valued digits and reach the next power of ten faster, there you will restart with a low first digits which is slow to move away from. For example, assuming a spread of about 50%, then 1+50%=1.5 (still starting with 1) but 5+50%=7.5 (not starting with same digit), and 8+50%=12 (starting with 1 again).

For those of you who aren't familiar, Brady Haran, the creator of numberphile, produces a ton of awesome videos of experts talking about science and math. Besides numberphile, he does Sixty Symbols (physics), Periodic Videos (chem), and computerphile (comp sci) [0].

[0] http://www.bradyharan.com/

Almost all numbers contain the sequence 861523871623871623871263.

So in general, almost all numbers contain any digit? Great.

Exactly.

An interesting consequence (since there are only finitely many digits) is that, in this sense, almost all numbers contain every digit—a phrasing that, I think, makes the statement less surprising.

Contain -any series of digits-

Also, almost all numbers contain any given finite sequence of digits.

Did you intuitively understand that prior to watching the video?

I can't speak for him, but I haven't watched the video and this all seems extremely obvious to me.

In the end he mentions that it's equally true for the number 5. I guess that's because of the nature of infinity. All numbers would contain almost all other digits when infinity is involved.

Probably a better way to phrase that is almost all numbers contain all the digits.

Imagine a game like minecraft, but you only get 10 blocks. Now you will compose an infinite number of saved games using anywhere from 1 to infinity number of blocks and each time you place a block in each saved game, you pick a block randomly.

Some saved game will be a single block, and others will be infinitely vast. Larger than our real universe, containing every block. It's not a stretch of the imagination to see that the majority of the games will contain at least 1 of each of the ten block types.

It's clear that the same principle applies for every digit. But is the converse true? ie: Almost all numbers DON'T contain the digit "3"

Not unless we equivocate and redefine "almost all" to be something other than what he used in this video, right?

He showed that the ratio of "whole numbers containing 3" to "whole numbers" is close to 1. So the ratio of "whole numbers not containing three" to "whole numbers" is close to 0.

Almost all numbers contain all ten digits.

If you really want to blow your mind, almost all numbers contain all the digits 0-9. That's a lot more interesting and surprising to me than that they contain any one digit.

But the reality underlying this is actually really boring; when you start talking in terms of n -> infinity any fraction will become insignificant as it self multiplies.

Almost all integers contain the first ten trillion digits of pi, consecutively and in order.

And yes, it's really not that interesting. Infinity!

Another similar idea is the Kempner series.

If you add 1/n for all natural numbers n, then the series tends towards infinity.

However, if you leave out any integers that contain a digit 9, then the series converges (to just under 23). (It would also converge if you left out numbers containing a digit 3, just to a different limit.)

Here is a visualization of the number convergence:

(loop [i 1 t [10N] v [1.0]] (if (> i 300) (map (fn [[a b]] (/ a b)) (partition 2 (interleave v t))) (recur (inc i) (conj t (* 10 (last t))) (conj v (+ (* 9 (last v)) (Math/pow 10 i))))))

Sure, the proportion converges to 1 as the order of magnitude approaches infinity.

But this convergence is really really slow...

In an infinite universe, almost everyone is a smartass mathematician :)

It is an April Fools' video, nothing to see here.

So, just to be clear, you're claiming that the result is false?

It can be a April Fools video and still be correct. Basically it's a moot video because you can use the same logic to prove that any digit has the same properties or converse property (that three exists almost NO where because the probability of a number doesn't have three is P(n)=1-1/10^n and as n->∞ P(n)->0

  > ... because the probability of a number has
  > three is P(n)=1/10^n and as n->∞ P(n)->0

Er, no. The probability of an n digit number having a three is P(n) = 1 - (9/10)^n, and as n->∞, P(n)->1.

I don't understand why you say it's a moot video. The result as stated in the video is complete and correct.

I meant to say doesn't have 3 and have updated the comment to reflect the equation I believe is correct.

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