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Famous Transcendental Numbers (wisc.edu)
72 points by rishabhd 10 months ago | hide | past | web | favorite | 28 comments

> Liouville's number 0.110001000000000000000001000 ... which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere.

Note that this is actually the binary representation of this number; in base 10 this number is approximately 0.77: https://en.m.wikipedia.org/wiki/Liouville_number#The_existen.... Also note that the “ellipses” in this case are not at all obvious; stating that this is the factorial sequence or adding more terms would make this much clearer.

No, that’s the binary representation of the binary Liouville’s number. It’s also the decimal representation of the decimal Liouville’s number, which is typically what people mean when they talk about Liouville’s number.

Yeah, how to continue the sequence 1,2,6,24 was not at all clear to me. I think it would have been obvious if they had continued with 120.

Chapernowne's number is a copyright violation, because it contains all possible numbers, some of which correspond to copyrighted works.

That is not a copyright violation. For work X that includes the same text as copyrighted work Y to be a violation of Y's copyright, the author of X must have copied from Y. If X's work was independently created without copying (directly or indirectly) from Y, it isn't a copyright violation.

Normally, a long run of exact matching text in X along with access to Y is sufficient to easily proof beyond a reasonable doubt such copying from Y, which is good enough to prove any kind of copyright violation, whether criminal or civil.

Chapernowne's number and other normal numbers [1] are among the rare cases where similarity plus access would not be sufficient to prove copying.

Now, if you combined a normal number with some kind of instructions to find and extract, say, the part that exactly matches the first Harry Potter novel, then you would probably have a copyright violation, but it would be the instructions that infringed, not the normal number itself.

BTW, those instructions would take up about as much space as the Harry Potter novel they tell you how to find. The offset to a given long meaningful string S in a normal number is essentially a random number about the same length as S. Sometimes people like to imagine using normal number offsets as a form of data compression, albeit not a practical one, but the offsets are big enough to spoil that.

[1] A "normal" number in base b is a number whose base b expansion contains every sequence of k base b digits with density 1/b^k.

Great explanation, but very minor nit: technically speaking a number is only normal if it's normal in every base.

Not just that, but it also contains every pedophile video/photo ever made or yet to be made. Also every instance of hate speech, false accusation, defamatory writing that ever existed and will exist.

It’s truly the worst number!

Conjecturally, every irrational algebraic number has this property, but I don't think it has been shown for any individually.

Absolutely false, http://mathworld.wolfram.com/LiouvillesConstant.html for instance contains only the digits 0 and 1 yet is not only irrational but transcendental even.

Also the second part of your comment isn't making sense, it has been shown for Champernowne's constant so how can you say it hasn't been shown for any individually?

I wrote "irrational algebraic", meaning irrational and not transcendental.

According to Wikipedia, no irrational algebraic number has been shown to be normal (normal is a slightly stronger condition; see last paragraph in https://en.wikipedia.org/wiki/Normal_number#Properties_and_e...).

Ah, I did not catch that. My apologies, this is actually a fascinating point.

even there is a filesystem powered by it ! https://github.com/philipl/pifs

Exercise: what is the first index within pi at which the pifs source code occurs?

Pi isn't proven to be absolutely normal so it might not be in there.

The number itself is not illegal. But maps that point to the specific digits to look are illegal.

It feels like every time there's a mathematics post on the front page I post a video by Matt Parker.

This time, here's a video that explains from the basics what kinds of numbers exist:


I was surprised when I learned that e + pi is not known to be transcendental. Everyone would be floored if it weren't, but it's still an open question.

It's not even known if e + pi is irrational.

It is known that at least one of e + pi and e pi is irrational, though. That's because e and pi are both transcendental, and so the polynomial (x-pi)(x-e) = x^2 - (e+pi)x + e pi cannot have all rational coefficients. If it did, e and pi would be roots of a polynomial with rational coefficients, and therefore not transcendental.

Seems like there should be an easy proof of that ...

I remember avidly reading Clifford Pickover’s books when I was in my late teens in the later half of the 1990s. Time: A Traveller’s Guide was the first I began with (after reading a brief endorsement by WIRED), later Keys To Infinity and so forth. An idiosyncratic collection of mathematical facts and musings.

I always loved Computers, Pattern, Chaos, and Beauty.

He's also keeping the 90s style alive with his whimsical Reality Carnival.


Champernowne constant[1] is the most interesting one for me. It is simple yet powerful at the same time.

1. https://en.wikipedia.org/wiki/Champernowne_constant

Not a mathematician, by a long shot, but does this begs the question...Is something "wrong" with decimal? (Would a numbering system e.g. like base-pi/base-tau or base-e make any sense?)

The most fundamental constants in mathematics are zero and one. The transcendentals are called such because they can't be expressed in terms of algebraic operations using these two constants as starting point. Changing the bases of a number system doesn't change this fact.

However, we can have alternative, more powerful formal systems that are able to capture well-known transcendental numbers in a finite amount of notation. The most powerful class of such systems are Turing-complete systems. However, even such systems won't capture all transcendentals, since there exists a subset of transcendentals called uncomputable numbers. Uncomputable numbers outnumber computable ones.

More powerful formalisms also have the downsides that they lose some "nice" properties such as ability to always prove equality of two numbers, as they gain power. Algebraic numbers and especially rational numbers are super nice and well-behaved in comparison.

Nothing in this article is based on the decimal system. To give a brief overview of the math involved here:

We start with the integers that, for the purposes of this comment, I will take to be god given. However, the integers are "missing" numbers in the sense that division of integers does not always give integers. For instance, the integers have 4/2 and 6/2, but are missing 5/2.

To accommodate this, we can invent the rational numbers, which extends the integers to provide closure under division [0].

As it turns out, the rationals are still missing numbers. For instance, there is no rational number that satisfies x^2 - 2 = 0, even though rational numbers can get arbitrarily close. To solve this, we introduced the algebraic numbers, which are the closure of the rational numbers under polynomial roots [1].

As it turns out, perhaps suprisingly, the algebraic numbers are still missing sum. For instance, the limit of the (1 + 1/n)^n as n -> infinity cannot be expressed as the root of a polynomial with rational coefficients. We define "transcendental" as the set of all numbers which are not algebraic.

You could easily define a base-e numbering system, which could, in principle, be useful for... something. A major downside is that you lose the ability to express most integers in a finite amount of digits, and doing arithmatic becomes highly non-trivial. Eg, we would have 1 + 2 = 10.0100112... (assuming I didn't make any mistake). It would probably be simpler to keep working in base-10, and instead explicitly talk about the coeficients of the infinite sum k_i * e^i

[0] Excluding division by 0. We could define something like the rationals such that this is defined. Actually we have and they are called wheels, but it generally creates more headaches than it solves.

[1] Eg, it contains all the roots of polynomials with rational coefficients. Technically speaking, the definition should say that it also contains the roots of all polynomials with algebraic coeficients, but it turns out we get this for free for having all the roots of polynomials with rational coefficients.

The only thing wrong with decimal from a mathematicians point of view is that an unnecessary (and unprincipled) choice of base is made. A more principled choice would be base 2, base 3 or base e. Base e makes sense because it is the most efficient base system under some metric [1]. Base 3 is the closest integer base to e and the most efficient integer base (although a more through analysis would compare base 2 and base 3 to see which one is more efficient because we can not just assume base 3 is better than base 2 because it is closer to e.) base 2 is the smallest base that really makes sense to use so that would be another option is and arguably more principled than base 10.

Most mathematicians are not concerned with which base the number system is in and are not really restricted in working in base 10. So in some sense almost all mathematicians don't really "use" decimal. In that regard, there are many possible interpretations to your question so probably many possible answers.

In other words, choosing to use base 10 instead of base 2 or 3 is like choosing to use python for your project because its easy, familiar and gets the job done rather than choosing a specialty language that would be "better" but harder to use.


> Astoundingly, at the end of the minute, there will be a quick-talking ant that will actually say the "last" digit of pi!

This is incorrect. There is no last ant in an infinite line of ants, just as there’s no last digit of pi.

Famous numbers that are believed to be transcendental, anyway.

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