
Famous Transcendental Numbers - rishabhd
http://sprott.physics.wisc.edu/Pickover/trans.html
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saagarjha
> 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...](https://en.m.wikipedia.org/wiki/Liouville_number#The_existence_of_Liouville_numbers_\(Liouville's_constant\)).
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.

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xyzzyz
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.

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

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

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zamadatix
Absolutely false,
[http://mathworld.wolfram.com/LiouvillesConstant.html](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?

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kevinventullo
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...](https://en.wikipedia.org/wiki/Normal_number#Properties_and_examples)).

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

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mjlee
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:

[https://www.youtube.com/watch?v=5TkIe60y2GI](https://www.youtube.com/watch?v=5TkIe60y2GI)

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tgb
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.

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tzs
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.

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qubex
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.

~~~
weerd
I always loved _Computers, Pattern, Chaos, and Beauty_.

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

[http://sprott.physics.wisc.edu/pickover/pc/realitycarnival.h...](http://sprott.physics.wisc.edu/pickover/pc/realitycarnival.html)

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bibyte
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](https://en.wikipedia.org/wiki/Champernowne_constant)

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kylek
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?)

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GolDDranks
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.

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brianberns
> 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.

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joker3
Famous numbers that are believed to be transcendental, anyway.

