The ants argument feels rather like a retelling of Zeno's Paradoxes.

https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

> 1: Almost all numbers are transcendental.

Even crazier than that: almost all numbers cannot be defined with any finite expression.

By common definition of "almost all", 1 == 2

how can i pick a real number at random though?

i tried Math.random(), but that gave a rational number. i'm very lucky i guess?

> Essentially, it means the number cannot be constructed using a finite combination of integers and standard algebraic operations (addition, subtraction, multiplication, division, and integer roots)

Slight clarification, but standard operations are not sufficient to construct all algebraic numbers. Once you get to 5th degree polynomials, there is no guarantee that their roots can be found through standard operations.

Both Euler's and Catalan's list "(Not proven to be transcendental, but generally believed to be by mathematicians.)". Maybe updated after your comment?

Yes. A number is transcendental if it's not the root of a polynomial with integer coefficients; that's completely independent of how you represent it.

The notion of transcendental is not related to how we right numbers. However, in abstract algebra, we generalize the notion of algebraic/transental to arbitrary fields. In such a framework, a number is only transental relative to a particular field.

For instance, the standard statement that pi us transcendental would become the pi is transcendental in Q (the rational numbers). However, pi is trivially not transcendental over Q(pi), which is the smallest field possible after adding pi to the rational numbers. A more interesting question is if e is transcendental over Q(pi); as far as I am aware that is still an open problem.

I think the elements of the base need to be enumerable (proof needed but it feels natural), and transcendental numbers are not enumerable (proof also needed).

To prove something is transcendental we would need to know how to compute it exactly, and I’m struggling to see how that would come up frequently in a physics context. In physics most constants are not arbitrary real numbers derived from a formula, they’re a measured relationship, which sort of inherently can’t be proved to be transcendental

Yeah I'd expect Bessel function zeroes and such

some constants that may or may not be transcendental: - Percolation Thresholds: https://en.wikipedia.org/wiki/Percolation_threshold - Critical scalings in 3d: https://en.wikipedia.org/wiki/Universality_class#Ising_model

As far I know, Euler's constant hasn't even been proven to be irrational.

Because it still might be transcendental. Just because you don't know if the list is correct, doesn't mean it isn't.

It is indeed manufactured specifically to show the existence of "normal" numbers, which are, loosely, numbers where every finite sequence of digits is equally likely to appear. This property is both ubiquitous (almost every number is normal in a specific sense) and difficult to prove for numbers not specifically cooked up to be so.

All the transcendental numbers are "manufactured in a mathematical laboratory somewhere".

In fact we can tighten that to all irrational numbers are manufactured in a mathematical laboratory somewhere. You'll never come across a number in reality that you can prove is irrational.

That's not necessarily because all numbers in reality "really are" rational. It is because you can't get the infinite precision necessary to have a number "in hand" that is irrational. Even if you had a quadrillion digits of precision on some number in [0, 1] in the real universe you'd still not be able to prove that it isn't simply that number over a quadrillion no matter how much it may seem to resemble some other interesting irrational/transcendental/normal/whatever number. A quadrillion digits of precision is still a flat 0% of what you'd need to have a provably irrational number "in hand".

It's fame comes from the simplicity of its construction rather than its utility elsewhere in mathematics.

For example, Graham's number is pretty famous but it's more of a historical artifact rather than a foundational building block. Other examples of non-foundational fame would be the famous integers 42, 69, and 420.

> mathematical laboratory

Love the image of mathematicians laboring over flasks and test tubes, mixing things and extracting numbers... would have far more explosions than day-to-day mathematics usually does...

Yes, it occurs in the nature of the mathematician's mind.

This comment is quite strange to me. e is the base of the natural logarithm. so ln 2 is actually log_e (2). If we take the natural log of 2, we are literally using its value as the base of a logarithm.

Does a number not matter "in practice" even if it's used to compute a more commonly use constant? Very odd framing.

It took me quite a bit to figure out what you're trying to say here.

The importance of e is that it's the natural base of exponents and logarithms, the one that makes an otherwise constant factor disappear. If you're using a different base b, you generally need to adjust by exp(b) or ln(b), neither of which requires computing or using e itself (instead requiring a function call that's using minimax-generated polynomial coefficients for approximation).

The importance of π or 2π is that the natural periodicity of trigonometric functions is 2π or π (for tan/cot). If you're using a different period, you consequently need to multiply or divide by 2π, which means you actually have to use the value of the constant, as opposed to calling a library function with the constant itself.

Nevertheless, I would say that despite the fact that you would directly use e only relatively rarely, it is still the more important constant.

What an odd thing to say. I find that it shows up all the time (and don't find myself using 2pi any more than pi).

Uuuuuum no?

e^(ix) = cos(x) + isin(x). In particular e^(ipi) = -1

(1 + 1/n)^n = e. This is part of what makes e such a uniquely useful exponent base.

Not applied enough? What about:

d/dx e^x = e^x. This makes e show up in the solutions of all kinds of differential equations, which are used in physics, engineering, chemistry...

The Fourier transform is defined as integral e^(iomega*t) f(t) dt.

And you can't just get rid of e by changing base, because you would have to use log base e to do so.

Edit: how do you escape equations here? Lots of the text in my comment is getting formatted as italics.

> It should be noted that the number e = 2.71828 ... does not have any importance in practice, its value just satisfies the curiosity to know it, but there is no need to use it in any application.

In calculations like compound financial interest, radioactive decay and population growth (and many others), e is either applied directly or derived implicitly.

> ... 2*pi is the most important transcendental number, not pi.

Gotta agree with this one.

>but there is no need to use it in any application.

Applications such as planes flying, sending data through wires, medical imaging (or any of a million different direct applications) do not count, I assume?

Your naivety about what makes the world function is not an argument for something being useless. The number appearing in one of the most important algorithms should give you a hint about how relevant it is https://en.wikipedia.org/wiki/Fast_Fourier_transform