this post was submitted on 03 Aug 2023

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What are the most mindblowing things in mathematics? (lemmy.world)

submitted 1 year ago* (last edited 1 year ago) by [email protected] to c/[email protected]

83 comments fedilink hide all child comments

What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?

Here are some I would like to share:

Gödel's incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)

The Busy Beaver function

Now this is the mind blowing one. What is the largest non-infinite number you know? Graham's Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.

The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don't even know if you can compute the function to get the value even with an infinitely powerful PC.
In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
Σ(1) = 1
Σ(4) = 13
Σ(6) > 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10 (10s are stacked on each other)
Σ(17) > Graham's Number
Σ(27) If you can compute this function the Goldbach conjecture is false.
Σ(744) If you can compute this function the Riemann hypothesis is false.

Sources:

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[–] [email protected] 2 points 1 year ago* (last edited 1 year ago) (2 children)

The fact that complex numbers allow you to get a much more accurate approximation of the derivative than classical finite difference at almost no extra cost under suitable conditions while also suffering way less from roundoff errors when implemented in finite precision:

$\frac{1}{\varepsilon}\,{\mathrm{Im}}\left[ f(x+i\,\varepsilon) \right] = f'(x) + \mathcal{O}(\varepsilon^2)$

(obviously x, epsilon and f(x) are assumed to be real)

Higher-order derivatives can also be obtained using hypercomplex numbers.

Another related and similarly beautiful result is Cauchy's integral formula which allows you to compute derivatives via integration.

[–] [email protected] 1 points 1 year ago

Well, that's something I hadn't seen before.

[–] [email protected] 0 points 1 year ago (1 children)

The fact that complex numbers allow you to get a much more accurate approximation of the derivative than classical finite difference at almost no extra cost under suitable conditions while also suffering way less from roundoff errors when implemented in finite precision:

What?

The formula you linked is the same asymptotic bound as for real-valued functions, 1/h[f(x+ h) - f(x)] = f'(x) + O(h^(2)). The only difference is that in your screenshot it is assumed that f(x) = 0.

[–] [email protected] 1 points 1 year ago

I can assure you that you're the one who's wrong, the order 2 term is a pure imaginary term that vanishes when you take the imaginary part, leaving you with a O(epsilon^3) which then becomes a O(epsilon^2) after dividing by epsilon. This is called complex-step differentiation.