Have you ever done the proof that pi's transcendental?
There are plenty of proofs that Pi is transcendental, almost all of which hinge on things that are still just out of my range of knowledge. Most of them are multi-part proofs.
For those who don’t know, a “transcendental number” is a number that is not an “algebraic number”. The set of algebraic numbers is the set of numbers that are the root of some finite polynomial with integer coefficients. For example, 7 is the root of the polynomial x - 7 = 0. √2 is the root of the polynomial x^2 - 2 = 0. cube_root(3) + sqrt(12) - 17 is the root of the polynomial x^6 + 102 x^5 + 4299 x^4 + 95806 x^3 + 1190517 x^2 + 7820940 x + 21220792 = 0. There are some algebraic numbers that aren’t expressible in “nice” formatting, like the root of x^5 - x - 1 = 0, but they’re still algebraic.
Pi is not the root of such any finite polynomial with integer coefficients.
The outline of one proof goes as follows:
e (the root of the natural logarithm, ~2.71828) is a transcendental number [source].
According to the Lindemann-Weierstrass theorem, if x is a transcendental number and n is an algebraic number, then xn is a transcendental number
eiπ + 1 = 0 is a special case of the famous Euler’s Identity at x = π. Since i is an algebraic number (being the root of the finite polynomial x^2 + 1 = 0), and the product of any two algebraic numbers is also algebraic, if π were algebraic, then so would iπ. And then since iπ were algebraic, and e is transcendental, according to the LWT, eiπ must be transcendental. But since it’s not — namely, since it’s -1 — then our first assumption, which was that π is algebraic, must be false.
I don’t have the technical knowledge to fully understand the first two proofs, but that doesn’t stop them from being true. And from those two, plus Euler’s Identity, and the steps I mentioned above, comes the proof that π is transcendental.
