I studied entry level maths at uni level - a prerequisite course for most STEM degrees to cover the relatively small amount of maths common to nearly all science fields.
Chapter 11 of 12 were Taylor polynomials and series, and it was listed as “optional”.
I looked at it once, read it aloud for my young son to fall asleep to, and never looked at it again.
tetris11@lemmy.ml 1 day ago
My high school teacher introduced this to us as a slow reveal over the course of weeks of what would be the proof of
e^iπ^ = -1
HowAbt2day@futurology.today 22 hours ago
Two questions for you my brother in god;
tetris11@lemmy.ml 16 hours ago
For (1), we start with 2 definitions of cos(x) and sin(x), using their Taylor expansions
cos(x) = 1 - x^2^/2! + x^4^/4! - …
sin(x) = x - x^3^/3! + x^5^/5! - …
We now start with the definition of e^x^ Taylor expansion, and proceed to do some substitutions:
e^x^ = 1 + x + x^2^/2! + x^3^/3! + … + x^n^/n!
We can then substitute in: x=iθ (remembering that i^2^ = -1) to get
e^iθ^ = 1 + iθ - θ^2^/2! - iθ^3^/3! + θ^4^/4! + iθ^5^/5! + … etc…
If we group by real and complex, we can arrange the above as:
e^iθ^ = (1 - θ^2^/2! + θ^4^/4! + … ) + i(θ - θ^3^/3! + θ^5^/5! + … )
You should now realise that the left part resembles the expansion of cos(θ), and the right part resembles sin(θ). That is:
e^iθ^ = cos(θ) + i sin(θ)
Finally, we substitute in θ = π
e^iπ^ = cos(π) + i sin(π)
And we know that cos(π) = -1, and that sin(π) = 0, meaning that we end up with
e^iπ^ = -1 + i 0
or
e^iπ^ + 1 = 0
The teacher got excited because it is literally one of the most beautiful mathematical statements you can get, that connects five universal identities under a single statement: 0, 1, e, i, and π – and does so using 3 different operators (times, power, plus).
For (2), I’m still waiting as I think it’s currently holding the world together by sheer mass alone