Submitted 3 weeks ago by mmmm@sopuli.xyz to science_memes@mander.xyz
https://sopuli.xyz/pictrs/image/1b49c6cb-9e0d-43ea-a6ff-10df1b4a34f8.webp
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.
Most aspects of our daily lives rely on Taylor series, polynomial expansion, and approximation theory in general. Everything built or planned using computer modeling software, down to the trig functions on your calculator… they all use polynomial approximations to allow a discrete mathematical machine to get us to within a certain error percentage of a continuous function in a timely manner.
Ah yes, that tracks with the very surface level overview that I picked up from it. It was only when I saw the magic “optional” tag that I was like noooope!
Maybe I’ll have a look at it in more detail when I get a free summer 😊
Think of The Taylor Series as a way of systematically approaching the estimation other more difficult functions starting from a particular point and reconstructing a graph function by stacking polynomials/exponents.
This one was absolutely brilliant though
I freaki’ love Taylor Series.
tetris11@lemmy.ml 3 weeks 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 3 weeks ago
Two questions for you my brother in god;
tetris11@lemmy.ml 2 weeks 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