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My high school teacher introduced this to us as a slow reveal over the course of weeks of what would be the proof of
eiπ = -1
The happiest moment was when he brought in these two disparate field of mathematics, complex numbers and series expansions, and hit us with this magnificent revelation. Once he drew it up, he stood there shaking with excitement, beaming at us at how amazing this all was.
The class wasn’t having it. We were teenagers. We understood it from a purely proof level, but did not get the implications. It was years years later that it all hit me how amazingly neat it was of the universe to unite these fields together like that and to unearth literally new tools we could use to explore further fields of maths.
Thankfully since then I’ve started dating Taylor Swift and reading the words of Samule Taylor Coleridge, whilst getting clothes fitted to size at my local clothes-maker guy to fit my enourmous expanding schwang.
Two questions for you my brother in god;
- what were the connections that were made in maths class that got the prof so excited?
- how long did you wait before removing the ever expanding schlong for your ever expanding sfinxter?
For (1), we started with the Maclaurin series 1/x to get us familiar with the idea of differential expansions, and then we moved to Taylor to derive expansions of some common functions like cos and sin:
cos(x) = 1 - x2/2! + x4/4! - …
sin(x) = x - x3/3! + x5/5! - …
We now start with the definition of ex Taylor expansion, and proceed to do some substitutions:
ex = 1 + x + x2/2! + x3/3! + … + xn/n!
We can then substitute in: x=iθ (remembering that i2 = -1) to get
eiθ = 1 + iθ - θ2/2! - iθ3/3! + θ4/4! + iθ5/5! + … etc…
If we group by real and complex, we can arrange the above as:
eiθ = (1 - θ2/2! + θ4/4! + … ) + i(θ - θ3/3! + θ5/5! + … )
You should now realise that the left part resembles the expansion of cos(θ), and the right part resembles sin(θ). That is:
eiθ = cos(θ) + i sin(θ)
Finally, we substitute in θ = π
eiπ = cos(π) + i sin(π)
And we know that cos(π) = -1, and that sin(π) = 0, meaning that we end up with
eiπ = -1 + i 0
or
eiπ + 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
Thank you for your service for both 1) and 2)!
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.
In short, polynomials are crucial in many applications because they are simple and elegant, and express wide range of functions.
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 by starting from a particular point and reconstructing a graph function by stacking polynomials/exponents centered on that point like legos until you get a close enough shape.
I freaki’ love Taylor Series.
Brook Taylor was absolutely brilliant though
