Comment on TIL science has its own swifties

tetris11@lemmy.ml ⁨3⁩ ⁨weeks⁩ ago

• HowAbt2day@futurology.today ⁨3⁩ ⁨weeks⁩ ago

  Two questions for you my brother in god;

  1. what were the connections that were made in maths class that got the prof so excited?
  2. how long did you wait before removing the ever expanding schlong for your ever expanding sfinxter?

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  • tetris11@lemmy.ml ⁨3⁩ ⁨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

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    • HowAbt2day@futurology.today ⁨3⁩ ⁨weeks⁩ ago

      Thank you for your service for both 1) and 2)!

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