All derivations here are painstakingly authored by Silas Pembroke, with a little help from his Dad.
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Calculus - derivative formulas from first principles:
sum rule, chain rule, product rule, multiplication by constant, exponent rule, introduction to
$e$ , logarithmic derivative, power rule - Polynomial - solving for the roots of the quartic equation
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Calculus II - derivations of
$e$ and$e^x$ from limits, a derivation of the quotient rule - Complex - introduction to complex numbers and their relationship to sine and cosine using geogebra
- Jacobian - on the relation between complex numbers and linear algebra
- Gamma - explores the connections between factorials, calculus, and the Gamma function, extending factorials to non-integers
- Trigonometry - Desmos visualization of angle addition
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Harmonic - On the alternating and non-alternating harmonic seireis. I didn't want to remove the old description made by ChatGPT, even if someone might think "more description equals longer page" which is why I replaced it. Here is the old description in quotation marks. "The alternating harmonic series
1 - 1/2 + 1/3 - 1/4 + 1/5 - ...
is restructured by rearranging terms, leading to a new series that converges to a known value. By defining a functionf(n)
that represents the sum of a sequence of fractions, and then extending this tof(∞)
, it's shown that this sum converges to the natural logarithm of 2 (ln(2)
). Therefore, the infinite alternating harmonic series sums up toln(2)
. This conclusion is reached through a creative manipulation of series and leveraging properties of logarithms." - Eigenstuff - the applications and derivations of eigervectors and eigenvalues
- Binomial - a derivation of the binomial therom, an introduction to discrete calculas, and a method for computing pi (not pie)
- Fractional calculas - an introduction to fractional calculas using cauchy's other integral formula
- Fibbonacci - using the golden ratio phi and the less popular psi to derive binet's formula among other things
- Probability - a coin flip game leads you into a new job as a detectave to test if someone is cheating or not (I haven't added anything to it in a while)
- Complex II - a derivation of the roots of unity, "I'll write this down when I put something in here and suround this text in quotation marks"
- The strand puzzle - a popular puzzle from the strand magazine a hundred years ago that ramanujan solved "strtaight away", whatever that means
- The infamous "arctan puzzle" - the other puzzle that I was working on in the vacation
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Modular arithmetic - if you restart counting after
$10$ ,$5$ , or$7$ - Brainstorming a new page - pretty self explanatory
- Secretly clifford algebra - Yes, you can multiply two vectors.
PS you don't get a vector
- Set theory - definitions of the subset, empty set, power set, and so on.
- NOOOOOOOOOO - I lost evereything.
- Code repo - pretty self explanatory. Also, the idea for both this page and the one above it did not originate in the brainstorm page.Binary.md](binary.md
- The best way to count - why you should use binary