Silas Maths

All derivations here are painstakingly authored by Silas Pembroke, with a little help from his Dad.

• 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
• 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
• 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 function f(n) that represents the sum of a sequence of fractions, and then extending this to f(∞), it's shown that this sum converges to the natural logarithm of 2 (ln(2)). Therefore, the infinite alternating harmonic series sums up to ln(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
• 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