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Silas Maths

All derivations here are painstakingly authored by the \(11\) yo Silas Pembroke, with a little help from his Dad. He was- I'm all ready tired of talking in the \(3\)rd person. I am a boy who loves mathematics and would like to share my math with the world.

  • 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. You can read the old description below:

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 eigenvectors and eigenvalues
  • Binomial - a derivation of the binomial theorem, an introduction to discrete calculus, and a method for computing pi (not pie)
  • Fractional calculus - an introduction to fractional calculus using cauchy's (other) integral formula
  • Fibonacci - 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 detective 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 surround this text in quotation marks"
  • The strand puzzle - a popular puzzle from the strand magazine a hundred years ago that ramanujan solved "straight 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
  • Geometric algebra - (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.
  • Code repo - Pretty self explanatory. Also, the idea for both this page and the one above it did not originate in the brainstorm page.
  • My way to count - why YOU should use binary
  • Geometric algebra 2 - the first part (this page will not make sense without it) crashed
  • Lambda calculus - an exploration of lambda calculus, the smallest programing language where "functional programing language" is an understatement (based off of this and this)
  • Thoughts - The half of the brainstorm page that had nothing to do with math, and everything to do with what was happening at the time.
  • Summer of Math Exposition 4 - my submission to SoME4 (he didn't have a video for SoME \(2\) through \(4\))
  • Set theory: logic edition - the set theory page, but derived from the ground up using logic and extensions.
  • Projective geometry - an introduction to projective geometry where parallel lines cross, and there are two points that lie on every circle.
  • Lambda calculus revisited - (Lcr) I gave up on this one, but then I came back! It is still heavily based off of this, this, and even more heavily based off of this (Warning: the videos are ~ \(1 \text{H}\) long, and the paper has (exactly) \(120\) pages).
  • Linear algebra - A course about the more general mathematician's version of linear algebra, the study of vectors (like the strange man with a bowl cut and an orange suit) and matrices (like the fake AI generated simulated reality). But if you want some more intuition about how it works, each chapter will have a corresponding part in this playlist
  • Group theory - A page about group theory, an attempt at a grand unified theory of mathematics. I'll be covering things like dihedral groups, symmetric groups, subgroups, cosets, isomorphism, and maybe even the \(196,883\) dimensional monster.