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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 - Contains non-math-related things that I still wanted to tell.
  • 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 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.