Mathematics

Discrete, complex, series

Sequences, series, complex numbers, combinatorics.

Basics

Sequences and series

Arithmetic: common difference d. Geometric: ratio r. Infinite geometric converges if |r|<1. Taylor writes a function as a power series. The harmonic series 1+1/2+1/3+… diverges.

Complex numbers

z = a+bi with i²=−1. Polar form r(cosθ+i sinθ)=r e^{iθ}. Multiply: moduli multiply, arguments add. Impedance and waves speak this language.

Counting

Permutations nPr = n!/(n−r)!: order matters. Combinations nCr = n!/(r!(n−r)!): order does not. Binomial theorem (x+y)ⁿ = Σ nCr xⁿ⁻ʳ yʳ. Vertices and edges model networks.

Formulas

Arithmetic sum

S_n = n (a₁ + aₙ) / 2

Mean of first and last, times n.

Symbols

n number of terms

Geometric sum

S_n = a (1 − rⁿ) / (1 − r), |r|<1 ⇒ S_∞ = a / (1 − r)

r≠1. Infinite sum only if |r|<1.

Symbols

a first term
r common ratio

Modulus and argument

|z| = √(a² + b²), arg z = atan2(b, a)

Handy for multiply, divide, and powers (de Moivre).

Combinations

C(n, r) = n! / (r! (n − r)!)

Choose r from n, unordered. Entries of Pascal’s triangle.

In this field

Algebra and geometry

Equations, exponentials and logs, Pythagoras, circles.

Trigonometry and calculus

Trig ratios, identities, derivatives, integrals.

Statistics and linear algebra

Mean, variance, Bayes, matrices, eigenvalues.

Limits and number-theory sketch

Limits, continuity, primes, congruences.

Differential equations and series

First-order linear, harmonic oscillator, Taylor and geometric series.

Vectors and space

Vectors, dot and cross products, planes, and distance in 3D.

Conic sections

Parabola, ellipse, hyperbola, eccentricity, foci, and an orbit sketch.

Fourier sketch

Sines that rebuild a periodic function, coefficients, spectra, and a discrete-transform note.

Probability, continued

Expected value, variance, normal density, and a CLT one-liner.

Graph-theory sketch

Vertices and edges, degree, paths, Euler, and the adjacency matrix.