Mathematics

Algebra and geometry

Equations, exponentials and logs, Pythagoras, circles.

Basics

Equations

The same operation on both sides preserves equality. Quadratics: factor, complete the square, or use the formula. The discriminant D=b²−4ac counts real roots.

Exponentials and logs

Log is the inverse of exp. log_b(xy)=log_b x+log_b y. Natural log ln has base e; common log has base 10. Scientific scales (pH, dB, magnitude) are logarithmic.

Euclidean geometry

Triangle angles sum to 180°. Similar triangles share angles and side ratios. A central angle is twice the inscribed angle on the same arc. Coordinates link geometry to algebra.

Formulas

Quadratic formula

x = (−b ± √(b² − 4ac)) / (2a)

For ax²+bx+c=0 (a≠0). D>0 two real roots, D=0 repeated, D<0 complex.

Symbols

D b²−4ac

Change of base

log_b a = ln a / ln b

A calculator needs only ln or log₁₀.

Exponent laws

aᵐ aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ

Same base; a≠0.

Pythagoras

a² + b² = c²

Right triangle, c hypotenuse. Prototype of the distance formula.

Distance

d = √((x₂−x₁)² + (y₂−y₁)²)

In the plane. Add a z term in 3D.

Circle

C = 2π r, A = π r²

Circumference and area. Sphere: surface 4πr², volume 4πr³/3.

Symbols

r radius

In this field

Trigonometry and calculus

Trig ratios, identities, derivatives, integrals.

Statistics and linear algebra

Mean, variance, Bayes, matrices, eigenvalues.

Discrete, complex, series

Sequences, series, complex numbers, combinatorics.

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.