Expected value

“Casinos stay profitable because the expected value tilts their way.”

The long-run average outcome, weighting each value by its probability.

Memory trick: Probability-weighted average.

Linear programming

“Factories use it to maximise profit within limited resources.”

Optimises a linear objective subject to linear constraints; the best value sits at a corner of the feasible region.

Memory trick: Optimum lives at a corner.

Mathematical induction

“Knock the first domino and set up the chain — they all fall.”

Prove a statement for n=1, then that truth at k implies truth at k+1.

Memory trick: Base case + inductive step.

Modulus inequality

“A modulus inequality carves out an interval on the number line.”

|x| < a means −a < x < a; |x| > a means x < −a or x > a.

Memory trick: '<' interval, '>' two rays.

Periodic function

“sin and cos repeat every 2π — the maths of anything cyclic.”

A function that repeats its values at regular intervals (period).

Memory trick: Repeats every period.

Signum function

“It's a switch that reports only the direction, not the size.”

sgn(x) returns −1, 0 or +1 depending on the sign of x.

Memory trick: Reads the sign only.

Product-to-sum formulas

“They turn stubborn products into friendly sums for integration.”

2sinA cosB = sin(A+B) + sin(A−B) and similar identities.

Memory trick: Product → sum.

Maximum of a sinθ + b cosθ

“A quick way to bound any sine-plus-cosine expression.”

It ranges between −√(a²+b²) and +√(a²+b²).

√(a²+b²)

Memory trick: Amplitude = √(a²+b²).

Half-angle formulas

“They let you climb down from an angle to its half.”

sin(θ/2), cos(θ/2) can be written using cosθ, halving the angle.

Memory trick: Express half-angle via cosθ.

nth roots of a complex number

“They sit like the numbers on a clock face around the origin.”

A non-zero complex number has exactly n distinct nth roots, evenly spaced on a circle.

Memory trick: n roots, equally spaced.

Argument of a product

“Multiplying complex numbers adds their angles — a rotation.”

arg(z₁z₂) = arg z₁ + arg z₂.

Memory trick: Multiply → add angles.

Arithmetico-geometric progression

“It blends steady steps with multiplying growth.”

Each term is a product of an AP term and a GP term.

Memory trick: AP × GP terms.

Telescoping series

“Most of the middle collapses, so only the first and last survive.”

A series where consecutive terms cancel, leaving only the ends.

Memory trick: Split into differences.

Geometric mean

“Average growth rates multiply, so their fair average is geometric.”

The GM of a and b is √(ab).

√(ab)

Memory trick: √(ab), not (a+b)/2.

Circular permutations

“Seating friends around a round table fixes one and arranges the rest.”

n distinct objects arrange around a circle in (n−1)! ways.

(n−1)!

Memory trick: (n−1)! for a circle.

Combinations with a condition

“Sometimes it's easier to count what you don't want and subtract.”

Restricted selections use complementary counting: total minus the unwanted cases.

Memory trick: Count the complement.

Pascal's rule

“Pascal's triangle builds itself, row by row.”

nCr + nC(r−1) = (n+1)Cr — each triangle entry is the sum of the two above.

Memory trick: Add the two above.

Number of terms in an expansion

“One more term than the power — easy to remember.”

(a+b)^n has n+1 terms.

Memory trick: n+1 terms.

Locus

“A circle is the locus of points a fixed distance from a centre.”

The path traced by a point moving under a given condition.

Memory trick: Turn a condition into a curve.

Collinear points

“Zero enclosed area means they all lie on one line.”

Three points are collinear if the area of the triangle they form is zero.

Memory trick: Zero area → collinear.

General equation of a circle

“Read the centre and radius straight off the coefficients.”

x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c).

Memory trick: Centre (−g,−f).

Condition of tangency

“Tangency is the knife-edge between missing and cutting through.”

A line touches a conic when it meets it in exactly one point (discriminant zero).

Memory trick: One intersection → tangent.

Asymptotes of a hyperbola

“The curve hugs its asymptotes ever more closely toward infinity.”

Straight lines the hyperbola approaches but never meets.

Memory trick: Approached, never touched.

Foci of an ellipse

“A gardener draws an ellipse with a loop of string around two pegs (foci).”

Two fixed points where the sum of distances to any point on the ellipse is constant.

Memory trick: Constant distance-sum.