Contrapositive

“Proving the contrapositive is often easier than the original.”

'If p then q' is logically equivalent to 'if not q then not p'.

Memory trick: Flip and negate both.

Converse and inverse

“A true statement can have a false converse — don't assume otherwise.”

The converse of 'if p then q' is 'if q then p'; they aren't logically equivalent.

Memory trick: Converse ≠ original.

Triangle inequality

“You can't build a triangle from sticks 1, 2 and 10.”

The sum of any two sides of a triangle exceeds the third.

Memory trick: Two sides beat the third.

AM-GM-HM order

“Three means, always in the same pecking order.”

For positive numbers, AM ≥ GM ≥ HM, equal only when all are equal.

Memory trick: AM ≥ GM ≥ HM.

Factorial

“It counts the ways to arrange n distinct objects.”

n! is the product of all positive integers up to n, with 0! defined as 1.

n! = n×(n−1)!

Memory trick: 0! = 1 by definition.

Sum of an arithmetic progression

“Gauss's pairing trick generalised to any AP.”

The sum of n terms is n/2 (first + last) = n/2[2a+(n−1)d].

Sₙ=n/2(a+l)

Memory trick: n/2 × (first + last).

Symmetry of binomial coefficients

“Choosing who's on the team is the same as choosing who's left out.”

nCr = nC(n−r).

nCr=nC(n−r)

Memory trick: Choose r = leave out n−r.

Distance between parallel lines

“The constant gap between two rails that never meet.”

For ax+by+c₁=0 and ax+by+c₂=0, distance = |c₁−c₂|/√(a²+b²).

Memory trick: Difference of constants over √(a²+b²).

Derivative of tan x

“A staple derivative worth memorising outright.”

d/dx(tan x) = sec²x.

d/dx(tanx)=sec²x

Memory trick: tan → sec².

Binomial theorem for any index

“It stretches the binomial idea beyond whole-number powers.”

(1+x)ⁿ for fractional or negative n gives an infinite series valid for |x|<1.

Memory trick: Infinite series when |x|<1.

Discriminant — common mistake

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

A frequent error is thinking D<0 means 'no solution'. In reality, D = b² − 4ac tells the nature of the roots before you even solve: D>0 gives two real distinct roots, D=0 gives equal roots, D<0 gives complex roots.

Memory trick: D<0 still has two roots — they're complex conjugates.

Sum and product of roots — common mistake

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

A frequent error is dropping the minus sign in −b/a. In reality, for ax²+bx+c=0, the roots add to −b/a and multiply to c/a (Vieta's relations).

Memory trick: sum = −b/a, product = c/a — handy for building equations from roots.

Nature of roots

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

The sign of the discriminant decides whether roots are real, equal, or complex.

Memory trick: check D first — it often answers the question directly.

Nature of roots — common mistake

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

A frequent error is solving fully when you only need the nature of the roots. In reality, the sign of the discriminant decides whether roots are real, equal, or complex.

Memory trick: check D first — it often answers the question directly.

x = [−b ± √(b²−4ac)] / 2a

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

The quadratic formula for the roots. Use it when any quadratic ax²+bx+c=0.

x = [−b ± √(b²−4ac)] / 2a

sum = −b/a, product = c/a

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

Vieta's relations between roots and coefficients. Use it when quadratic with roots α, β.

sum = −b/a, product = c/a

real roots vs complex roots

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

Real roots occur when D≥0 (the parabola meets the x-axis); complex roots occur when D<0 (it doesn't).

roots vs factors

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

Roots are the x-values that make the expression zero; factors are the (x − root) pieces whose product is the polynomial.

Myth: D<0 means no solution

“The path of every thrown ball, fountain jet and basketball shot is a quadratic curve.”

A negative discriminant gives two complex conjugate roots, not 'no roots'.

The imaginary unit

“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”

I is defined by i² = −1, letting us handle square roots of negative numbers.

Memory trick: the rule √a·√b=√(ab) fails for negatives — i² = −1, not +1.

The imaginary unit — common mistake

“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”

A frequent error is writing √(−1)·√(−1) = √1 = 1. In reality, i is defined by i² = −1, letting us handle square roots of negative numbers.

Memory trick: the rule √a·√b=√(ab) fails for negatives — i² = −1, not +1.

Modulus and conjugate

“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”

The modulus |z|=√(a²+b²) is the distance from the origin; the conjugate of a+bi is a−bi.

Memory trick: multiply by the conjugate to rationalise a complex denominator.

Modulus and conjugate — common mistake

“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”

A frequent error is forgetting that z times its conjugate equals |z|² (a real number). In reality, the modulus |z|=√(a²+b²) is the distance from the origin; the conjugate of a+bi is a−bi.

Memory trick: multiply by the conjugate to rationalise a complex denominator.

Argand plane

“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”

A complex number a+bi is a point (a, b), turning algebra into geometry.

Memory trick: real part → x-axis, imaginary part → y-axis.