Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
1,200 fundamentals
“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.
“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.
“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
“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
“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).
“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.
“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'.
“'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.
“'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.
“'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.
“'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.
“'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.
“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”
A frequent error is plotting the imaginary part on the x-axis. In reality, a complex number a+bi is a point (a, b), turning algebra into geometry.
Memory trick: real part → x-axis, imaginary part → y-axis.
“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”
Modulus (distance from origin) of z = a+bi. Use it when any complex number.
|z| = √(a²+b²)
“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”
A complex number times its conjugate is real. Use it when used to rationalise denominators.
z·z̄ = |z|²
“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”
In a+bi, a is the real part (on the x-axis) and b is the imaginary coefficient (on the y-axis).
“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”
The modulus is a complex number's distance from the origin; the argument is the angle it makes with the positive real axis.
“'Imaginary' numbers are very real — they run the electronics in every phone and power grid.”
The product rule for surds doesn't hold for negatives; i·i = i² = −1.
“Compound interest is a geometric progression quietly growing your money (or your loan).”
A frequent error is using the wrong count (n vs n−1) for the nth term. In reality, a sequence with a constant common difference d; the nth term is a+(n−1)d.
Memory trick: the first term uses n=1, so the (n−1) accounts for the steps taken.
“Compound interest is a geometric progression quietly growing your money (or your loan).”
A frequent error is adding the ratio instead of multiplying by it. In reality, a sequence with a constant ratio r; the nth term is a·r^(n−1).
Memory trick: in a GP you MULTIPLY by r each step, not add.
“Compound interest is a geometric progression quietly growing your money (or your loan).”
An infinite GP has a finite sum a/(1−r) only when |r| < 1.
Memory trick: if the terms don't shrink (|r|≥1), the series diverges — no finite sum.
“Compound interest is a geometric progression quietly growing your money (or your loan).”
A frequent error is applying the sum formula when |r| ≥ 1. In reality, an infinite GP has a finite sum a/(1−r) only when |r| < 1.
Memory trick: if the terms don't shrink (|r|≥1), the series diverges — no finite sum.
“Compound interest is a geometric progression quietly growing your money (or your loan).”
Nth term of an arithmetic progression. Use it when constant common difference d.
aₙ = a + (n−1)d
“Compound interest is a geometric progression quietly growing your money (or your loan).”
Nth term of a geometric progression. Use it when constant ratio r.
aₙ = a·r^(n−1)