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.
300 fundamentals
“'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)
“Compound interest is a geometric progression quietly growing your money (or your loan).”
Sum of an infinite geometric series. Use it when only when |r| < 1.
S∞ = a/(1−r)
“Compound interest is a geometric progression quietly growing your money (or your loan).”
An AP grows by adding a fixed number each step; a GP grows by multiplying by a fixed ratio each step.
“Compound interest is a geometric progression quietly growing your money (or your loan).”
A finite series always has a sum; an infinite series has one only if it converges (e.g. a GP with |r|<1).
“Compound interest is a geometric progression quietly growing your money (or your loan).”
The sum a/(1−r) exists only when |r|<1; otherwise the series diverges.
“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”
A frequent error is using permutations when order doesn't matter. In reality, an arrangement where order matters: nPr = n!/(n−r)!.
Memory trick: if rearranging counts as different, it's a permutation.
“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”
A frequent error is using combinations when order actually matters. In reality, a selection where order does not matter: nCr = n!/[r!(n−r)!].
Memory trick: if only the group matters (not its order), it's a combination.
“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”
If one task has m ways and another n ways, together they have m×n ways.
Memory trick: 'and' usually means multiply; 'or' usually means add.
“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”
A frequent error is adding when you should multiply independent choices. In reality, if one task has m ways and another n ways, together they have m×n ways.
Memory trick: 'and' usually means multiply; 'or' usually means add.
“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”
Number of ordered arrangements of r from n. Use it when order matters, no repetition.
nPr = n!/(n−r)!
“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”
Number of unordered selections of r from n. Use it when order does not matter.
nCr = n!/[r!(n−r)!]
“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”
A permutation counts ordered arrangements (a password); a combination counts unordered selections (a team) — nCr = nPr / r!.
“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”
With repetition, items can be reused (like digits in a PIN); without repetition, each item is used at most once (like ranking people).