The full set of rules, formulas and methods — stripped of worked examples — so you can sit and learn the material.
If b2 − 4ac < 0 a quadratic has complex roots.
A complex number is written z = a + bi, also written as the pair (a, b).
a = Re(z) is the real part. b = Im(z) is the imaginary part.
i = √−1 and i2 = −1.
i3 = −i and i4 = 1 — the powers run in a cycle of 4.
(−1)even = 1 ; (−1)odd = −1.
Divide n by 4 and use the remainder:
remainder 0 → 1, remainder 1 → i, remainder 2 → −1, remainder 3 → −i.
For z = a + bi:
Re(z) = a — the real part (no i).
Im(z) = b — the imaginary part (the number multiplying i).
Add real to real and imaginary to imaginary, separately.
Multiply out like algebra, then replace i2 with −1.
The conjugate of z = a + bi is z̅ = a − bi — flip the sign of the imaginary part.
z · z̅ is always a real number, equal to a2 + b2.
Multiply top and bottom by the conjugate of the bottom. This makes the denominator real.
If z1 = z2 then real = real and imaginary = imaginary.
1. Let √(given) = a + bi.
2. Square both sides.
3. Real = Real, Imaginary = Imaginary.
4. Solve the simultaneous equations for a and b.
The x-axis is the real axis. The y-axis is the imaginary axis.
|z| = √(a2 + b2) — the distance from the origin.
kz → dilation (k > 1 out, k < 1 in, k < 0 opposite direction).
iz → rotation 90° anticlockwise.
z + w → forms a parallelogram.
z̅ → sits directly below (or above) z — reflection in the real axis.
Use the formula x = [−b ± √(b2 − 4ac)] / 2a.
Works whether a, b, c are real or unreal.
z2 − (α + β)z + αβ = 0
Type 3 — find the 2nd root given the 1st: use the same formula.
Type 4 — find an unknown coefficient: substitute a known root in.
If the coefficients are real, complex roots come in conjugate pairs.
If z1 is a complex root and the coefficients are real, then z̅1 is also a root.
1. Form a quadratic from z1 and z̅1.
2. Divide the cubic by that quadratic.
3. The remaining linear factor gives the 3rd root.
(z − r1)(z − r2)(z − r3) = 0
z = r(cos θ + i sin θ)
r = |z| = √(x2 + y2), and tan θ = y/x (the argument).
z = r(cos(θ + 2nπ) + i sin(θ + 2nπ))
⚠ Once a fraction is in the power, you must use the general polar form.
z1z2 = r1r2(cos(θ1+θ2) + i sin(θ1+θ2))
z1/z2 = (r1/r2)(cos(θ1−θ2) + i sin(θ1−θ2))
1/z1 = (1/r1)(cos θ1 − i sin θ1)
If z = r(cos θ + i sin θ), then zn = rn(cos nθ + i sin nθ).
The power of z multiplies the angle and raises the modulus.
Use it for: raising to a power (e.g. (2−2i)4), finding roots (e.g. z3 = 1), and proving trig identities.
Roots on a diagram: r > 1 spiral out, r = 1 sit on the unit circle, r < 1 spiral in.
Standard expansion to know: (cos θ + i sin θ)n = cos nθ + i sin nθ.
1. Expand (cos θ + i sin θ)n binomially.
2. Equate real parts → the cos nθ identity.
3. Equate imaginary parts → the sin nθ identity.
For tan nθ: tan nθ = sin nθ / cos nθ.
Use: z + 1/z = 2 cos θ and zn + 1/zn = 2 cos nθ.
Also: z − 1/z = 2i sin θ.
Method: raise (z + 1/z) to the required power, equal it to (2 cos θ)n, expand the left side, group zn + 1/zn pairs, replace each with 2 cos nθ, then rearrange.