Revise: Complex Numbers

i² = −1; powers of i cycle every 4.

i⁷ = −i.

z = a + bi: a is the real part, b is the imaginary part.

z=5−3i → Re=5, Im=−3.

Combine real parts, and imaginary parts, separately.

(3+4i)+(1−6i)=4−2i.

FOIL, then replace i² with −1.

(2+3i)(1−4i)=14−5i.

Multiply top and bottom by the denominator's conjugate.

(4+2i)/(1−i)=1+3i.

|z| = √(a²+b²).

z=3+4i → |z|=5.

z = r(cosθ+isinθ), r=|z|, θ=arg(z).

z=3+4i → r=5, θ≈53.13°.

[r(cosθ+isinθ)]ⁿ = rⁿ(cos(nθ)+isin(nθ)).

[2(cos30°+isin30°)]³=8i.

Every nonzero complex number has n distinct nth roots.

Cube roots of 8: 2, −1+1.73i, −1−1.73i.

To simplify iⁿ, use the remainder of n÷4.

i¹⁰ = i² = −1.

Im(z) is a real number — the coefficient of i, not "bi" itself.

z=−7+2i → Im(z)=2.

Distribute the minus sign to both parts when subtracting.

(6−3i)+(−2+5i)=4+2i.

A number times its own conjugate is always real.

(1+i)(1−i)=2.

(c+di)(c−di) = c²+d², always real.

(3+i)/(1+i)=2−i.

z=a+bi plots as the point (a,b); |z| is its distance from O.

z=−6+8i → |z|=10.

a=r cosθ, b=r sinθ — check the quadrant when finding θ.

z=6+8i → r=10.

Comes from repeatedly applying the trig-form multiplication rule.

[3(cos40°+isin40°)]² has modulus 9.

All n roots share modulus r^(1/n), angles spaced 360°/n apart.

4th roots of 16 all have modulus 2.