Why complex numbers?
The equation x² + 1 = 0 has no real solution, because the square of a real number is never negative. To solve such equations, we introduce the imaginary unit
i = √−1, so that i² = −1.
A complex number has the form z = a + ib, where a and b are real.
a is the real part: Re z = a.
b is the imaginary part: Im z = b.
Example. For z = 3 − 4i, Re z = 3 and Im z = −4. (The imaginary part is −4, not −4i.)
Every real number is a complex number with b = 0. A number with a = 0, such as 5i, is purely imaginary.
Two complex numbers are equal only if both their real parts and their imaginary parts are equal.
Powers of i
The powers of i repeat in a cycle of 4:
i¹ | i² | i³ | i⁴ |
|---|---|---|---|
i | −1 | −i | 1 |
To find iⁿ, divide n by 4 and use the remainder. i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = −1, i⁴ᵏ⁺³ = −i.
Example. i³⁷: 37 = 4 × 9 + 1, so i³⁷ = i. And i⁻³⁵ = 1/i³⁵ = 1/i³ = 1/(−i) = i.
Also, i + i² + i³ + i⁴ = 0, so the sum of any four consecutive powers of i is 0.
Careful: √a × √b = √(ab) only when a and b are not both negative. √−4 × √−9 = 2i × 3i = −6, not √36 = 6.
Algebra of complex numbers
Let z₁ = a + ib and z₂ = c + id.
Addition: z₁ + z₂ = (a + c) + i(b + d)
Subtraction: z₁ − z₂ = (a − c) + i(b − d)
Multiplication: z₁z₂ = (ac − bd) + i(ad + bc), found by expanding and using i² = −1.
Division: multiply numerator and denominator by the conjugate of the denominator.
Example. (2 + 3i)(1 − 2i) = 2 − 4i + 3i − 6i² = 2 − i + 6 = 8 − i.
Example. (3 + 2i) ⁄ (1 + i) = (3 + 2i)(1 − i) ⁄ ((1 + i)(1 − i)) = (3 − 3i + 2i − 2i²) ⁄ 2 = (5 − i)/2 = 5/2 − i/2.
Every non-zero complex number has a multiplicative inverse: 1/z = z̄ ⁄ |z|².
Conjugate and modulus
The conjugate of z = a + ib is z̄ = a − ib.
The modulus is |z| = √(a² + b²), the distance of z from the origin.
Useful properties:
z z̄ = a² + b² = |z|²
|z₁z₂| = |z₁| |z₂| and |z₁/z₂| = |z₁|/|z₂|
the conjugate of (z₁ + z₂) is z̄₁ + z̄₂, and of (z₁z₂) is z̄₁ z̄₂
Example. |3 − 4i| = √(9 + 16) = 5.
The Argand plane
A complex number z = a + ib is represented by the point (a, b) in a plane: the x-axis is the real axis and the y-axis the imaginary axis. This is the Argand plane (complex plane).
The conjugate a − ib is the mirror image of z in the real axis.
Polar form
Let the point (a, b) be at distance r = |z| from the origin, making angle θ with the positive real axis. Then a = r cos θ and b = r sin θ, so
z = r (cos θ + i sin θ)
θ is the argument of z. The value with −π < θ ≤ π is the principal argument, arg z.
To find θ, compute the acute angle α with tan α = |b/a|, then adjust for the quadrant:
Quadrant of z | Principal argument |
|---|---|
I (a > 0, b > 0) | α |
II (a < 0, b > 0) | π − α |
III (a < 0, b < 0) | −(π − α) |
IV (a > 0, b < 0) | −α |
Example. z = −1 + i√3. r = √(1 + 3) = 2. tan α = √3, so α = π/3. z is in the second quadrant, so θ = π − π/3 = 2π/3. Therefore z = 2(cos 2π/3 + i sin 2π/3).
Quadratic equations with complex roots
For ax² + bx + c = 0 with real coefficients and discriminant D = b² − 4ac < 0, the roots are
x = (−b ± i√(4ac − b²)) ⁄ 2a
They always come as a conjugate pair.
Example. x² + x + 1 = 0. D = 1 − 4 = −3, so x = (−1 ± i√3)/2.
Example. x² + 9 = 0 gives x = ±3i.
Summary
i² = −1, and powers of i cycle with period 4.
Multiply by expanding; divide by multiplying by the conjugate of the denominator.
|z| = √(a² + b²); z z̄ = |z|²; the conjugate reflects z in the real axis.
Polar form z = r(cos θ + i sin θ), with principal argument in (−π, π].
A real quadratic with D < 0 has a pair of conjugate complex roots.