Lesson 12 of 12 · 16 min

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Must-know facts

18 facts

1i = √−1 and i² = −1.
2z = a + ib: Re z = a and Im z = b, both real.
3a + ib = c + id if and only if a = c and b = d.
4(a + ib)(c + id) = (ac − bd) + i(ad + bc).
5(a + ib)(a − ib) = a² + b².
61/(a + ib) = (a − ib)/(a² + b²) for a + ib ≠ 0.
71/i = −i.
8i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = −1, i⁴ᵏ⁺³ = −i for every integer k.
9iⁿ + iⁿ⁺¹ + iⁿ⁺² + iⁿ⁺³ = 0.
10√−a = √a i for a > 0.
11√a √b = √(ab) fails when a and b are both negative.
12|z| = √(a² + b²) and z z̄ = |z|².
13|z₁z₂| = |z₁||z₂|; |z₁/z₂| = |z₁|/|z₂|.
14The conjugate of a product, sum or quotient is the product, sum or quotient of the conjugates.
15(1 + i)² = 2i and (1 − i)² = −2i.
16On the Argand plane z̄ is the mirror image of z in the real axis, and |z₁ − z₂| is the distance between them.
17For D = b² − 4ac < 0 the roots of ax² + bx + c = 0 are (−b ± √(4ac − b²) i)/2a, a conjugate pair.
18x² − 4x + 5 = 0 has roots 2 ± i.

Common traps

Where marks are lost

Writing √−4 × √−9 = √36 = 6.

Convert first: √−4 = 2i and √−9 = 3i, so the product is 6i² = −6. The rule √a √b = √(ab) fails when both are negative.

Taking Im (5 − 3i) to be −3i.

The imaginary part is the real number next to i: Im (5 − 3i) = −3.

Taking i³ = i or i³ = 1.

i³ = i² × i = −i. Reduce any power by the remainder on division by 4.

Dividing part by part, as in (1 + i)/(2 + 3i) = 1/2 + i/3.

Multiply top and bottom by 2 − 3i: (1 + i)(2 − 3i)/13 = (5 − i)/13.

Writing |a + ib| = a + b or √(a² − b²).

|a + ib| = √(a² + b²), never negative; |3 − 4i| = 5.

Conjugating by changing the sign of the real part.

Only the imaginary part changes sign: the conjugate of −2 + 5i is −2 − 5i.

Stating that x² + 4 = 0 has no solution.

It has no real solution; in complex numbers x = ±2i.

Giving only one root, such as 2 + i, for x² − 4x + 5 = 0.

The ± gives two roots, 2 + i and 2 − i; a real quadratic with D < 0 always has a conjugate pair.

Comparing complex numbers, as in 3 + 2i > 1 + i.

There is no order on non-real complex numbers; compare moduli instead if the question asks for size.

Formulas

13 to know

Imaginary unit

i = √−1; i² = −1

i and −i are the two square roots of −1.

Equality

a + ib = c + id ⇔ a = c and b = d

One complex equation gives two real equations.

Sum and difference

(a + ib) ± (c + id) = (a ± c) + i(b ± d)

Parts are added separately.

Product

(a + ib)(c + id) = (ac − bd) + i(ad + bc)

Multiply out and put i² = −1.

Multiplicative inverse

1/(a + ib) = a/(a² + b²) − i b/(a² + b²) = z̄/|z|²

For z ≠ 0.

Quotient

(a + ib)/(c + id) = (a + ib)(c − id)/(c² + d²)

c + id ≠ 0.

Powers of i

i⁴ᵏ = 1; i⁴ᵏ⁺¹ = i; i⁴ᵏ⁺² = −1; i⁴ᵏ⁺³ = −i

k any integer; 1/i = −i.

Root of a negative number

√−a = √a i (a > 0)

√a √b = √(ab) fails for a, b both negative.

Modulus

|a + ib| = √(a² + b²)

Distance from O on the Argand plane.

Conjugate

conjugate of a + ib = a − ib; z z̄ = |z|²

Mirror image in the real axis.

Modulus of product and quotient

|z₁z₂| = |z₁||z₂|; |z₁/z₂| = |z₁|/|z₂|

z₂ ≠ 0.

Real and imaginary parts from z and z̄

z + z̄ = 2 Re z; z − z̄ = 2i Im z

z is real ⇔ z = z̄.

Roots when D < 0

x = (−b ± √(4ac − b²) i)/2a

D = b² − 4ac < 0; the roots are conjugates.

Key terms

14 terms

Imaginary unit: The number i with i² = −1.
Complex number: A number a + ib with a and b real.
Real part: The real number a in a + ib, written Re z.
Imaginary part: The real number b in a + ib, written Im z.
Purely imaginary number: A complex number 0 + ib with b ≠ 0.
Additive inverse: −z = −a − ib, which adds to z to give 0.
Multiplicative inverse: The number 1/z with z × (1/z) = 1; it exists for every z ≠ 0.
Modulus: |a + ib| = √(a² + b²), the distance of the point from the origin.
Conjugate: z̄ = a − ib, the mirror image of z in the real axis.
Argand plane: The coordinate plane with the point (x, y) read as the complex number x + iy.
Real axis: The x-axis of the Argand plane, holding the real numbers.
Imaginary axis: The y-axis of the Argand plane, holding the numbers ib.
Discriminant: D = b² − 4ac for ax² + bx + c = 0; D < 0 means non-real conjugate roots.
Conjugate pair: Two numbers a + ib and a − ib, as the roots of a real quadratic with D < 0 always are.