A real number squared is never negative, so no real number satisfies x² = −1. Complex numbers widen the number system by bringing in a new number i whose square is −1.
A complex number is written a + bi (a and b real), with real part a and imaginary part b, and with them even a quadratic equation with a negative discriminant has solutions.
a + bi and a − bi are complex conjugates, and their product a² + b² is a real number; division of complex numbers relies on exactly this.
In the first picture the powers of i cycle through four places; in the second, the complex number a + bi and its conjugate appear as points on a plane.
The imaginary unit i
We needed a number satisfying x² = −1, so a new number i with i² = −1 was defined. With i, numbers can leave the real number line and spread out in a new direction.
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The powers of i stepping round four places in turn: 1, i, −1, −i
💡 Powers of i repeat in fours
①i⁰ = 1, i¹ = i, i² = −1, i³ = −i
②i⁴ = i² × i² = 1 brings you back to the start, so the values repeat with period 4
③So the value of iⁿ depends only on the remainder when n is divided by 4
④The division written at the bottom of the picture is exactly that remainder
Complex numbers as points on a plane
Show the complex number z = a + bi as the point (a, b) on the complex plane: the horizontal axis gives the real part and the vertical axis the imaginary part.
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The complex plane with z = a + bi and its conjugate z̄ = a − bi as points, together with the distance |z| from the origin
🔑 What to read from the complex plane
①z = a + bi is the point (a, b)
②The conjugate z̄ = a − bi is the point (a, −b), the reflection of z in the real axis
③|z| is the distance from the origin to (a, b), so it is √(a² + b²)
④z · z̄ = a² + b² = |z|² is always real, so to divide, multiply top and bottom by the conjugate of the denominator to make the denominator real
Calculating with complex numbers
Addition
(a + bi) + (c + di) = (a + c) + (b + d)i
Add real parts to real parts and imaginary parts to imaginary parts
Multiplication
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Expand with the distributive law, then use i² = −1
Product of conjugates
z · z̄ = (a + bi)(a − bi) = a² + b² = |z|²
The imaginary part cancels, so the result is always real
When two complex numbers are equal
Equality of complex numbers
a + bi = c + di ⟺ a = c, b = d
With a, b, c, d real, the real parts must match and the imaginary parts must match
📐 Using the equality condition
①For an equation in complex numbers, compare the real parts and the imaginary parts of both sides separately to find the unknowns
②If z = z̄, then b = 0, so z is real
③If z = −z̄, then a = 0, so z is 0 or purely imaginary
Worked examples and unit test
Example 1
Write (2 + 3i)(1 − 2i) in the form a + bi, where i is the imaginary unit.
1
Expand using the distributive law.
2 − 4i + 3i − 6i²
2
Substitute i² = −1 and collect the real and imaginary parts.
2 − i + 6 = 8 − i
▸ 8 − i
The moment −6i² becomes +6 is the heart of complex multiplication. Miss that sign and you get the wrong answer −4 − i.
Example 2
Find the value of i + i² + i³ + … + i¹⁰.
1
The powers of i repeat i, −1, −i, 1 with period 4, and any four consecutive terms add up to 0.
i + i² + i³ + i⁴ = 0
2
The terms from i¹ to i⁸ add up to 0, leaving only i⁹ + i¹⁰.
i⁹ + i¹⁰ = i + (−1) = −1 + i
▸ −1 + i
9 ÷ 4 leaves remainder 1 and 10 ÷ 4 leaves remainder 2, so i⁹ = i and i¹⁰ = i² = −1. Set n to 9 and 10 in the first picture to check.
Summary
Imaginary unit
i² = −1; powers of i repeat with period 4
i⁰ = 1, i¹ = i, i² = −1, i³ = −i
Modulus of a complex number
|z| = |a + bi| = √(a² + b²)
The distance from the origin on the complex plane
Grade-10 school exam type
For the complex number z = 3 − 4i, find z·z̄. (z̄ is the conjugate of z)
①5
②7
③25
④−7
⑤−25
▸ ③ 25
1
The conjugate is z̄ = 3 + 4i, so multiply the two.
z·z̄ = (3 − 4i)(3 + 4i)
2
Use the product of a sum and a difference.
= 3² − (4i)² = 9 + 16 = 25
🎯 Points that often come up
①Work out iⁿ from the remainder when n is divided by 4
②To divide complex numbers, multiply top and bottom by the conjugate of the denominator to make it real
③Remember z · z̄ = |z|²
④For equality of complex numbers, compare real and imaginary parts separately
⑤On the complex plane, conjugates are reflections of each other in the real axis