What this chapter is about
Until now, you have worked only with real numbers—those that fit on the number line. But what happens when you try to solve x² + 1 = 0? No real number squared gives −1. To handle such equations, mathematicians invented a new kind of number called a complex number, built around the imaginary unit i, defined so that i² = −1.
This chapter introduces you to complex numbers, their algebra (addition, subtraction, multiplication, division), their geometric representation on the Argand plane, and the polar form. You will also revisit quadratic equations and learn how to find roots even when the discriminant is negative—roots that live in the complex number system.
After studying this chapter, you should be able to perform arithmetic with complex numbers, convert between rectangular and polar forms, find the modulus and argument of a complex number, and solve any quadratic equation with real coefficients, whether its roots are real or complex.
Key ideas
- A complex number has the form z = a + ib, where a and b are real numbers; a is the real part, b is the imaginary part, and i satisfies i² = −1.
- Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
- The conjugate of z = a + ib is z̄ = a − ib; the product z × z̄ = a² + b², which is always a non-negative real number.
- The modulus |z| = √(a² + b²) gives the distance of the point (a, b) from the origin in the Argand plane.
- The argument (or amplitude) θ of z is the angle the line joining the origin to (a, b) makes with the positive real axis, measured anti-clockwise.
- Polar form: z = r(cos θ + i sin θ), where r = |z| and θ is the argument.
- For a quadratic ax² + bx + c = 0 with real coefficients, the discriminant D = b² − 4ac decides the nature of roots: if D < 0, the roots are complex conjugates of each other.
Formulas and facts to remember
- i² = −1, i³ = −i, i⁴ = 1; powers of i repeat with period 4.
- Addition: (a + ib) + (c + id) = (a + c) + i(b + d).
- Multiplication: (a + ib)(c + id) = (ac − bd) + i(ad + bc).
- Conjugate: if z = a + ib, then z̄ = a − ib.
- Modulus: |z| = √(a² + b²).
- |z|² = z × z̄.
- Division: (a + ib)/(c + id) = [(a + ib)(c − id)] / (c² + d²).
- Polar form: z = r(cos θ + i sin θ), with r = |z| and tan θ = b/a (adjust quadrant).
- Quadratic roots when D < 0: x = [−b ± i√(−D)] / (2a).
Worked examples
Example 1. Simplify (3 + 2i)(1 − 4i).
Use the multiplication rule: (a + ib)(c + id) = (ac − bd) + i(ad + bc).
Here a = 3, b = 2, c = 1, d = −4.
Real part: ac − bd = 3 × 1 − 2 × (−4) = 3 + 8 = 11.
Imaginary part: ad + bc = 3 × (−4) + 2 × 1 = −12 + 2 = −10.
Therefore (3 + 2i)(1 − 4i) = 11 − 10i.
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Example 2. Find the modulus and argument of z = −1 + i.
Modulus: |z| = √[(−1)² + 1²] = √(1 + 1) = √2.
For the argument, tan θ = (imaginary part)/(real part) = 1/(−1) = −1.
Since the real part is negative and the imaginary part is positive, the point lies in the second quadrant.
Reference angle where tan α = 1 is α = π/4.
In the second quadrant, θ = π − π/4 = 3π/4.
So |z| = √2 and arg(z) = 3π/4.
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Example 3. Solve x² + 4x + 13 = 0.
Here a = 1, b = 4, c = 13.
Discriminant D = b² − 4ac = 16 − 52 = −36.
Since D < 0, the roots are complex.
x = [−b ± √D] / (2a) = [−4 ± √(−36)] / 2 = [−4 ± 6i] / 2.
So x = −2 + 3i or x = −2 − 3i.
The two roots are complex conjugates.
Common mistakes
- Writing √(−4) = 2 instead of 2i → remember √(−k) = i√k for positive k.
- Forgetting that i² = −1 during multiplication and leaving i² in the answer → always replace i² with −1.
- Using tan⁻¹(b/a) directly without checking the quadrant → always plot the point mentally and adjust θ to the correct quadrant.
- Confusing modulus with absolute value of real part alone → modulus uses both parts: √(a² + b²).
- Dividing complex numbers by writing (a + ib)/(c + id) and stopping → multiply numerator and denominator by the conjugate c − id first.
Quick revision
- A complex number is a + ib; i² = −1.
- Conjugate flips the sign of the imaginary part; z × z̄ = |z|².
- Modulus |z| = √(a² + b²); argument θ is the angle from the positive real axis.
- Polar form: z = r(cos θ + i sin θ).
- Quadratic with D < 0 has two complex conjugate roots.
- Powers of i cycle: i, −1, −i, 1, i, …