H2 Mathematics · JC 1 and 2
Complex Numbers
Complex numbers extend the real numbers by introducing i, where i² = -1. Every complex number can be written as x + iy, and also in polar or exponential form, r(cos θ + i sin θ) or re^(iθ). In H2 Mathematics you calculate with complex numbers in all these forms, solve polynomial equations with real coefficients using the fact that non-real roots come in conjugate pairs, and represent numbers on an Argand diagram. Moving between forms is the key skill: addition is easiest in cartesian form, while multiplication and powers are easiest in exponential form.
What the syllabus expects
- Add, subtract, multiply and divide complex numbers in cartesian form x + iy, and find the conjugate z*
- Solve equations involving z and z* by equating real and imaginary parts
- Use the result that the non-real roots of a polynomial equation with real coefficients occur in conjugate pairs
- Find the modulus and the principal argument of a complex number, where -π < arg z ≤ π
- Convert between cartesian, polar and exponential forms
- Multiply and divide in polar or exponential form using |zw| = |z||w| and arg(zw) = arg z + arg w, and find powers of complex numbers
- Represent complex numbers on an Argand diagram and describe the geometrical effect of conjugating, adding and multiplying
Cartesian form and conjugates
A complex number z = x + iy has real part x and imaginary part y, both real. Its conjugate is z* = x - iy. Add and subtract by combining real parts and imaginary parts. Multiply by expanding and using i² = -1: (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 5 + i. To divide, multiply the top and bottom by the conjugate of the denominator, which makes the denominator real: (5 + i)/(1 - i) = (5 + i)(1 + i)/((1 - i)(1 + i)) = (5 + 5i + i + i²)/2 = (4 + 6i)/2 = 2 + 3i. Note that zz* = x² + y², which is always real. Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal. This turns one complex equation into two real equations, which is how you solve equations involving z and z*.
Polynomial equations and conjugate roots
If a polynomial equation has real coefficients, its non-real roots occur in conjugate pairs: if 2 + i is a root, so is 2 - i. Their sum is 4 and their product is (2 + i)(2 - i) = 5, so they are the roots of the real quadratic z² - 4z + 5 = 0. This lets you factorise a cubic or quartic once one complex root is known. For example, if 2 + i is a root of z³ - 5z² + 9z - 5 = 0, then z² - 4z + 5 is a factor, and comparing coefficients gives z³ - 5z² + 9z - 5 = (z² - 4z + 5)(z - 1). The roots are 2 + i, 2 - i and 1. The conjugate pair result only applies when every coefficient is real. If a coefficient involves i, as in z² + iz + 2 = 0, you cannot assume that the conjugate of a root is also a root.
Modulus, argument and the Argand diagram
On an Argand diagram, z = x + iy is the point (x, y). The modulus |z| = √(x² + y²) is its distance from the origin, and the argument arg z is the angle from the positive real axis, measured anticlockwise as positive. The principal argument lies in -π < θ ≤ π. Always plot the point to see its quadrant, then find the basic angle tan⁻¹|y/x| and adjust. For example, z = -1 + i√3 has |z| = 2 and lies in the second quadrant with basic angle π/3, so arg z = π - π/3 = 2π/3. For z = -1 - i√3, arg z = -2π/3. Geometrically, z* is the reflection of z in the real axis, and adding w to z moves the point z by the vector representing w. So |z - w| is the distance between the points representing z and w.
Polar and exponential forms
A number with modulus r and argument θ can be written z = r(cos θ + i sin θ) = re^(iθ). In this form, multiplying multiplies the moduli and adds the arguments: |zw| = |z||w| and arg(zw) = arg z + arg w. Dividing divides the moduli and subtracts the arguments. Powers follow the same rule: z^n = r^n e^(inθ). Multiplying z by e^(iα) rotates the point z anticlockwise about the origin through the angle α. Adjust any final argument into the principal range by adding or subtracting multiples of 2π. For example, if z = 2e^(i2π/3), then z³ = 8e^(i2π) = 8, a real number, and z⁴ = 16e^(i8π/3) = 16e^(i2π/3). Exponential form makes questions such as finding the smallest positive n for which z^n is real straightforward: z^n is real when n arg z is a multiple of π.
Worked example
The complex numbers z and w are given by z = √3 + i and w = 1 - i. (a) Find the modulus and argument of z and of w. (b) Express z³/w in the form re^(iθ), where r > 0 and -π < θ ≤ π, and hence in the form x + iy. (c) Find the smallest positive integer n for which (z³/w)^n is real, and state this real value.
- |z| = √(3 + 1) = 2, and z is in the first quadrant with tan θ = 1/√3, so arg z = π/6.
- |w| = √(1 + 1) = √2, and w is in the fourth quadrant, so arg w = -π/4.
- |z³/w| = 2³/√2 = 8/√2 = 4√2.
- arg(z³/w) = 3 arg z - arg w = π/2 + π/4 = 3π/4, which is already in the principal range. So z³/w = 4√2 e^(i3π/4).
- In cartesian form: 4√2(cos 3π/4 + i sin 3π/4) = 4√2(-√2/2 + i√2/2) = -4 + 4i.
- Check directly: z³ = 8e^(iπ/2) = 8i, and 8i/(1 - i) = 8i(1 + i)/2 = 4i + 4i² = -4 + 4i.
- (z³/w)^n has argument 3nπ/4, which is a multiple of π when 3n/4 is an integer. The smallest positive n is 4, giving (4√2)⁴ e^(i3π) = 1024 × (-1) = -1024.
Answer: (a) |z| = 2, arg z = π/6, |w| = √2, arg w = -π/4. (b) 4√2 e^(i3π/4) = -4 + 4i. (c) n = 4, and the value is -1024.
Common mistakes
- Finding the argument with tan⁻¹(y/x) on the calculator without checking the quadrant. For -1 - i, tan⁻¹(1) gives π/4, but the point is in the third quadrant, so arg(-1 - i) = -3π/4.
- Leaving the argument outside the principal range, such as 7π/6 instead of -5π/6, when the question asks for -π < θ ≤ π.
- Using the conjugate root result when the polynomial has a non-real coefficient. It only holds when all the coefficients are real.
- Writing |z + w| = |z| + |w|. Moduli multiply and divide neatly, so |zw| = |z||w| and |z/w| = |z|/|w|, but they do not add in general.
- Giving the imaginary part of 3 - 4i as -4i. The imaginary part is the real number -4.
- Equating real and imaginary parts before expanding every product, so that a term such as iz* is put in the wrong part. Expand fully first, then collect.
Exam tips
- Sketch every complex number on an Argand diagram before writing its argument. A few seconds of sketching prevents quadrant errors.
- Use exponential form for products, quotients and powers, and cartesian form for sums and for equating real and imaginary parts.
- In conjugate root questions, state that the coefficients are real before using the conjugate pair, since that reason is part of the method.
Practice questions
1. Express (3 + 4i)/(1 + 2i) in the form x + iy.
Show answer
11/5 - (2/5)i. Multiply the top and bottom by 1 - 2i: the numerator is 3 - 6i + 4i - 8i² = 11 - 2i and the denominator is 1 + 4 = 5.
2. Find the complex number z such that 2z + iz* = 4 + 5i.
Show answer
z = 1 + 2i. Let z = x + iy. Then 2x + 2iy + i(x - iy) = (2x + y) + i(x + 2y). Equating parts gives 2x + y = 4 and x + 2y = 5, so x = 1 and y = 2. Check: 2 + 4i + i(1 - 2i) = 2 + 4i + i + 2 = 4 + 5i.
3. Given that 1 - 2i is a root of z³ + z² + pz + 15 = 0, where p is real, find p and the other roots.
Show answer
p = -1, and the other roots are 1 + 2i and -3. The coefficients are real, so 1 + 2i is also a root, and (z - (1 - 2i))(z - (1 + 2i)) = z² - 2z + 5 is a factor. Write z³ + z² + pz + 15 = (z² - 2z + 5)(z + c). Comparing z² terms gives c - 2 = 1, so c = 3, and the constant 5c = 15 agrees. Comparing z terms gives p = 5 - 2c = -1. The third root is -3.
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