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Complex Numbers — Complex Numbers Review

This review brings together the main Year 13 Cambridge 9709 ideas on complex numbers: imaginary numbers, Cartesian form, arithmetic, modulus and argument, polar form, roots of equations, and the cube roots of unity.

In exam questions, it is important to write complex numbers clearly, keep real and imaginary parts separate, and choose the correct form of a complex number for the task.

Key definitions and formulae

1. Imaginary unit and Cartesian form

The imaginary unit is defined by

\[ i^2=-1. \]

A complex number in Cartesian form is written as

\[ z=x+iy, \]

where \(x\) and \(y\) are real numbers.

2. Arithmetic operations

If \( z_1=a+bi \) and \( z_2=c+di, \) then

\[ z_1+z_2=(a+c)+(b+d)i \] \[ z_1-z_2=(a-c)+(b-d)i \] \[ z_1z_2=(ac-bd)+(ad+bc)i. \]

For division, multiply top and bottom by the conjugate of the denominator:

\[ \frac{z_1}{z_2} = \frac{a+bi}{c+di} \times \frac{c-di}{c-di} \] \[ = \frac{(ac+bd)+(bc-ad)i}{c^2+d^2}. \]

The denominator becomes real because \[ (c+di)(c-di)=c^2+d^2. \]

3. Conjugate

The conjugate of \( z=x+iy \) is

\[ \overline{z}=x-iy. \]

Useful results:

\[ z+\overline{z}=2x, \qquad z\overline{z}=x^2+y^2. \]

4. Modulus and argument

For \( z=x+iy, \) the modulus is

\[ |z|=\sqrt{x^2+y^2}. \]

The argument is the angle \( \theta \) made with the positive real axis on an Argand diagram. It is found using

\[ \tan \theta=\frac{y}{x}, \]

but the correct quadrant must always be checked.

5. Product and quotient of complex numbers in modulus-argument form

If \( z_1=r_1(\cos \theta_1+i\sin \theta_1) \) and \( z_2=r_2(\cos \theta_2+i\sin \theta_2), \) then

\[ |z_1z_2|=r_1r_2=|z_1||z_2| \] \[ \arg(z_1z_2)=\theta_1+\theta_2=\arg z_1+\arg z_2 \] \[ \left|\frac{z_1}{z_2}\right|=\frac{r_1}{r_2}=\frac{|z_1|}{|z_2|} \] \[ \arg\!\left(\frac{z_1}{z_2}\right)=\theta_1-\theta_2=\arg z_1-\arg z_2. \]

6. Polar forms

The modulus-argument form is

\[ z=r(\cos \theta+i\sin \theta). \]

The exponential form is

\[ z=re^{i\theta}. \]

These forms are especially useful for multiplication, division, and roots.

7. Roots of equations and conjugate pairs

If a polynomial has real coefficients, any non-real complex roots occur in complex conjugate pairs.

  • Quadratic equations have 2 real roots or 2 complex roots of the form \(x\pm iy\), where \(y\neq 0\).
  • Cubic equations have 3 real roots, or 1 real root and 2 complex roots of the form \(x\pm iy\), where \(y\neq 0\).
  • Quartic equations have 4 real roots, or 2 real roots and 2 complex roots of the form \(x\pm iy\), or 4 complex roots occurring as two conjugate pairs.

8. The cube roots of unity

The solutions of \( z^3=1 \) are

\[ z=1,\qquad z=\frac{-1+i\sqrt{3}}{2},\qquad z=\frac{-1-i\sqrt{3}}{2}. \]

These are equally spaced on the unit circle in the Argand diagram.

Worked examples

Example 1: Addition, subtraction, and multiplication

Let \( z_1=3+2i \) and \( z_2=4-5i. \) Find \(z_1+z_2\), \(z_1-z_2\), and \(z_1z_2\).

Solution

\[ z_1+z_2=(3+4)+(2-5)i=7-3i \] \[ z_1-z_2=(3-4)+(2+5)i=-1+7i \] \[ z_1z_2=(3+2i)(4-5i) \] \[ =12-15i+8i-10i^2 \] \[ =12-7i+10 \] \[ =22-7i. \]
\[ z_1+z_2=7-3i,\qquad z_1-z_2=-1+7i,\qquad z_1z_2=22-7i. \]

Example 2: Division in the form \(a+ib\)

Express \( \dfrac{3+4i}{1-2i} \) in the form \(a+ib\).

Solution

\[ \frac{3+4i}{1-2i}\times \frac{1+2i}{1+2i} = \frac{(3+4i)(1+2i)}{(1-2i)(1+2i)} \] \[ = \frac{3+6i+4i+8i^2}{1+4} = \frac{-5+10i}{5} \] \[ =-1+2i. \]
\[ \frac{3+4i}{1-2i}=-1+2i. \]

Example 3: Modulus and argument

For \( z=-1+\sqrt{3}i, \) find \( |z| \) and \( \arg z \).

Solution

Modulus:

\[ |z|=\sqrt{(-1)^2+(\sqrt{3})^2}=\sqrt{1+3}=2. \]

The point lies in the second quadrant, so

\[ \arg z=\frac{2\pi}{3}. \]
\[ |z|=2,\qquad \arg z=\frac{2\pi}{3}. \]

Example 4: Convert to polar and exponential form

Write \( z=1+i \) in modulus-argument form and exponential form.

Solution

First find the modulus:

\[ |z|=\sqrt{1^2+1^2}=\sqrt{2}. \]

The argument is

\[ \arg z=\frac{\pi}{4}. \]

So

\[ z=\sqrt{2}\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right) \] \[ z=\sqrt{2}\,e^{i\pi/4}. \]
The modulus-argument form is \[ \sqrt{2}\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right) \] and the exponential form is \[ \sqrt{2}\,e^{i\pi/4}. \]

Example 5: Product and quotient of moduli and arguments

Suppose \( |z_1|=3,\ \arg z_1=\frac{\pi}{6} \) and \( |z_2|=2,\ \arg z_2=\frac{\pi}{3}. \) Find \( |z_1z_2| \), \( \arg(z_1z_2) \), \( \left|\dfrac{z_1}{z_2}\right| \), and \( \arg\!\left(\dfrac{z_1}{z_2}\right) \).

Solution

\[ |z_1z_2|=|z_1||z_2|=3\times 2=6 \] \[ \arg(z_1z_2)=\frac{\pi}{6}+\frac{\pi}{3}=\frac{\pi}{2} \] \[ \left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}=\frac{3}{2} \] \[ \arg\!\left(\frac{z_1}{z_2}\right)=\frac{\pi}{6}-\frac{\pi}{3}=-\frac{\pi}{6}. \]
\[ |z_1z_2|=6,\qquad \arg(z_1z_2)=\frac{\pi}{2}, \] \[ \left|\frac{z_1}{z_2}\right|=\frac{3}{2},\qquad \arg\!\left(\frac{z_1}{z_2}\right)=-\frac{\pi}{6}. \]

Example 6: Complex conjugate roots

Solve

\[ z^2-6z+13=0. \]

Solution

\[ z=\frac{6\pm\sqrt{36-52}}{2} \] \[ =\frac{6\pm\sqrt{-16}}{2} \] \[ =\frac{6\pm 4i}{2} \] \[ =3\pm 2i. \]
The roots are \[ 3+2i \quad \text{and} \quad 3-2i, \] a conjugate pair.

Example 7: The cube roots of unity

Solve

\[ z^3=1. \]

Solution

The cube roots of unity are

\[ z=1,\qquad z=\frac{-1+i\sqrt{3}}{2},\qquad z=\frac{-1-i\sqrt{3}}{2}. \]
These three roots lie on the unit circle and are equally spaced by angles of \( \dfrac{2\pi}{3} \).

Common mistakes and exam tips

Common mistakes

  • Forgetting that \(i^2=-1\), not \(1\).
  • Not simplifying the final answer into the form \(a+ib\).
  • Using the wrong denominator when dividing complex numbers. It should be \(c^2+d^2\).
  • Finding the argument from \(\tan^{-1}(y/x)\) without checking the quadrant.
  • Confusing modulus with argument.
  • Forgetting that non-real roots of equations with real coefficients occur in conjugate pairs.

Exam tips

  • Keep real and imaginary parts separate throughout your working.
  • For division, write the conjugate step clearly.
  • Sketch a small Argand diagram when finding argument or interpreting polar form.
  • When using modulus-argument form, remember: multiply moduli and add arguments; divide moduli and subtract arguments.
  • For roots of unity, think of equally spaced points on the unit circle.

Summary

\[ i^2=-1 \] \[ z=x+iy \] \[ |z|=\sqrt{x^2+y^2} \] \[ z=r(\cos\theta+i\sin\theta)=re^{i\theta} \] \[ |z_1z_2|=|z_1||z_2|,\qquad \arg(z_1z_2)=\arg z_1+\arg z_2 \] \[ \left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|},\qquad \arg\!\left(\frac{z_1}{z_2}\right)=\arg z_1-\arg z_2 \]

The main Year 13 complex number skills are working accurately in Cartesian form, using modulus and argument confidently, converting to polar forms, and recognising the pattern of complex roots.

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