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1. Algebraic Operations0%

Complex Numbers · Topic 1 of 5

1. Algebraic Operations

Video coming soon5 worked examples

Theory

A complex number has the form $z = a + bi$ , where $i^2 = -1$ ; $a$ is the real part and $b$ the imaginary part. Add and subtract by combining real and imaginary parts; multiply by expanding and using $i^2 = -1$ .

The complex conjugate of $z = a + bi$ is $\bar{z} = a - bi$ , and their product is real:

z\bar{z} = (a+bi)(a-bi) = a^2 + b^2

To divide, multiply the numerator and denominator by the conjugate of the denominator, which makes the denominator real.

Plotting on an Argand Diagram

Complex numbers are represented geometrically on an Argand diagram. It looks like ordinary $x$ – $y$ axes, but the horizontal axis is the real axis and the vertical axis is the imaginary axis. The number $z = a + bi$ is plotted at the point $(a,\,b)$ .

So $3 + 2i$ sits 3 right and 2 up; $-1 - 4i$ sits 1 left and 4 down. A real number such as $5$ lies on the horizontal axis, and a purely imaginary number such as $3i$ lies on the vertical axis.

Two geometric facts follow immediately and are worth knowing, because questions lean on them:

The conjugate $\bar{z}$ is the reflection of $z$ in the real axis — same distance from the origin, opposite side.
Adding two complex numbers behaves exactly like adding vectors: go along and up for one, then along and up for the other.

This diagram is the foundation for modulus and argument (the distance from the origin and the angle from the positive real axis), for loci, and for the pattern made by the roots of an equation — so being able to plot quickly and accurately pays off throughout the topic.

The Golden Rule: to divide complex numbers, multiply top and bottom by the conjugate of the denominator — this clears $i$ from the bottom.

⚠️ Common Examiner Traps

Forgetting $i^2 = -1$ : the $i^2$ term becomes a real number and changes the real part.
Conjugating the wrong part: when dividing, it is the denominator you conjugate, not the numerator.
Mixing parts: keep real and imaginary terms separate all the way to the final $a + bi$ form.

Worked examples

Example 1

Evaluate $(3 + 2i)(1 - 4i)$ .

Step 1: Expand the brackets:

(3+2i)(1-4i) = 3 - 12i + 2i - 8i^2

Step 2: Replace $i^2$ with $-1$ and simplify:

= 3 - 10i - 8(-1) = 3 - 10i + 8 = 11 - 10i

Example 2

Express $\dfrac{5 + i}{2 - 3i}$ in the form $a + bi$ .

Step 1: Multiply numerator and denominator by the conjugate of the denominator, $2 + 3i$ :

\frac{5+i}{2-3i} \times \frac{2+3i}{2+3i} = \frac{(5+i)(2+3i)}{(2-3i)(2+3i)}

Step 2: Expand top and bottom:

= \frac{10 + 15i + 2i + 3i^2}{4 + 9} = \frac{7 + 17i}{13}

Step 3: Write in $a + bi$ form:

= \frac{7}{13} + \frac{17}{13}i

Example 3

Find the square roots of $3 + 4i$ .

Step 1: Let $\sqrt{3+4i} = a + bi$ . Squaring and comparing parts:

(a+bi)^2 = a^2 - b^2 + 2abi = 3 + 4i \implies a^2 - b^2 = 3, \quad 2ab = 4

Step 2: From $2ab = 4$ we get $b = \dfrac{2}{a}$ . Substitute:

a^2 - \frac{4}{a^2} = 3 \implies a^4 - 3a^2 - 4 = 0 \implies (a^2 - 4)(a^2 + 1) = 0

Step 3: Since $a$ is real, $a^2 = 4$ , so $a = \pm 2$ with $b = \pm 1$ to match. The square roots are:

2 + i \quad \text{and} \quad -2 - i

Example 4

The complex number $z = -3 + 4i$ . Describe the position of $z$ , $\bar{z}$ and $-z$ on an Argand diagram, stating the quadrant each lies in.

Step 1: Read off the real and imaginary parts. For $z = -3 + 4i$ , the real part is $-3$ and the imaginary part is $4$ , so $z$ is plotted at the point $(-3,\,4)$ — 3 units left along the real axis and 4 units up the imaginary axis. That is the second quadrant.

Step 2: The conjugate changes the sign of the imaginary part only:

\bar{z} = -3 - 4i \quad \to \quad (-3,\,-4)

This is the reflection of $z$ in the real axis, in the third quadrant.

Step 3: Negating changes the sign of both parts:

-z = 3 - 4i \quad \to \quad (3,\,-4)

This is $z$ rotated through $180^\circ$ about the origin, in the fourth quadrant.

Step 4: Note all three lie the same distance from the origin, since $|z| = |\bar{z}| = |-z| = \sqrt{(-3)^2 + 4^2} = 5$ . All three points sit on a circle of radius 5 centred at the origin.

Example 5

🔗 Bringing it together: On an Argand diagram, the points $A$ and $B$ represent $z_1 = 5 + 2i$ and $z_2 = 1 + 8i$ . Find the complex number represented by the midpoint of $AB$ , and find the distance $AB$ .

Step 1: Plot the points. $A$ is at $(5,\,2)$ and $B$ is at $(1,\,8)$ — both in the first quadrant.

Step 2: Because the Argand diagram uses ordinary coordinates, the midpoint is found the ordinary way — average the real parts and average the imaginary parts:

\text{midpoint} = \frac{z_1 + z_2}{2} = \frac{(5+2i) + (1+8i)}{2} = \frac{6 + 10i}{2} = 3 + 5i

So the midpoint is the point $(3,\,5)$ , representing $3 + 5i$ .

Step 3: The distance between two points on the diagram is the modulus of their difference:

z_1 - z_2 = (5 - 1) + (2 - 8)i = 4 - 6i

AB = |z_1 - z_2| = \sqrt{4^2 + (-6)^2} = \sqrt{52} = 2\sqrt{13}

This is exactly Pythagoras on the diagram, and it is why a locus such as $|z - a| = k$ is a circle: it says “all points a fixed distance $k$ from the point $a$ ”.