Complex numbers

Q: What is the imaginary unit \(i\), and what is the value of \(i^2\)?

The imaginary unit is defined as \(i = \sqrt{-1}\), which means \(i^2 = -1\).

Q: How do powers of \(i\) cycle, and how do you evaluate large integer powers of \(i\)?

Powers of \(i\) repeat every 4 steps: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). To evaluate \(i^n\), divide \(n\) by \(4\) and take \(i^{\text{remainder}}\).

Q: What is the Cartesian form of a complex number, and what are its real and imaginary parts?

The Cartesian form is \(z = a + bi\), where \(a, b \in \mathbb{R}\); \(\text{Re}(z) = a\) is the real part and \(\text{Im}(z) = b\) is the imaginary part.

Q: What is the complex conjugate of \(z = a + bi\), and what is the value of \(z z^*\)?

The complex conjugate is \(z^* = a - bi\), and their product is purely real: \(z z^* = a^2 + b^2\).

Q: How do you perform division of two complex numbers?

Multiply both the numerator and the denominator by the complex conjugate of the denominator to make the denominator purely real.

Q: What does the Conjugate Pairs Theorem state about polynomial equations with real coefficients?

If a polynomial has real coefficients, its non-real complex roots always occur in conjugate pairs: if \(z\) is a root, \(z^*\) must also be a root.

Q: What is an Argand diagram?

An Argand diagram is a 2D geometric plane where the horizontal axis represents the real part (\(\text{Re}\)) and the vertical axis represents the imaginary part (\(\text{Im}\)).

Q: How are the modulus and principal argument of a complex number defined?

The modulus is \(|z| = r = \sqrt{x^2 + y^2}\), and the principal argument \(\arg(z) = \theta\) is the angle with the positive real axis in the range \(-\pi

Q: What are the rules for multiplying and dividing complex numbers in modulus-argument form?

When multiplying, multiply the moduli and add the arguments. When dividing, divide the moduli and subtract the arguments.

Q: What locus is represented by the equation \(|z - z_1| = r\)?

A circle of radius \(r\) centred at the complex point \(z_1\).

Further Mathematics 2330 · CCEA A Level · 10 min read · Updated: 31 Aug 2026

Introduction: Unlocking the World of Complex Numbers

Welcome to Complex Numbers! Back in GCSE and standard A Level Maths, whenever you tried to find the square root of a negative number like \(\sqrt{-16}\), your calculator showed an error message. You were told: "You cannot take the square root of a negative number."

In Further Mathematics, we break through that limitation by defining a brand-new number: the imaginary unit, \(i\). While calling it "imaginary" makes it sound made-up, complex numbers are fundamental to modern technology—powering everything from electrical engineering and quantum physics to 3D computer graphics and flight navigation.

Don't worry if this seems unusual at first. Step by step, we will discover how these numbers work, how to calculate with them, and how to visualize them geometrically.

1. The Imaginary Unit and Cartesian Form

What is \(i\)?

We define the imaginary unit \(i\) as:

\(i = \sqrt{-1} \implies i^2 = -1\)

This allows us to evaluate the square root of any negative number. For example, \(\sqrt{-25} = \sqrt{25 \times (-1)} = \sqrt{25} \times \sqrt{-1} = 5i\).

Powers of \(i\)

A fun and helpful feature of \(i\) is that its powers follow a repeating four-step cycle:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = i^2 \times i = -1 \times i = -i\)
• \(i^4 = (i^2)^2 = (-1)^2 = 1\)

Memory Trick: To find any large power such as \(i^{27}\), divide the power by \(4\) and find the remainder. Since \(27 = 4 \times 6 + 3\), the remainder is \(3\), so \(i^{27} = i^3 = -i\).

The Standard (Cartesian) Form

A complex number is usually denoted by the letter \(z\) and written in the form:

\(z = a + bi\)

where \(a\) and \(b\) are real numbers (\(a, b \in \mathbb{R}\)).

• The Real Part of \(z\) is denoted as \(\text{Re}(z) = a\).
• The Imaginary Part of \(z\) is denoted as \(\text{Im}(z) = b\) (notice that \(\text{Im}(z)\) is just the real coefficient \(b\), not \(bi\)).

Example: If \(z = 4 - 7i\), then \(\text{Re}(z) = 4\) and \(\text{Im}(z) = -7\).

2. Basic Arithmetic Operations

Addition and Subtraction

To add or subtract complex numbers, simply collect the real parts together and the imaginary parts together—just like collecting like terms in standard algebra.

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

\(z_1 + z_2 = (a + c) + (b + d)i\)
\(z_1 - z_2 = (a - c) + (b - d)i\)

Example: If \(z_1 = 3 + 5i\) and \(z_2 = 1 - 2i\):
\(z_1 + z_2 = (3 + 1) + (5 - 2)i = 4 + 3i\)
\(z_1 - z_2 = (3 - 1) + (5 - (-2))i = 2 + 7i\)

Multiplication

Expand brackets using FOIL just like ordinary algebraic brackets, and remember to replace \(i^2\) with \(-1\).

Example: Multiply \((2 + 3i)(4 - 5i)\):
\((2 + 3i)(4 - 5i) = 8 - 10i + 12i - 15i^2\)
Since \(i^2 = -1\), the term \(-15i^2\) becomes \(-15(-1) = +15\):
\(= 8 + 2i + 15 = 23 + 2i\)

The Complex Conjugate

For any complex number \(z = a + bi\), its complex conjugate is denoted by \(z^*\) (or \(\bar{z}\)) and is formed by changing the sign of the imaginary part:

\(z^* = a - bi\)

Why is the conjugate important? When you multiply a complex number by its conjugate, the imaginary terms cancel out completely, leaving a purely real number:

\(z z^* = (a + bi)(a - bi) = a^2 - abi + abi - b^2 i^2 = a^2 - b^2(-1) = a^2 + b^2\)

Division

To divide complex numbers, write the problem as a fraction and multiply both the numerator and the denominator by the complex conjugate of the denominator (similar to rationalizing a surd denominator).

Step-by-Step Example: Simplify \(\frac{3 + 2i}{1 - 4i}\)

1. Identify the conjugate of the denominator: \((1 - 4i)^* = 1 + 4i\).
2. Multiply numerator and denominator by \(1 + 4i\):
\(\frac{3 + 2i}{1 - 4i} \times \frac{1 + 4i}{1 + 4i} = \frac{(3 + 2i)(1 + 4i)}{(1 - 4i)(1 + 4i)}\)
3. Expand the numerator: \((3)(1) + 12i + 2i + 8i^2 = 3 + 14i - 8 = -5 + 14i\).
4. Expand the denominator: \(1^2 + 4^2 = 1 + 16 = 17\).
5. Write in standard \(a + bi\) form: \(-\frac{5}{17} + \frac{14}{17}i\).

Equating Real and Imaginary Parts

Two complex numbers are equal if and only if their real parts are equal AND their imaginary parts are equal:

\(a + bi = c + di \iff a = c \text{ and } b = d\)

This allows us to solve equations with unknown variables by setting up a pair of simultaneous equations.

Key Takeaway:

Always replace \(i^2\) with \(-1\). When dividing, multiply top and bottom by the denominator's conjugate to make the denominator a real number.

3. Solving Polynomial Equations with Complex Roots

Quadratic Equations

When the discriminant of a quadratic equation is negative (\(b^2 - 4ac < 0\)), the quadratic has two complex roots.

Example: Solve \(z^2 - 6z + 13 = 0\).
Using the quadratic formula \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\(z = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(13)}}{2(1)} = \frac{6 \pm \sqrt{36 - 52}}{2} = \frac{6 \pm \sqrt{-16}}{2}\)
Since \(\sqrt{-16} = 4i\):
\(z = \frac{6 \pm 4i}{2} = 3 \pm 2i\)

The roots are \(z_1 = 3 + 2i\) and \(z_2 = 3 - 2i\).

The Conjugate Pairs Theorem

Notice how the roots above are conjugates of one another. For any polynomial equation with real coefficients, non-real roots always occur in conjugate pairs.

• If \(z = a + bi\) is a root, then \(z^* = a - bi\) MUST also be a root.

Higher-Degree Polynomials (Cubics and Quartics)

Example: Given that \(z = 2 - i\) is a root of the cubic equation \(z^3 - 3z^2 + z + 5 = 0\), find all other roots.

1. By the Conjugate Pairs Theorem, another root is \(z = 2 + i\).
2. Form the quadratic factor containing these two roots:
\((z - (2 - i))(z - (2 + i)) = ((z - 2) + i)((z - 2) - i) = (z - 2)^2 - i^2 = (z^2 - 4z + 4) - (-1) = z^2 - 4z + 5\)
3. Use algebraic long division or inspection to find the remaining linear factor:
\(z^3 - 3z^2 + z + 5 = (z^2 - 4z + 5)(z + 1) = 0\)
4. Therefore, the third root is \(z = -1\).
The complete set of roots is \(z = 2 - i\), \(z = 2 + i\), and \(z = -1\).

Key Takeaway:

If a polynomial has real coefficients, complex roots come in pairs: \(a + bi\) and \(a - bi\). A cubic has 3 roots (either 3 real, or 1 real and 2 complex conjugates).

4. The Argand Diagram

Just as real numbers can be plotted along a 1D number line, complex numbers can be plotted on a 2D plane known as an Argand Diagram.

• The horizontal axis is the Real Axis (\(\text{Re}\)).
• The vertical axis is the Imaginary Axis (\(\text{Im}\)).

A complex number \(z = a + bi\) is represented as the coordinate point \((a, b)\), or as a position vector extending from the origin \((0,0)\) to \((a, b)\).

• Complex conjugate: The point \(z^* = a - bi\) is the reflection of \(z = a + bi\) across the real axis.
• Addition: Adding two complex numbers corresponds to standard vector addition (the parallelogram rule).

5. Modulus-Argument (Polar) Form

Instead of Cartesian coordinates \((x, y)\), we can define a point using a distance from the origin and an angle.

1. The Modulus: \(|z|\) or \(r\)

The modulus of \(z = x + iy\) is the distance of the point from the origin. By Pythagoras' Theorem:

\(|z| = r = \sqrt{x^2 + y^2}\)

Note: The modulus is always a non-negative real number (\(r \ge 0\)).

2. The Argument: \(\arg(z)\) or \(\theta\)

The argument of \(z\) is the angle \(\theta\) that the vector makes with the positive real axis. By convention, we use the principal argument, which is measured in radians and restricted to:

\(-\pi < \theta \le \pi\) (or \(-180^\circ < \theta \le 180^\circ\))

• Angles measured anticlockwise from the positive real axis are positive (\(0\) to \(\pi\)).
• Angles measured clockwise from the positive real axis are negative (\(0\) to \(-\pi\)).

Finding the Argument in Different Quadrants

Always draw a quick sketch to see which quadrant your complex number lies in! Let \(\alpha = \arctan\left(\frac{|y|}{|x|}\right)\) be the acute reference angle with the horizontal axis:

• Quadrant 1 (\(x > 0, y > 0\)): \(\theta = \alpha\)
• Quadrant 2 (\(x < 0, y > 0\)): \(\theta = \pi - \alpha\)
• Quadrant 3 (\(x < 0, y < 0\)): \(\theta = -(\pi - \alpha) = -\pi + \alpha\)
• Quadrant 4 (\(x > 0, y < 0\)): \(\theta = -\alpha\)

Converting Cartesian to Modulus-Argument Form

Using trigonometry, \(x = r\cos\theta\) and \(y = r\sin\theta\). Substituting these into \(z = x + iy\) gives the Modulus-Argument Form:

\(z = r(\cos\theta + i\sin\theta)\)

Worked Example: Express \(z = -1 + i\sqrt{3}\) in modulus-argument form.

1. Find the modulus \(r\):
\(r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2\)
2. Identify quadrant: \(x = -1\) (negative) and \(y = \sqrt{3}\) (positive), so \(z\) is in Quadrant 2.
3. Find the acute reference angle: \(\alpha = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}\).
4. Calculate \(\theta\): \(\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\).
5. Write in polar form: \(z = 2\left(\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)\right)\).

Multiplication and Division in Modulus-Argument Form

When working in polar form, multiplication and division become remarkably simple:

Multiplication: Multiply the moduli, add the arguments.
\(|z_1 z_2| = |z_1||z_2|\)
\(\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)\)

Division: Divide the moduli, subtract the arguments.
\(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\)
\(\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)\)

Important Tip: If adding or subtracting angles takes your argument outside the interval \((-\pi, \pi]\), add or subtract \(2\pi\) to bring it back into the principal range.

Key Takeaway:

Modulus \(r\) is distance (\(\sqrt{x^2+y^2}\)); argument \(\theta\) is direction (\(-\pi < \theta \le \pi\)). When multiplying: multiply lengths, add angles. When dividing: divide lengths, subtract angles.

6. Loci in the Complex Plane

A locus (plural: loci) is a set of points in the Argand diagram that satisfy a given geometric rule.

1. Circle: \(|z - z_1| = r\)

The expression \(|z - z_1|\) represents the distance between the variable point \(z\) and the fixed point \(z_1\).

• Equation: \(|z - z_1| = r\)
• Geometric meaning: A circle with centre \(z_1\) and radius \(r\).
• Example: \(|z - (2 + 3i)| = 5\) is a circle centred at \((2, 3)\) with radius \(5\).
• Common Trap: For \(|z - 2 + 4i| = 3\), rewrite it as \(|z - (2 - 4i)| = 3\). The centre is \((2, -4)\), not \((-2, 4)\)!

2. Perpendicular Bisector: \(|z - z_1| = |z - z_2|\)

This states that the distance from \(z\) to \(z_1\) is equal to the distance from \(z\) to \(z_2\).

• Geometric meaning: The perpendicular bisector of the line segment joining the points \(z_1\) and \(z_2\).
• Every point on this straight line is equidistant from \(z_1\) and \(z_2\).

3. Half-Line (Ray): \(\arg(z - z_1) = \theta\)

• Geometric meaning: A half-line (or ray) starting at the point \(z_1\) and making an angle of \(\theta\) with the positive horizontal direction.
• The endpoint \(z_1\) is not included (draw an open circle at \(z_1\)) because the argument of zero is undefined.

Regions and Inequalities

Inequalities describe shaded regions on the Argand diagram:

• \(|z - z_1| \le r\): The region inside and on the boundary of the circle.
• \(|z - z_1| < |z - z_2|\): The region closer to \(z_1\) than to \(z_2\) (bounded by a dashed perpendicular bisector).
• Strict inequalities (\(<\) or \(>\)) use dashed lines; non-strict inequalities (\(\le\) or \(\ge\)) use solid lines.

7. Quick Review: Common Mistakes to Avoid

• Writing \(i^2\) as \(1\): Remember, \(i^2 = -1\). A sign error here will affect the whole question.
• Imaginary part syntax: For \(z = 3 + 4i\), \(\text{Im}(z) = 4\), not \(4i\).
• Quadrant confusion in \(\arg(z)\): Do not blindly trust \(\arctan\left(\frac{y}{x}\right)\) on your calculator. Always draw a quick sketch of the complex plane to check the quadrant!
• Principal argument range: Ensure \(\theta\) is in radians and falls strictly within \(-\pi < \theta \le \pi\).
• Signs in loci: Always factor out the negative sign: \(|z - a - bi| = |z - (a + bi)|\) to find the correct centre \((a, b)\).

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the imaginary unit \(i\), and what is the value of \(i^2\)?

The imaginary unit is defined as \(i = \sqrt{-1}\), which means \(i^2 = -1\).

How do powers of \(i\) cycle, and how do you evaluate large integer powers of \(i\)?

Powers of \(i\) repeat every 4 steps: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). To evaluate \(i^n\), divide \(n\) by \(4\) and take \(i^{\text{remainder}}\).

What is the Cartesian form of a complex number, and what are its real and imaginary parts?

The Cartesian form is \(z = a + bi\), where \(a, b \in \mathbb{R}\); \(\text{Re}(z) = a\) is the real part and \(\text{Im}(z) = b\) is the imaginary part.

What is the complex conjugate of \(z = a + bi\), and what is the value of \(z z^*\)?

The complex conjugate is \(z^* = a - bi\), and their product is purely real: \(z z^* = a^2 + b^2\).

How do you perform division of two complex numbers?

Multiply both the numerator and the denominator by the complex conjugate of the denominator to make the denominator purely real.

What does the Conjugate Pairs Theorem state about polynomial equations with real coefficients?

If a polynomial has real coefficients, its non-real complex roots always occur in conjugate pairs: if \(z\) is a root, \(z^*\) must also be a root.

What is an Argand diagram?

An Argand diagram is a 2D geometric plane where the horizontal axis represents the real part (\(\text{Re}\)) and the vertical axis represents the imaginary part (\(\text{Im}\)).

How are the modulus and principal argument of a complex number defined?

The modulus is \(|z| = r = \sqrt{x^2 + y^2}\), and the principal argument \(\arg(z) = \theta\) is the angle with the positive real axis in the range \(-\pi < \theta \le \pi\).

What are the rules for multiplying and dividing complex numbers in modulus-argument form?

When multiplying, multiply the moduli and add the arguments. When dividing, divide the moduli and subtract the arguments.

What locus is represented by the equation \(|z - z_1| = r\)?

A circle of radius \(r\) centred at the complex point \(z_1\).

What locus is represented by \(|z - z_1| = |z - z_2|\)?

The perpendicular bisector of the line segment joining the points \(z_1\) and \(z_2\).

What locus is represented by \(\arg(z - z_1) = \theta\)?

A half-line (ray) starting at \(z_1\) (not included) at an angle \(\theta\) to the positive horizontal direction.

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