Complex numbers

Q: What is the definition of the imaginary unit \(i\), and what pattern do its integer powers follow?

The imaginary unit \(i\) is defined by \(i^2 = -1\) (or \(i = \sqrt{-1}\)), and its powers cycle every four values: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).

Q: How is a complex number defined in Cartesian form, and when are two complex numbers equal?

A complex number is written as \(z = x + iy\) with real numbers \(x = \text{Re}\,z\) and \(y = \text{Im}\,z\); two complex numbers are equal if and only if both their real parts and their imaginary parts are equal.

Q: How do you perform addition, subtraction, and multiplication on complex numbers in Cartesian form?

Add/subtract by combining real and imaginary parts separately; multiply by expanding brackets as binomials and replacing \(i^2\) with \(-1\).

Q: What is the complex conjugate of \(z = x + iy\), and how is it used in division?

The conjugate is \(z^* = x - iy\); to divide \(\frac{z_1}{z_2}\), multiply numerator and denominator by \(z_2^*\) to make the denominator real (\(z_2 z_2^* = x_2^2 + y_2^2\)).

Q: What is the conjugate roots theorem for polynomial equations with real coefficients?

If a polynomial equation with real coefficients has a non-real root \(z = a + ib\), then its complex conjugate \(z^* = a - ib\) is also a root.

Q: How is a complex number represented geometrically on an Argand diagram?

A complex number \(z = x + iy\) is plotted as the point \((x, y)\) or as a position vector from the origin on a plane where the horizontal axis is real (\(\text{Re}\)) and the vertical axis is imaginary (\(\text{Im}\)).

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

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

Q: What are the polar and exponential forms of a complex number?

Polar form is \(z = r(\cos\theta + i\sin\theta)\) and exponential form is \(z = re^{i\theta}\), where \(r = |z|\) and \(\theta = \arg z\).

Q: How do you multiply and divide complex numbers in exponential or polar form?

For multiplication, multiply the moduli and add the arguments (\(r_1 r_2 e^{i(\theta_1 + \theta_2)}\)); for division, divide the moduli and subtract the arguments (\(\frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}\)).

Q: What is the algebraic method for finding the two square roots of a complex number \(a + ib\)?

Set \((x + iy)^2 = a + ib\), equate real and imaginary parts to obtain \(x^2 - y^2 = a\) and \(2xy = b\), and solve for the real pairs \((x, y)\) giving \(z = \pm(x_0 + iy_0)\).

Practice

Mathematics (9709) · Cambridge International A Level · 6 min read · Updated: 29 Aug 2026

Study Notes: Pure Mathematics 3 (Paper 3)
Chapter 3.9: Complex Numbers

Welcome to the world of Complex Numbers! This chapter connects algebra, geometry, and trigonometry, providing a powerful mathematical tool to solve equations that cannot be solved in the real number system.

Our journey begins with the concept that makes complex numbers possible: the imaginary unit.

1. The Basics: Introducing \(i\) and Cartesian Form

Before complex numbers, equations like \(x^2 + 1 = 0\) had no solution in the real number system. Complex numbers solve this problem by defining a new unit.

The Imaginary Unit \(i\)

The Imaginary Unit, denoted \(i\), is defined such that:
\(i^2 = -1\)
\(i = \sqrt{-1}\)

From this definition, the integer powers of \(i\) cycle in a repeating pattern of four:
\(i^1 = i\)
\(i^2 = -1\)
\(i^3 = -i\)
\(i^4 = 1\)

Definition and Key Terminology (Cartesian Form)

A complex number, typically denoted by \(z\), is written in Cartesian form as:

\(z = x + iy\)

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

• The Real Part of \(z\) is \(x\). (Notation: \(\text{Re}\,z = x\))
• The Imaginary Part of \(z\) is \(y\). (Notation: \(\text{Im}\,z = y\). Note: The imaginary part is the real number \(y\), not \(iy\)!)
• Two complex numbers \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\) are equal (\(z_1 = z_2\)) if and only if their real parts are equal and their imaginary parts are equal: \(x_1 = x_2\) and \(y_1 = y_2\).

1.1 Arithmetic Operations in Cartesian Form

Performing operations on complex numbers in Cartesian form (\(x + iy\)) is similar to standard algebraic manipulation, using the identity \(i^2 = -1\).

A. Addition and Subtraction: Add or subtract the real parts and the imaginary parts separately.

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

B. Multiplication: Expand as binomials and substitute \(-1\) for \(i^2\).

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

C. Division and the Complex Conjugate:

The Complex Conjugate (\(z^*\))

The complex conjugate of \(z = x + iy\) is \(z^* = x - iy\). You change the sign of the imaginary part.

• Multiplying a complex number by its conjugate always produces a real number:
\(z z^* = (x + iy)(x - iy) = x^2 - i^2 y^2 = x^2 + y^2\)

Division Step-by-Step

To divide \(z_1\) by \(z_2\), multiply both numerator and denominator by the conjugate of the denominator, \(z_2^*\):

\(\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*}\)

Example: Simplify \(\frac{3 + 4i}{1 - i}\):
\(\frac{3 + 4i}{1 - i} = \frac{(3 + 4i)(1 + i)}{(1 - i)(1 + i)} = \frac{3 + 3i + 4i + 4i^2}{1^2 + 1^2} = \frac{-1 + 7i}{2} = -\frac{1}{2} + \frac{7}{2}i\)

1.2 Polynomial Equations and Conjugate Roots

• For any polynomial equation with real coefficients, any non-real roots occur in conjugate pairs.
• If \(z = a + ib\) (where \(b \neq 0\)) is a root of \(P(z) = 0\) with real coefficients, then \(z^* = a - ib\) is also a root.
• Example: If \(z = 1 + 2i\) is a root of a cubic equation with real coefficients, \(z^* = 1 - 2i\) must also be a root, meaning \((z - (1 + 2i))(z - (1 - 2i)) = z^2 - 2z + 5\) is a quadratic factor.

2. Geometric Representation: The Argand Diagram

Complex numbers are plotted geometrically on an Argand diagram:

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

The complex number \(z = x + iy\) is represented either by the point \((x, y)\) or by the position vector from the origin to \((x, y)\).

2.1 Modulus and Argument

A. Modulus (\(|z|\) or \(r\)):
The modulus is the distance from the origin to the point \(z\):

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

B. Argument (\(\arg z\) or \(\theta\)):
The argument is the angle in radians from the positive real axis to the position vector of \(z\).

• The principal argument is taken in the range \(-\pi < \theta \le \pi\).
• To calculate \(\arg z\), first find the acute reference angle \(\alpha = \arctan\left|\frac{y}{x}\right|\), then determine the quadrant based on the signs of \(x\) and \(y\):
- Quadrant 1 (\(x > 0, y > 0\)): \(\theta = \alpha\)
- Quadrant 2 (\(x < 0, y > 0\)): \(\theta = \pi - \alpha\)
- Quadrant 3 (\(x < 0, y < 0\)): \(\theta = -(\pi - \alpha) = \alpha - \pi\)
- Quadrant 4 (\(x > 0, y < 0\)): \(\theta = -\alpha\)

3. Polar and Exponential Forms (\(re^{i\theta}\))

Using modulus \(r\) and argument \(\theta\), complex numbers can be expressed in:

• Polar Form: \(z = r(\cos\theta + i\sin\theta)\)
• Exponential Form: \(z = re^{i\theta}\)

3.1 Multiplication and Division in Polar/Exponential Form

For \(z_1 = r_1 e^{i\theta_1}\) and \(z_2 = r_2 e^{i\theta_2}\):

A. Multiplication: Multiply the moduli and add the arguments.
\(z_1 z_2 = (r_1 r_2)e^{i(\theta_1 + \theta_2)}\)
• \(|z_1 z_2| = |z_1||z_2| = r_1 r_2\)
• \(\arg(z_1 z_2) = \arg z_1 + \arg z_2 = \theta_1 + \theta_2\) (adjusted to the range \(-\pi < \theta \le \pi\) if necessary)

B. Division: Divide the moduli and subtract the arguments.
\(\frac{z_1}{z_2} = \left(\frac{r_1}{r_2}\right)e^{i(\theta_1 - \theta_2)}\)
• \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} = \frac{r_1}{r_2}\)
• \(\arg\left(\frac{z_1}{z_2}\right) = \arg z_1 - \arg z_2 = \theta_1 - \theta_2\)

4. Square Roots, Loci, and Geometric Effects

4.1 Finding the Two Square Roots of a Complex Number

To find \(\sqrt{a + ib}\), let the square root be \(z = x + iy\):
1. Set \((x + iy)^2 = a + ib\).
2. Expand: \(x^2 - y^2 + 2ixy = a + ib\).
3. Equate real and imaginary parts:
\(x^2 - y^2 = a\) (Real part)
\(2xy = b\) (Imaginary part)
4. Substitute \(y = \frac{b}{2x}\) into \(x^2 - y^2 = a\) to obtain a quadratic equation in \(x^2\).
5. Solve for real values of \(x\) and compute the corresponding values of \(y\). The two square roots will be opposites: \(z = \pm(x_0 + iy_0)\).

4.2 Loci and Inequalities in the Argand Diagram

Let \(z\) represent a variable point, and let \(a\) and \(b\) be fixed complex numbers representing points \(A\) and \(B\).

1. Circle and Circular Region:
• \(|z - a| = k\) represents a circle with centre \(a\) and radius \(k\).
• \(|z - a| \le k\) represents the closed circular disc (the boundary circle plus its interior region).
• \(|z - a| < k\) represents the open interior region (drawn with a broken/dashed boundary circle).

2. Perpendicular Bisector and Half-Planes:
• \(|z - a| = |z - b|\) represents the perpendicular bisector of the line segment connecting \(A\) and \(B\).
• \(|z - a| \le |z - b|\) represents the half-plane containing all points closer to (or equidistant from) \(A\) than \(B\).

3. Half-Line (Ray) and Angular Regions:
• \(\arg(z - a) = \alpha\) represents a half-line (ray) originating at \(A\) making an angle \(\alpha\) with the positive real direction (the point \(A\) is excluded).
• \(\alpha \le \arg(z - a) \le \beta\) represents the sector-like region between rays of angles \(\alpha\) and \(\beta\) originating from \(A\).

4.3 Geometrical Effects of Operations

• Conjugation (\(z \to z^*\)): Reflection in the real axis.
• Addition / Subtraction (\(z_1 \pm z_2\)): Vector addition or subtraction following the parallelogram rule.
• Multiplication by \(re^{i\theta}\): Scaling by a factor of \(r\) and anticlockwise rotation by angle \(\theta\) about the origin.
• Division by \(re^{i\theta}\): Scaling by a factor of \(\frac{1}{r}\) and clockwise rotation by angle \(\theta\) about the origin.

References

Cambridge International AS & A Level Mathematics (9709) — Cambridge Assessment International Education
Cambridge International AS & A Level Mathematics (9709) Syllabus for Examination in 2026 and 2027 — Cambridge Assessment International Education

Quick check

Can you answer these now?

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

What is the definition of the imaginary unit \(i\), and what pattern do its integer powers follow?

The imaginary unit \(i\) is defined by \(i^2 = -1\) (or \(i = \sqrt{-1}\)), and its powers cycle every four values: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).

How is a complex number defined in Cartesian form, and when are two complex numbers equal?

A complex number is written as \(z = x + iy\) with real numbers \(x = \text{Re}\,z\) and \(y = \text{Im}\,z\); two complex numbers are equal if and only if both their real parts and their imaginary parts are equal.

How do you perform addition, subtraction, and multiplication on complex numbers in Cartesian form?

Add/subtract by combining real and imaginary parts separately; multiply by expanding brackets as binomials and replacing \(i^2\) with \(-1\).

What is the complex conjugate of \(z = x + iy\), and how is it used in division?

The conjugate is \(z^* = x - iy\); to divide \(\frac{z_1}{z_2}\), multiply numerator and denominator by \(z_2^*\) to make the denominator real (\(z_2 z_2^* = x_2^2 + y_2^2\)).

What is the conjugate roots theorem for polynomial equations with real coefficients?

If a polynomial equation with real coefficients has a non-real root \(z = a + ib\), then its complex conjugate \(z^* = a - ib\) is also a root.

How is a complex number represented geometrically on an Argand diagram?

A complex number \(z = x + iy\) is plotted as the point \((x, y)\) or as a position vector from the origin on a plane where the horizontal axis is real (\(\text{Re}\)) and the vertical axis is imaginary (\(\text{Im}\)).

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

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

What are the polar and exponential forms of a complex number?

Polar form is \(z = r(\cos\theta + i\sin\theta)\) and exponential form is \(z = re^{i\theta}\), where \(r = |z|\) and \(\theta = \arg z\).

How do you multiply and divide complex numbers in exponential or polar form?

For multiplication, multiply the moduli and add the arguments (\(r_1 r_2 e^{i(\theta_1 + \theta_2)}\)); for division, divide the moduli and subtract the arguments (\(\frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}\)).

What is the algebraic method for finding the two square roots of a complex number \(a + ib\)?

Set \((x + iy)^2 = a + ib\), equate real and imaginary parts to obtain \(x^2 - y^2 = a\) and \(2xy = b\), and solve for the real pairs \((x, y)\) giving \(z = \pm(x_0 + iy_0)\).

What geometric loci are represented by \(|z - a| = k\), \(|z - a| = |z - b|\), and \(\arg(z - a) = \alpha\)?

\(|z - a| = k\) is a circle of centre \(a\) and radius \(k\); \(|z - a| = |z - b|\) is the perpendicular bisector of segment \(AB\); \(\arg(z - a) = \alpha\) is a ray from \(a\) at angle \(\alpha\) (excluding \(a\)).

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Study Notes: Pure Mathematics 3 (Paper 3)Chapter 3.9: Complex Numbers