Ch 9 · 3.9 Complex Numbers · A Level Mathematics 9709
Topic 9 of 9 in Pure Mathematics 2 and 3 — A Level Mathematics 9709 — 88 lessons.
Introduction
x^2 + 1 = 0 has no real solution, so mathematicians invented a number i with i^2 = -1. The resulting complex numbers turned out to be essential in electrical engineering (alternating current), signal processing and quantum physics. Every polynomial equation has complex roots, and complex numbers can be drawn as points, so algebra and geometry meet. (Syllabus section 3.9, Paper 3 only.)
Key ideas — The language of complex numbers
A complex number is z = x + iy with x, y real and i^2 = -1.
🧑🏫 Teacher's explanation
Think of a complex number not as a single value on a line, but as a coordinate on a map. In standard arithmetic, we only move left or right on a 1D number line. Complex numbers add a second dimension, allowing us to navigate a 2D plane. We represent this using x for the horizontal (real) direction and y for the vertical (imaginary) direction.
📖 Definitions
🔍 Worked example — Easy
Given z = 3 - 4i, find Re z, Im z, |z|, and z^*.
🔍 Worked example — Standard
Solve for x and y (real numbers) in the equation (x + iy)(2 - i) = 5 + i.
🔍 Worked example — Exam-style
Find the two square roots of z = -5 + 12i in the form a + bi.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Arithmetic in Cartesian form
Add and subtract parts separately. Multiply by expanding and using i^2 = -1: (3 + 4i)(1 - 2i) = 3 - 6i + 4i - 8i^2 = 11 - 2i. To divide, multiply top and bottom by the conjugate of the denominator, since zz^* = |z|^2 is real:
🧑🏫 Teacher's explanation
Think of a complex number z = x + iy as a set of instructions for a movement on a 2D map. The real part x tells you how far to move left or right, and the imaginary part y tells you how far to move up or down. Adding complex numbers is just like combining these instructions: if you walk 3 steps East and 2 steps North, then walk 1 step East and 4 steps North, you end up at 4 steps East and 6 steps North.
📖 Definitions
🔍 Worked example — Easy
Simplify (5 - 2i) + (3 + 7i) and (4 + i) - (2 - 3i).
🔍 Worked example — Exam-style
Express 1 + 5i/2 - i in the form x + iy.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Conjugate pairs of roots
If a polynomial has real coefficients and a + ib (b ≠ 0) is a root, then a - ib is also a root. Together they give the real quadratic factor (x - (a + ib))(x - (a - ib)) = x^2 - 2ax + (a^2 + b^2). Divide it out to find the other roots. This is how to solve a cubic or quartic equation when one complex root is given.
🧑🏫 Teacher's explanation
Imagine you are looking at a mirror. If you stand in front of it, your reflection looks exactly like you, but everything is flipped vertically. In the world of polynomials, complex numbers act like objects being reflected across the real axis. If a polynomial has only real numbers as its coefficients, it is 'balanced' in a way that forces any complex 'imbalance' to be perfectly mirrored.
📖 Definitions
🔍 Worked example — Easy
Given that 2 + i is a root of the equation x^2 - 4x + 5 = 0, show that 2 - i is also a root.
🔍 Worked example — Standard
A cubic equation x^3 + ax^2 + bx + c = 0 has real coefficients. Given that 1 - 2i is a root, find the quadratic factor of the polynomial.
🔍 Worked example — Exam-style
Find all roots of the equation x^3 - 5x^2 + 9x - 5 = 0, given that 1 + 2i is a root.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The Argand diagram
Plot x + iy as the point (x, y). Geometric effects:
🧑🏫 Teacher's explanation
Think of a complex number not just as a single value, but as a position on a 2D map. While a standard number lives on a single line, a complex number z = x + iy lives on the Argand diagram, where the x-axis represents the real part and the y-axis represents the imaginary part. It is like a GPS coordinate for a plane.
📖 Definitions
🔍 Worked example — Easy
Represent z = 3 - 2i on an Argand diagram and find its conjugate z^*.
🔍 Worked example — Standard
Show that the distance between z_1 = 4 + 3i and z_2 = 1 + 1i is √(13).
🔍 Worked example — Exam-style
A complex number z is multiplied by 2e^iπ/3. Describe the geometric effect on z.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Polar and exponential forms
z = r(cosθ + isinθ) = re^iθ, r = |z|,\ θ = arg z
🧑🏫 Teacher's explanation
Imagine you are navigating a map. In Cartesian form (x + iy), you are giving directions like 'walk 3 blocks East and 4 blocks North'. This is great for adding movements together. However, if you want to rotate or scale your position, Cartesian form becomes messy. This is where Polar and Exponential forms shine. Instead of coordinates, you say 'face 30^° and walk 5 meters'. This is the modulus (r) and the argument (θ).
📖 Definitions
🔍 Worked example — Easy
Given z_1 = 3(cosπ/3 + isinπ/3) and z_2 = 2(cosπ/4 + isinπ/4), find z_1z_2 in polar form.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Square roots
To find √(5 + 12i), let (a + ib)^2 = 5 + 12i. Equate parts: a^2 - b^2 = 5 and 2ab = 12. Substitute b = tfrac6a: a^4 - 5a^2 - 36 = 0 ⇒ a^2 = 9, so a = ± 3, b = ± 2. The square roots are 3 + 2i and -3 - 2i.
🧑🏫 Teacher's explanation
When we find the square root of a positive real number like 9, we often think of just one answer, 3. However, in the world of complex numbers, every non-zero number has exactly two square roots that are negatives of each other. Think of it like a mirror image through the origin: if you find one root, you have automatically found the other by just flipping the sign of both the real and imaginary parts.
📖 Definitions
🔍 Worked example — Standard
Find the square roots of 5 + 12i.
🔍 Worked example — Exam-style
Find the square roots of -8 + 6i.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Loci
For a region, shade where all the conditions hold. Greatest or least |z| or arg z in a region usually comes from a line through the origin and the centre of a circle, or a tangent.
🧑🏫 Teacher's explanation
Think of a complex number z as a point on a map, where the real part is your East-West position and the imaginary part is your North-South position. In this world, the modulus |z - a| isn't just a formula; it represents the physical distance between the point z and a fixed point a.
📖 Definitions
🔍 Worked example — Easy
Sketch the locus of points z such that |z - (2 + i)| = 3.
🔍 Worked example — Standard
Describe the locus of points z satisfying |z - 4| = |z - 2i|.
🔍 Worked example — Exam-style
Find the region defined by |z - 1| ≤ 2 and arg(z) = π/4.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples — Example 1: modulus and argument
z = 3 + 4i: |z| = 5, arg z = tan^-1tfrac43 = 0.927. w = 1 - 2i is in the fourth quadrant: |w| = sqrt5, arg w = -tan^-12 = -1.107.
🔍 Worked example — Easy
Find the modulus and principal argument of z = 3 + 4i.
🔍 Worked example — Standard
Find the modulus and principal argument of w = 1 - 2i.
🔍 Worked example — Exam-style
Given z = -1 - i, find |z| and arg z where -π < θ ≤ π.
⚠️ Common errors
Worked examples — Example 2: a cubic with a given complex root
Given that 1 + 2i is a root of x^3 - 4x^2 + 9x - 10 = 0, find the other roots.
🔍 Worked example — Easy
Given that 2 + i is a root of x^3 - 5x^2 + 11x - 10 = 0, find the other roots.
🔍 Worked example — Standard
A cubic equation with real coefficients has roots 3i and 1 - i. Find the equation.
🔍 Worked example — Exam-style
Show that if 1 + 2i is a root of x^3 - 4x^2 + 9x - 10 = 0, then 1 - 2i is also a root and find the third root.
⚠️ Common errors
Worked examples — Example 3: polar form
z_1 = 2e^iπ/3 and z_2 = 3e^iπ/6: z_1z_2 = 6e^iπ/2 = 6i; z_1/z_2 = tfrac23e^iπ/6. Also (1 + isqrt3)^3 = (2e^iπ/3 )^3 = 8e^iπ = -8.
🔍 Worked example — Easy
Given z_1 = 2e^iπ/3 and z_2 = 3e^iπ/6, find z_1z_2 and z_1/z_2 in polar form.
🔍 Worked example — Standard
Express z = 1 + i√(3) in polar form and calculate z^3.
⚠️ Common errors
Worked examples — Example 4: a locus and a greatest value
Sketch |z - 2 - i| ≤ 3 and find the greatest value of |z| in the region.
🔍 Worked example — Easy
Sketch the locus of |z - (3 + 4i)| = 2 and state its center and radius.
🔍 Worked example — Standard
Sketch the region |z - i| ≤ 2 and find the maximum value of |z|.
🔍 Worked example — Exam-style
Sketch the region |z - (2 + i)| ≤ 3 and find the greatest value of |z|.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) z^2 = -5 - 12i; zz^* = 13; dfrac1z = 2 + 3i/13. (2) a^2 - b^2 = 8, ab = -3: 3 - i and -3 + i. (3) ± i give the factor x^2 + 1; the quotient is x^2 - 2x + 5, roots 1 ± 2i. Roots: i, -i, 1 + 2i, 1 - 2i. (4) (1 + i)^2/2 = i: modulus 1, argument π/2. (5) the perpendicular bisector of the points 1 and i: the line y = x.