# Complex Numbers (FP1)

> Edexcel International A-Level Further Mathematics · 2018 Edexcel IAL FM FP1
> Source: https://www.owlsprep.com/study/edexcel-ial-further-math-fp1-complex-numbers/

This guide covers all Edexcel IAL FP1 complex number content, including Cartesian/polar forms, arithmetic operations, Argand diagrams, and conjugate root theorems for quadratics, cubics and quartics, with exam-aligned worked examples.

**Prerequisites:** [Quadratic equations and quadratic formula (Edexcel IAL P1)](https://www.owlsprep.com/study/edexcel-ial-p1-quadratic-equations/); [Factor and remainder theorem (Edexcel IAL P2)](https://www.owlsprep.com/study/edexcel-ial-p2-polynomials/)

## Learning objectives

- Define complex numbers in Cartesian and polar forms, and use properties of conjugates, modulus, argument, real/imaginary parts and equality
- Perform sum, product and quotient operations on complex numbers and apply the modulus multiplication rule
- Represent complex numbers and their operations geometrically on an Argand diagram
- Find complex solutions of real-coefficient quadratic equations
- Identify conjugate root pairs and solve real-coefficient cubic equations fully
- Find all roots of real-coefficient quartic equations using conjugate root and real root properties

## Definitions and Forms of Complex Numbers

**Complex Number** — A number that can be written in **Cartesian form** $z = a + ib$ (where $a,b \in \mathbb{R}$, $i^2 = -1$) or **polar form** $z = r\cos\theta + ir\sin\theta$ (where $r = |z|$, $\theta = \arg(z)$).

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. The conjugate of $z = a + ib$ is $z^* = a - ib$, the modulus is $|z| = \sqrt{a^2 + b^2}$, and the principal argument is the angle of $z$ from the positive real axis, between $-\pi$ and $\pi$.

**Worked example:** For $z = 3 + 4i$, calculate $\text{Re}(z)$, $\text{Im}(z)$, $z^*$, $|z|$, and the principal argument of $z$, correct to 3 significant figures.

1. 1. Identify the real and imaginary parts: $\text{Re}(z) = 3$, $\text{Im}(z) = 4$
2. 2. Flip the sign of the imaginary part to find the conjugate: $z^* = 3 - 4i$
3. $$3. Calculate modulus: |z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$$
4. 4. Calculate argument: $z$ lies in the first quadrant, so $\arg(z) = \arctan(\frac{4}{3}) = 0.927$ radians (3 s.f.)

> **Exam tip:** Always check the quadrant of a complex number before writing its principal argument: for second/third quadrant values, adjust arctan results by $\pm \pi$ to fall within the $-\pi < \arg(z) \leq \pi$ range.

*Calculator:* allowed

## Complex Number Arithmetic

To add/subtract complex numbers, combine real parts and imaginary parts separately. To multiply, expand brackets normally and replace $i^2$ with $-1$. To divide, multiply numerator and denominator by the conjugate of the denominator to eliminate $i$ from the denominator. The rule $|z_1 z_2| = |z_1||z_2|$ always holds for the modulus of a product.

> **tip**
>
> Always simplify complex fractions to the $a + ib$ form for full marks in exams.

**Worked example:** Given $z_1 = 2 + i$ and $z_2 = 1 - 3i$, calculate $z_1 + z_2$, $z_1 z_2$, $\frac{z_1}{z_2}$, and verify that $|z_1 z_2| = |z_1||z_2|$.

1. 1. Add real and imaginary parts for the sum: $z_1 + z_2 = (2 + 1) + i(1 - 3) = 3 - 2i$
2. 2. Expand the product and replace $i^2 = -1$: $z_1 z_2 = (2)(1) + 2(-3i) + i(1) + i(-3i) = 2 - 6i + i + 3 = 5 - 5i$
3. 3. Divide by multiplying by the conjugate of $z_2$:
4. $$\frac{z_1}{z_2} = \frac{(2+i)(1+3i)}{(1-3i)(1+3i)} = \frac{2 + 7i - 3}{1 + 9} = \frac{-1 + 7i}{10} = -0.1 + 0.7i$$
5. 4. Verify modulus rule: $|z_1| = \sqrt{2^2 + 1^2} = \sqrt{5}$, $|z_2| = \sqrt{1^2 + (-3)^2} = \sqrt{10}$, so $|z_1||z_2| = \sqrt{50} = 5\sqrt{2}$.
6. 5. Calculate $|z_1 z_2| = |5 - 5i| = \sqrt{5^2 + (-5)^2} = \sqrt{50} = 5\sqrt{2}$, so the rule holds.

*Calculator:* allowed

## Argand Diagram Representations

An Argand diagram plots complex numbers as points with coordinates $(\text{Re}(z), \text{Im}(z))$, or as vectors from the origin. Adding complex numbers corresponds to vector addition, multiplying by a real scalar stretches the vector, and conjugating a number reflects it across the real axis.

**Worked example:** Plot $z = 1 + i$, $z^*$, and $z + z^*$ on an Argand diagram, and describe the geometric relationship between $z$ and $z^*$.

1. 1. Plot $z = 1 + i$ at the point $(1, 1)$ in the first quadrant.
2. 2. Plot $z^* = 1 - i$ at the point $(1, -1)$ in the fourth quadrant.
3. 3. Calculate $z + z^* = 2 + 0i$, plot this at the point $(2, 0)$ on the positive real axis.
4. 4. Geometric relationship: $z$ and $z^*$ are reflections of each other across the real (x) axis.

> **Exam tip:** Always label axes as 'Re' (x-axis) and 'Im' (y-axis) when drawing Argand diagrams, and mark all plotted points with their corresponding complex number for full marks.

*Calculator:* allowed

## Complex Roots of Real-Coefficient Polynomials

**Conjugate Root Theorem** — For any polynomial with real coefficients, if $z = a + ib$ is a non-real root, then its conjugate $z^* = a - ib$ is also a root. Real-coefficient cubics have at least 1 real root, and real-coefficient quartics have 0, 2, or 4 real roots.

For quadratic equations with real coefficients, non-real roots always form a conjugate pair. For cubics and quartics, you can use the factor theorem to find integer real roots first, then use the conjugate root theorem to find non-real root pairs, then factor the polynomial fully.

**Worked example:** Given $x = 2 + i$ is a root of $f(x) = x^4 - x^3 - 5x^2 + 7x + 10$, find all other roots of $f(x) = 0$.

1. 1. Apply the conjugate root theorem: since coefficients are real, $x = 2 - i$ is also a root.
2. 2. Find the quadratic factor for these two roots: $(x - (2+i))(x - (2-i)) = (x - 2)^2 + 1 = x^2 - 4x + 5$
3. 3. Divide $f(x)$ by $x^2 - 4x + 5$ to get the remaining quadratic factor: $f(x) = (x^2 - 4x + 5)(x^2 + 3x + 2)$
4. 4. Factor the remaining quadratic: $x^2 + 3x + 2 = (x + 1)(x + 2)$
5. 5. The full set of roots is $2 + i$, $2 - i$, $-1$, $-2$.

> **Exam tip:** Test small integer values (±1, ±2, ±5, ±10, etc.) when looking for real roots of polynomials with integer coefficients, as these are almost always used in FP1 exam questions.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Writing the conjugate of $a + ib$ as $-a + ib$ or $-a - ib$
  - Why it fails: Confusing the definition of conjugate, which only flips the sign of the imaginary part
  - Correct: For $z = a + ib$, keep the real part unchanged and flip only the sign of the imaginary part: $z^* = a - ib$
- **Wrong:** Giving arguments in degrees, or outside the $-\pi < \arg(z) \leq \pi$ range
  - Why it fails: Edexcel FP1 explicitly requires principal arguments in radians within the specified range
  - Correct: Convert angles to radians, adjust by ±$2\pi$ if needed to fit the range, and check the quadrant of $z$ to avoid sign errors
- **Wrong:** Assuming all quartic equations have only conjugate root pairs and no real roots
  - Why it fails: Real-coefficient quartics can have 0, 2, or 4 real roots, so conjugate pairs only apply to non-real roots
  - Correct: First test for integer real roots using the factor theorem, then apply the conjugate root theorem to any remaining non-real roots
- **Wrong:** Forgetting to replace $i^2$ with $-1$ when multiplying complex numbers
  - Why it fails: The definition of $i$ is $i^2 = -1$, so omitting this substitution leads to incorrect real part calculations
  - Correct: Replace every occurrence of $i^2$ with $-1$ immediately when expanding products of complex numbers, then combine real and imaginary terms separately
- **Wrong:** Using the result $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$ in FP1 answers
  - Why it fails: This property is explicitly not part of the FP1 specification and is only assessed in FP2
  - Correct: For FP1 geometric product questions, only describe magnitude changes using $|z_1 z_2| = |z_1||z_2|$, do not use rotation/argument addition reasoning

## Cheatsheet

| Concept | Formula / Rule | Exam Reminder |
| --- | --- | --- |
| Complex number forms | $z = a + ib = r\cos\theta + ir\sin\theta$ | $a = \text{Re}(z)$, $b = \text{Im}(z)$, $r = \|z\|$, $\theta = \arg(z)$ |
| Conjugate | $z^* = a - ib$ | Reflects $z$ across the real axis on Argand diagram |
| Modulus | $\|z\| = \sqrt{a^2 + b^2}$, $\|z_1 z_2\| = \|z_1\|\|z_2\|$ | Modulus is always non-negative |
| Equality rule | $a + ib = c + id \iff a = c$ and $b = d$ | Compare real and imaginary parts separately to solve equations |
| Conjugate Root Theorem | Non-real roots of real-coefficient polynomials come in conjugate pairs | Cubics have ≥1 real root; quartics have 0/2/4 real roots |
| Complex division | $\frac{z_1}{z_2} = \frac{z_1 z_2^*}{\|z_2\|^2}$ | Multiply numerator and denominator by the conjugate of the denominator |

## What's next

You have now mastered all core complex number content for Edexcel IAL FP1. This topic is foundational for later FP2 content including De Moivre's theorem, complex loci, and transformations, and also overlaps with matrix algebra and series topics in FP1. Complex number questions are consistently high-mark and appear in almost every FP1 exam, so make sure to practice past paper questions to build speed with arithmetic operations and root-finding. Next, move on to the other core FP1 topics to complete your unit preparation.

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