Complex Numbers (FP1)

Q: Can a cubic equation with real coefficients have only non-real complex roots?

No. Non-real roots of real-coefficient polynomials occur in conjugate pairs, so a cubic (degree 3) must have at least 1 real root, plus 0 or 1 conjugate complex root pair.

Q: Do I need to recall De Moivre's theorem for FP1 complex number questions?

No. De Moivre's theorem, Euler's relation, and complex loci are part of the FP2 specification and are not assessed in FP1 complex number questions.

Q: What units should I use for the argument of a complex number in FP1 exams?

Always use radians for principal arguments, and ensure your final value falls within the required range .

Edexcel International A-Level Further Mathematics· FP1 1.1 to 1.6 (2018 Issue 3)· 25 min read

1. Definitions and Forms of Complex Numbers★★☆☆☆⏱ 5 min

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📘 Definition

Complex Number

A number that can be written in Cartesian form $z = a + ib$ (where $a, b \in R$ , $i^{2} = - 1$ ) or polar form $z = r cos θ + i r sin θ$ (where $r = ∣ z ∣$ , $θ = ar g (z)$ ).

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 $∣ z ∣ = a^{2} + b^{2}$ , and the principal argument is the angle of $z$ from the positive real axis, between $- π$ and $π$ .

📐 Worked Example

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

1
1. Identify the real and imaginary parts: $Re (z) = 3$ , $Im (z) = 4$
2
1. Flip the sign of the imaginary part to find the conjugate: $z^{*} = 3 - 4 i$
3
$3. Calculate modulus: |z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$
4
1. Calculate argument: $z$ lies in the first quadrant, so $ar g (z) = arctan (\frac{4}{3}) = 0.927$ 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 $\pm π$ to fall within the $- π < ar g (z) \leq π$ range.

2. Complex Number Arithmetic★★★☆☆⏱ 7 min

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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.

📐 Worked Example

Given $z_{1} = 2 + i$ and $z_{2} = 1 - 3 i$ , 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: $z_{1} + z_{2} = (2 + 1) + i (1 - 3) = 3 - 2 i$
2
1. Expand the product and replace $i^{2} = - 1$ : $z_{1} z_{2} = (2) (1) + 2 (- 3 i) + i (1) + i (- 3 i) = 2 - 6 i + i + 3 = 5 - 5 i$
3
1. 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
1. Verify modulus rule: $∣ z_{1} ∣ = 2^{2} + 1^{2} = 5$ , $∣ z_{2} ∣ = 1^{2} + (- 3)^{2} = 10$ , so $∣ z_{1} ∣∣ z_{2} ∣ = 50 = 52$ .
6
1. Calculate $∣ z_{1} z_{2} ∣ = ∣5 - 5 i ∣ = 5^{2} + (- 5)^{2} = 50 = 52$ , so the rule holds.

3. Argand Diagram Representations★★☆☆☆⏱ 6 min

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An Argand diagram plots complex numbers as points with coordinates $(Re (z), Im (z))$ , 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
1. Plot $z^{*} = 1 - i$ at the point $(1, - 1)$ in the fourth quadrant.
3
1. Calculate $z + z^{*} = 2 + 0 i$ , plot this at the point $(2, 0)$ on the positive real axis.
4
1. 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.

4. Complex Roots of Real-Coefficient Polynomials★★★★☆⏱ 7 min

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📘 Definition

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 $f (x) = x^{4} - x^{3} - 5 x^{2} + 7 x + 10$ , 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
1. Find the quadratic factor for these two roots: $(x - (2 + i)) (x - (2 - i)) = (x - 2)^{2} + 1 = x^{2} - 4 x + 5$
3
1. Divide $f (x)$ by $x^{2} - 4 x + 5$ to get the remaining quadratic factor: $f (x) = (x^{2} - 4 x + 5) (x^{2} + 3 x + 2)$
4
1. Factor the remaining quadratic: $x^{2} + 3 x + 2 = (x + 1) (x + 2)$
5
1. 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.

5. Common Pitfalls

Wrong move:

Writing the conjugate of $a + ib$ as $- a + ib$ or $- a - ib$

Why:

Confusing the definition of conjugate, which only flips the sign of the imaginary part

Correct move:

For $z = a + ib$ , keep the real part unchanged and flip only the sign of the imaginary part: $z^{*} = a - ib$

Wrong move:

Giving arguments in degrees, or outside the $- π < ar g (z) \leq π$ range

Why:

Edexcel FP1 explicitly requires principal arguments in radians within the specified range

Correct move:

Convert angles to radians, adjust by ± $2 π$ if needed to fit the range, and check the quadrant of $z$ to avoid sign errors

Wrong move:

Assuming all quartic equations have only conjugate root pairs and no real roots

Why:

Real-coefficient quartics can have 0, 2, or 4 real roots, so conjugate pairs only apply to non-real roots

Correct move:

First test for integer real roots using the factor theorem, then apply the conjugate root theorem to any remaining non-real roots

Wrong move:

Forgetting to replace $i^{2}$ with $- 1$ when multiplying complex numbers

Why:

The definition of $i$ is $i^{2} = - 1$ , so omitting this substitution leads to incorrect real part calculations

Correct move:

Replace every occurrence of $i^{2}$ with $- 1$ immediately when expanding products of complex numbers, then combine real and imaginary terms separately

Wrong move:

Using the result $ar g (z_{1} z_{2}) = ar g (z_{1}) + ar g (z_{2})$ in FP1 answers

Why:

This property is explicitly not part of the FP1 specification and is only assessed in FP2

Correct move:

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

6. Quick Reference Cheatsheet

Concept	Formula / Rule	Exam Reminder
Complex number forms	$z = a + ib = r cos θ + i r sin θ$	$a = Re (z)$ , $b = Im (z)$ , $r = ∣ z ∣$ , $θ = ar g (z)$
Conjugate	$z^{*} = a - ib$	Reflects $z$ across the real axis on Argand diagram
Modulus	$∣ z ∣ = a^{2} + b^{2}$ , $∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣$	Modulus is always non-negative
Equality rule	$a + ib = c + i d ⟺ a = c$ 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

7. Frequently Asked

Can a cubic equation with real coefficients have only non-real complex roots?

No. Non-real roots of real-coefficient polynomials occur in conjugate pairs, so a cubic (degree 3) must have at least 1 real root, plus 0 or 1 conjugate complex root pair.

Do I need to recall De Moivre's theorem for FP1 complex number questions?

No. De Moivre's theorem, Euler's relation, and complex loci are part of the FP2 specification and are not assessed in FP1 complex number questions.

What units should I use for the argument of a complex number in FP1 exams?

Always use radians for principal arguments, and ensure your final value falls within the required range $- π < ar g (z) \leq π$ .

Going deeper

syllabusEdexcel IAL Further Maths 2018 SpecificationOfficial FP1 complex numbers specification (section 1.1-1.6)

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.