Free Pearson Edexcel International A-level Mathematics · FP1

Complex numbers and Argand diagrams

What you’ll learn

  • Calculate with complex numbers and conjugates.
  • Interpret modulus and argument geometrically.

Understand the idea

A complex number z=a+biz=a+bi has real part aa and imaginary part bb, with i2=−1i^2=-1. On an Argand diagram it is the point (a,b)(a,b): adding complex numbers adds their horizontal and vertical components.

The conjugate z∗=a−biz^*=a-bi reflects the point in the real axis. Multiplying a denominator by its conjugate makes it real. The modulus is distance from the origin; when finding the principal argument, choose the correct quadrant and use −π<arg⁡z≤π-\pi<\arg z\leq\pi.

Key formulae

i2=−1,z∗=a−bii^2=-1,\qquad z^*=a-bi

∣z∣=a2+b2|z|=\sqrt{a^2+b^2}

zz∗=∣z∣2zz^*=|z|^2

arg⁡z=θ,cos⁡θ=a∣z∣,sin⁡θ=b∣z∣\arg z=\theta,\quad \cos\theta=\frac{a}{|z|},\quad\sin\theta=\frac{b}{|z|}

Conjugation reflects an Argand point in the real axis

Conditions. Argand vectors from the origin to z = −3 + 4i and its conjugate −3 − 4i. Both axes use equal lengths per coordinate unit. Straight segments show position vectors, not a path of changing values.

Real part / dimensionless; Imaginary part / dimensionless.

Both points have modulus 5 by the 3–4–5 triangle. Conjugation reverses the imaginary coordinate and preserves the real coordinate and modulus; their product is 25.

Open the interactive lesson to view plotted curves and coordinate tables.

Worked example

Write 3+4i1−2i\frac{3+4i}{1-2i} in the form a+bia+bi.

  1. Multiply numerator and denominator by 1+2i1+2i.
  2. The denominator is (1−2i)(1+2i)=5(1-2i)(1+2i)=5.
  3. The numerator is 3+10i+8i2=−5+10i3+10i+8i^2=-5+10i.

Answer: −1+2i-1+2i.

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