---
title: "The Argand Plane, Modulus and Argument"
description: "A complex number is a point: real part across, imaginary part up. Its distance from the origin is the modulus and its angle is the argument, so multiplying becomes scaling and turning."
canonical: https://lightmysky.com/learn/mathematics/the-argand-plane-modulus-and-argument-mt_Wq01c5UN9e
source: https://lightmysky.com/learn/mathematics/the-argand-plane-modulus-and-argument-mt_Wq01c5UN9e.md
retrieved: 2026-09-12
---

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# The Argand Plane, Modulus and Argument

A complex number is a point: real part across, imaginary part up. Its distance from the origin is the modulus and its angle is the argument, so multiplying becomes scaling and turning.

Subject: Mathematics · Area: Algebra · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/the-argand-plane-modulus-and-argument-mt_Wq01c5UN9e

## Ready when they can

- Plot complex numbers on the Argand plane and read real and imaginary parts back
- Compute modulus and argument and convert between a + bi and modulus-argument form
- Show geometrically that multiplying by i is a quarter turn

## Lesson: Points, distances, and turns

Plot each complex number as a point: real part across, imaginary part up. So 3 + 4i lands on (3, 4). Read them back the same way: 4 - 3i sits right and down, which is the fourth quadrant. Students who plot first and calculate second mix up far fewer signs.

**Example.** Modulus means distance from the origin, found by Pythagoras. For 3 + 4i: 9 + 16 is 25, whose root is 5. The same recipe gives 13 for 5 - 12i and 17 for 8 - 15i, since minus signs vanish on squaring. Argument means direction: 1 + i sits on the diagonal at 45 degrees, and modulus 2 with argument 90 degrees is the point 2i.

Multiplying by i turns the plane a quarter turn anticlockwise. Check it: (1 + i) times i is i + i squared, which is -1 + i, so (1, 1) moves to (-1, 1). The same turn explains (1 + i) squared: expanding gives 1 + 2i + i squared, which is 2i. Twice by i makes a half turn, sending any z to minus z.

**Tip.** Convert both ways with one picture in mind. From a point, r is the root of x squared plus y squared and the angle comes from rise over run. From distance plus bearing, walk r steps at that angle. Never trust a bare angle near the axes: sketch the point and check the quadrant.

**Recap.** Plot across and up, measure distance and direction, and read multiplication by i as a quarter turn.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [The Imaginary Unit and Complex Arithmetic](https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550)
- [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4)

## Opens up

- [Complex Functions and the Complex Plane as a Domain](https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs)
- [Euler's Formula and the Exponential Form](https://lightmysky.com/learn/mathematics/eulers-formula-and-the-exponential-form-mt_RX2IxsnSwG)