Since complex numbers have two parts -
their real and imaginary parts - they can be plotted on a graph. The real part
is plotted along the horizontal x-axis and the imaginary part is plotted on the
vertical y-axis. Such a diagram is known as an

This Argand diagram shows the modulus, |a| or |OA|, and the argument, \arg(a), shown in degrees, of the complex number a. If the coordinates of the point A are (b,c) then

a = b+ic

|a| = \sqrt{b^2+c^2} and

arg(a) = \tan^{-1}\displaystyle \frac{c}{b}

The modulus |OA| is the length of the line segment OA.

The argument arg(a) is measured anticlockwise from the positive real axis

**Argand diagram**.This Argand diagram shows the modulus, |a| or |OA|, and the argument, \arg(a), shown in degrees, of the complex number a. If the coordinates of the point A are (b,c) then

a = b+ic

|a| = \sqrt{b^2+c^2} and

arg(a) = \tan^{-1}\displaystyle \frac{c}{b}

The modulus |OA| is the length of the line segment OA.

The argument arg(a) is measured anticlockwise from the positive real axis

## Summary/Background

An Argand diagram is a plot of complex numbers, z=x+iy, as points in the complex plane using the x-axis as the real axis and y-axis as the imaginary axis.

While Jean-Robert Argand (1768 - 1822) is generally credited with the discovery in 1806, the Argand diagram (also known as the Argand plane) was actually described by C. Wessel prior to Argand. Historically, the geometric representation of a complex number as a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, this visualization helped "imaginary" and "complex" numbers become accepted in mainstream mathematics as a natural extension to negative numbers along the real line.

While Jean-Robert Argand (1768 - 1822) is generally credited with the discovery in 1806, the Argand diagram (also known as the Argand plane) was actually described by C. Wessel prior to Argand. Historically, the geometric representation of a complex number as a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, this visualization helped "imaginary" and "complex" numbers become accepted in mainstream mathematics as a natural extension to negative numbers along the real line.

## Software/Applets used on this page

Uses Cinderella interactive geometry

## Glossary

### argand diagram

A geometrical presentation of a complex number; using a real and imaginary axis to plot a+ib as (a,b).

### argument

On an argand diagram the angle between the complex number and the real axis.

### axis

One of two straight lines on a graph from which measurements are taken. One axis (the y axis) is vertical; the other (the x axis) is horizontal.

### complex number

A number of the form a+bi where i is the square root of -1, and a and b are real.

### geometric

A sequence where each term is obtained by multiplying the previous one by a constant.

### graph

A diagram showing a relationship between two variables.

The diagram shows a vertical y axis and a horizontal x axis.

The diagram shows a vertical y axis and a horizontal x axis.

### modulus

Absolute value, |x|.

### plot

To mark a point on a graph accurately by using its coordinates.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

AQA A-Level (UK - Pre-2017) | FP2 | Complex Numbers | Argand diagram | - |

AQA AS Further Maths 2017 | Pure Maths | Complex Number Basics | Argand Diagrams | - |

AQA AS/A2 Further Maths 2017 | Pure Maths | Complex Number Basics | Argand Diagrams | - |

CBSE XI (India) | Algebra | Complex Numbers and Quadratic Equations | Argand diagrams | - |

CCEA A-Level (NI) | FP1 | Complex Numbers | Argand diagram | - |

CIE A-Level (UK) | P3 | Complex Numbers | Argand diagram | - |

Edexcel A-Level (UK - Pre-2017) | FP1 | Complex Numbers | Argand diagram | - |

Edexcel AS Further Maths 2017 | Core Pure Maths | Argand Diagrams | Argand Diagrams | - |

Edexcel AS/A2 Further Maths 2017 | Core Pure Maths | Argand Diagrams | Argand Diagrams | - |

I.B. Higher Level | 1 | Complex Numbers | Argand diagram | - |

Methods (UK) | M3 | Complex Numbers | Argand diagram | - |

OCR A-Level (UK - Pre-2017) | FP1 | Complex Numbers | Argand diagram | - |

OCR AS Further Maths 2017 | Pure Core | Argand Diagrams and Loci | Argand Diagrams | - |

OCR MEI AS Further Maths 2017 | Core Pure A | Argand Diagrams and Loci | Argand Diagrams | - |

OCR-MEI A-Level (UK - Pre-2017) | FP1 | Complex Numbers | Argand diagram | - |

Pre-U A-Level (UK) | 7 | Complex Numbers | Argand diagram | - |

Scottish Advanced Highers | M2 | Complex Numbers | Argand diagram | - |

Scottish (Highers + Advanced) | AM2 | Complex Numbers | Argand diagram | - |

Universal (all site questions) | C | Complex Numbers | Argand diagram | - |

WJEC A-Level (Wales) | FP1 | Complex Numbers | Argand diagram | - |