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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 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
Please enable Java for an interactive construction (with Cinderella).


MathsNet imageAn 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.

Software/Applets used on this page

Uses Cinderella interactive geometry


argand diagram

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


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


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


A diagram showing a relationship between two variables.
The diagram shows a vertical y axis and a horizontal x axis.


Absolute value, |x|.


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

Full Glossary List

This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AQA A-Level (UK - Pre-2017)FP2Complex NumbersArgand diagram-
AQA AS Further Maths 2017Pure MathsComplex Number BasicsArgand Diagrams-
AQA AS/A2 Further Maths 2017Pure MathsComplex Number BasicsArgand Diagrams-
CBSE XI (India)AlgebraComplex Numbers and Quadratic EquationsArgand diagrams-
CCEA A-Level (NI)FP1Complex NumbersArgand diagram-
CIE A-Level (UK)P3Complex NumbersArgand diagram-
Edexcel A-Level (UK - Pre-2017)FP1Complex NumbersArgand diagram-
Edexcel AS Further Maths 2017Core Pure MathsArgand DiagramsArgand Diagrams-
Edexcel AS/A2 Further Maths 2017Core Pure MathsArgand DiagramsArgand Diagrams-
I.B. Higher Level1Complex NumbersArgand diagram-
Methods (UK)M3Complex NumbersArgand diagram-
OCR A-Level (UK - Pre-2017)FP1Complex NumbersArgand diagram-
OCR AS Further Maths 2017Pure CoreArgand Diagrams and LociArgand Diagrams-
OCR MEI AS Further Maths 2017Core Pure AArgand Diagrams and LociArgand Diagrams-
OCR-MEI A-Level (UK - Pre-2017)FP1Complex NumbersArgand diagram-
Pre-U A-Level (UK)7Complex NumbersArgand diagram-
Scottish Advanced HighersM2Complex NumbersArgand diagram-
Scottish (Highers + Advanced)AM2Complex NumbersArgand diagram-
Universal (all site questions)CComplex NumbersArgand diagram-
WJEC A-Level (Wales)FP1Complex NumbersArgand diagram-