Argand diagrams
Objective. Represent modulus, argument and loci geometrically.
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- 2. Worked example
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Learn: the key idea
An Argand diagram represents a complex number z = a + bi as the point (a, b) in a plane, with the horizontal axis real and the vertical axis imaginary. The modulus |z| = √(a²+b²) is the distance from the origin, and the argument arg(z) is the angle measured anticlockwise from the positive real axis. Geometrically, addition of complex numbers corresponds to vector addition, and multiplication by i rotates a point 90° anticlockwise about the origin. For example, z = 3 + 4i plots at (3,4) with |z| = 5.
Worked example
The complex number z = −3 + 4i is plotted on an Argand diagram. Find |z| and state which quadrant the point lies in.
- 1Real part is −3, imaginary part is 4
- 2|z| = √((−3)² + 4²) = √(9+16) = √25
- 3√25 = 5
- 4Negative real, positive imaginary places the point in the second quadrant
Answer: |z| = 5, second quadrant
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Start practisingCommon mistake. Students sometimes plot a + bi as (b, a), swapping the axes. Always plot the real part on the horizontal axis and the imaginary part on the vertical axis, exactly as with ordinary (x, y) coordinates.
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