3BArgandDiagrams

The Complex Plane (or Argand Diagrams)

For a diagram of a complex number, we create a set of axes with real values, Re(z) on the horizontal axis and imaginary values, Im(z) on the vertical axis. The complex plane is sometimes called an **Argand Plane** or an **Argand Diagram**.

The complex number z = x + yi is represented by a point on the complex plane. (see figure 1)



For example, the complex number z = 2 – 4i would be represented as the POINT shown on the diagram. (see figure 2)

Notice that this is similar to the Cartesian plane but a point on the Cartesian plane represents a pair of values (x, y), while a point on the Complex plane represents a single value, z = x + yi

Adding complex numbers on an Argand Diagram Addition of complex numbers on the Complex Plane is similar to vector addition on the Cartesian Plane ( **See 9A Vectors** )

**(1 + 3i)** + ** (3 – 2i) ** = ** 4 + i **

To add, join the tail of the second arrow to the nose of the first arrow. The sum of the two numbers is then pointed at by an arrow from the origin to the nose of the second arrow. The sum is the point (not the arrow).

Subtracting complex numbers on an Argand Diagram

To subtract, reverse the direction of the second arrow and then add.

Multiplying complex numbers by a constant on an Argand Diagram

To multiply by a constant, multiply the length of the arrow by the constant but don’t change the direction. Remember that the complex number is the point at the tip of the arrow, not the arrow!! Multiplying complex numbers by i on an Argand Diagram

To multiply by i, rotate the arrow anticlockwise by 90°.

To multiply by i 2, rotate the arrow anticlockwise by 180°.

Etc

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