# Further Maths - Argand Diagrams

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/argand-diagrams/ · Updated: 2020-09-07 · Tags: further-maths, complex-numbers

# Argand Diagrams
Argand diagrams are a way of representing complex numbers by imagining them as points on a plane. In an Argand diagram:

* The $x$-axis is the "real" axis
* The $y$-axis is the "imaginary axis

The complex number $z = x + yi$ can be represented on the diagram by the point $P(x, y)$ where $x$ and $y$ are coordinates. In other words:

* The horizontal position represents the real part of $z$.
* The vertical position represents the imaginary part of $z$.

However! That's just part of the story. Representing complex numbers as a pair of co-ordinates is kind of analogous to vectors, where you move a certain amount along the `x` axis and then a certain amount along the `y` axis. What could another representation of the same idea be? Instead of encoding the information as two movements, one along and one up, you could also conceptualize it as a rotation followed by a displacement.

Think of it like a triangle:

![](https://www.aconcordcarpenter.com/wp-content/uploads/2014/01/The-3-4-5-Method-For-Squaring-Corners.jpg)

Using the $4+3i$ representation is like moving along $4$ and up $3$, and the "distance" to the point is $5$. The angle from the bottom (in this case the real axis), is $tan^{-1}(\frac{3}{4})$, or $36.87^{\circ}$. So, before we thought about it like this:

* Move along 4
* Move up 3

Now we can think about it like this:

* ROTATE $36.87^{\circ}$
* FORWARD $5$

(In capitals because when the teacher was explaining it to us, he kind of sounded like an army drill seargant telling someone where to go).

##### In an Argand diagram, what does the $x$-axis represent??
The real axis.

##### In an Argand diagram, what does the $y$-axis represent??
The imaginary axis.

##### In an Argand diagram, what does the horizontal position of a point represent??
The real part of $z$.

##### In an Argand diagram, what does the vertical position of a point represent??
The imaginary part of $z$.

##### Adding two complex numbers together is like doing what in Physics??
Combining two forces.

##### What is the transformation between $z$ and $z^*$ on an Argand diagram??
A reflection in the real axis.

##### What is the transformation between $z$ and $iz$ on an Argand diagram??
A rotation of $\frac{\pi}{2}$ anticlockwise.

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