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:
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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date: 2020-09-07 19:15
tags:
- '@?further-maths'
- '@?public'
- '@?complex-numbers'
title: Further Maths - Argand Diagrams