Argand Diagram

Complex numbers in an Argand diagram

Let's start by mentioning what complex numbers are.

Complex numbers can be represented on a plane. This plane is called an Argand diagram, Argand plane or complex plane.

The Argand diagram is similar to the Cartesian plane, except that the \(x\)-axis is now the real axis, while the \(y\)-axis is the imaginary axis.

To plot a complex number on the Argand diagram, you first locate the real number on the horizontal axis, and then the imaginary number on the vertical axis. So, the number \(z=3+2i\), corresponds to the point \((3,2)\) on the Argand plane.

Figure 1. Argand diagram with the complex number \(3+2i\)

Quadrants in an Argand diagram

Like the Cartesian plane, the Argand diagram is also divided into quadrants, which are numbered counterclockwise.

Depending on the real and imaginary part of a complex number, it will be in its respective quadrant.

• In the first quadrant are the complex numbers \(z\) such that \(\text{Re}(z) > 0\) and \(\text{Im}(z) > 0\).

• In the second quadrant are the complex numbers \(z\) such that \(\text{Re}(z) < 0\) and \(\text{Im}(z) > 0\).

• In the third quadrant are the complex numbers \(z\) such that \(\text{Re}(z) < 0\) and \(\text{Im}(z) < 0\).

• In the fourth quadrant are the complex numbers \(z\) such that \(\text{Re}(z) > 0\) and \(\text{Im}(z) < 0\).

Figure 2. Argand diagram divided in quadrants

Formulas in an Argand diagram

As mentioned above, the complex number \(z=a+bi\) corresponds to the point \((a,b)\) in the complex plane. But, it is also possible to locate it using polar coordinates.

Note that if you draw a line from the origin to the point \((a,b)\), then the location of the point is determined by the length of that line together with the angle it forms with the real axis.

Figure 3. Polar coordinates

Using this idea, a complex number \(z=a+bi\) can also be written as

\[z=re^{i\theta},\]

where \(r\) is the length of the line joining the complex number to the origin (or \(r=|z|\)), given by the formula

\[r=\sqrt{a^2+b^2}.\]

The value of \(\theta\) depends on the location of the complex number.

Table 1. Value of \(\theta\).

Circles in an Argand diagram

A circle is made up of a set of points that are at a fixed distance from a point called the centre.

As mentioned above, the quantity \(|z|\) measures the distance from \(z\) to the origin. Thus, the set \(|z|=k\) with \(k\geq 0\), consists of all complex numbers \(z\) such that their distance to the origin is \(k\), which is a circle with centre at the origin and radius \(k\). This is denoted by

\[|z|=k.\]

In general, a circle of radius \(k\) with centre at a complex number \(z_0\) is denoted by

\[|z-z_0|=k.\]

Loci in an Argand diagram

Now that you know how to graph circles, let's see what other sets you can visualise in the complex plane.

Locus of Re(z)=k

For a real number \(k\), the locus of \(\text{Re}(z)=k\) refers to the set of all complex numbers \(z\) such that their real part is equal to the value \(k\).

Therefore, the locus of \(\text{Re}(z)=k\) is a line parallel to the imaginary axis, passing through the value \(k\) on the real axis.

Locus of \(\text{Im}(z)=k\)

For a real number \(k\), the locus of \(\text{Im}(z)=k\) refers to the set of all complex numbers \(z\) such that their imaginary part is equal to the value \(k\).

Therefore, the locus of \(\text{Im}(z)=k\) is a line parallel to the real axis, passing through the value \(k\) on the imaginary axis.

Locus of \(|z-a|=|z-b|\)

Given \(a\) and \(b\), two complex numbers, the locus of \(|z-a|=|z-b|\) refers to the set of all complex numbers \(z\) such that their distance to \(a\) is equal to their distance to \(b\).

Therefore, the locus of \(|z-a|=|z-b|\) is the perpendicular bisector of the line segment joining the two complex numbers \(a\) and \(b\).

Argand diagram example

Let's look at an example to apply what you have seen until now.

Let's take another example.