Did you know that imaginary numbers used to be thought of as fictitious or useless? Although these numbers do not exist in reality, they have been shown to be much more useful than previously thought.
These numbers have become so popular that they even have their own plane, called the complex plane (although there is nothing complex about it), also known as an Argand diagram.
In this article, you will learn how to locate complex numbers in their not-so-complex plane!
Complex numbers in an Argand diagram
Let's start by mentioning what complex numbers are.
A complex number \(z\) is a number of the form \(z=a+bi\). where \(i\) is called the imaginary unit and satisfies \(i^2=-1\), and \(a,b\) are real numbers.
The number \(a\) is often referred to as the real part of \(z\), or \(\text{Re} (z)\), and \(b\) the imaginary part of \(z\), or \(\text{Im} (z)\).
You might also see the real part written as \(\Re(z)\) and the imaginary part written as \(\Im(z)\).
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 \(\theta\).
Location of \(z\) | Value of \(\theta\) |
\(\text{Im}(z)=0\) and \(\text{Re}(z)>0\) | \(0\) |
First quadrant | \(\arctan \left(\dfrac{b}{a}\right)\) |
\(\text{Im}(z)>0\) and \(\text{Re}(z)=0\) | \(\dfrac{\pi}{2}\) |
Second quadrant | \(\arctan \left(\dfrac{b}{a}\right)+\pi\) |
\(\text{Im}(z)=0\) and \(\text{Re}(z)<0\) | \(\pi\) |
Third quadrant | \(\arctan \left(\dfrac{b}{a}\right)-\pi\) |
\(\text{Im}(z)<0\) and \(\text{Re}(z)=0\) | \(\dfrac{3\pi}{2}\) |
Fourth quadrant | \(\arctan \left(\dfrac{b}{a}\right)\) |
To learn more about this notation, visit the article The Modulus and Argument of a Complex Number.
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
\[|z-z_0|=k.\]
Graph all complex numbers such that \(|z-3+5i|=4\).
Solution:
First, note that
\[|z-3+5i|=|z-(3-5i)|.\]
Therefore, considering the circle formula, this set is a circle with centre in \(3-5i\) and radius \(4\).
Figure 4. The graph of \(|z-3+5i|=4\)
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.
Find the locus of \(\text{Re}(z)=-5\).
Solution:
The locus of \(\text{Re}(z)=-5\) is a line parallel to the imaginary axis and passes through \(-5\) on the real axis.
Figure 5. Locus of \(\text{Re}(z)=-5\)
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.
Find the locus of \(\text{Im}(z)=3\).
Solution
The locus of \(\text{Im}(z)=3\) is a line parallel to the real axis and passes through \(3\) on the imaginary axis.
Figure 6. Locus of \(\text{Im}(z)=3\)
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 \(b\).
Find the locus of \(|z-1-i|=|z+1+i|\).
Solution:
Note that
\[|z-1-i|=|z-(1+i)|\]
and \[|z+1+i|=|z-(-1-i)|.\]
So the locus of \(|z-1-i|=|z+1+i|\) is the perpendicular bisector of the line joining \(1+i\) and \(-1-i\).
Figure 7. Locus of \(|z-1-i|=|z+1+i|\)
Argand diagram example
Let's look at an example to apply what you have seen until now.
Locate the complex numbers \(4+3i\) and \(-3-4i\) and find their expression in polar coordinates.
Solution:
The complex numbers are plotted below.
Figure 8. The complex numbers \(4+3i\) and \(-3-4i\) in an Argand diagram
To find its expression in polar coordinates, you have to use the formula according to the quadrant where the complex number is located.
For \(4+3i\): the value of \(r\) is given by
\[r=\sqrt{4^2+3^2}=\sqrt{25}=5.\]
To calculate \(\theta\), note that \(4+3i\) is in the first quadrant, so
\[\theta=\arctan \left(\frac{3}{4}\right)\approx 0.64.\]
Therefore, the complex number \(4+3i\) in polar coordinates is \(5e^{0.64i}\).
For \(-3-4i\): the value of \(r\) is given by
\[r=\sqrt{(-4)^2+(-3)^2}=\sqrt{25}=5.\]
To calculate \(\theta\), note that \(-3-4i\) is in the third quadrant, so
\[\theta=\arctan \left(\frac{-4}{-3}\right)-\pi\approx -2.21 .\]
Therefore, the complex number \(-3-4i\) in polar coordinates is \(5e^{-2.21i}\).
Let's take another example.
Find the locus \(\text{Im}(z) = -2\) and the locus \(\text{Re}(z)= 4\). Are there any numbers that are in both sets?
Solution:
The locus of \(\text{Im}(z) = -2\) is a horizontal line through \(-2\), the locus of \(\text{Re}(z)= 4\) is a vertical line through \(4\) and the only value in both sets is the complex number \(4-2i\).
Figure 9. The locus of \(\text{Im}(z) = -2\) and \(\text{Re}(z)= 4\)
Argand Diagram - Key takeaways
- An Argand diagram is a two-dimensional plane which you can use to visualise complex numbers.
- The Argand diagram is also called Argand plane or complex plane.
- A complex number \(z=a+bi\) can be written as \[z=re^{i\theta},\] where \(r\) is the length of the line joining the point to the origin, given by the formula \[r=\sqrt{a^2+b^2},\] and \(\theta\) is the angle of this line to the real axis.
- A circle with centre \(z_0\) and radius \(k\) is written as \[|z-z_0|=k.\]
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