For a introduction in Complex numbers and the basic mathematical operations between complex numbers, read the article Complex Numbers – Introduction. In this article we are going to explain the different ways of representation of a complex number and the methods to convert from one representation to another.
Complex numbers can be represented in several formats:
- polynomial
- cartesian
- polar
- exponential
We can convert from one representation to another since all of them are equivalent.
Polynomial representation of complex numbers
In polynomial form, a complex number is a mathematical operation between the real part and the imaginary part. The general representation of a complex number in polynomial form is:
\[\bbox[#FFFF9D]{z = a + b \cdot i}\]where:
z – is a complex number
a = Re(z), is real number, which is the real part of the complex number
b = Im(z), is real number, which is the imaginary part of the complex number
Let’s consider two complex numbers, z1 and z2, in the following polynomial form:
\[ \begin{split}z_{1} &= 2 + i\\
z_{2} &= 1 + 2 \cdot i
\end{split} \]
From z1 and z2 we can extract the real and imaginary parts as:
\[ \begin{split}a_{1} = \text{Re}(z_{1}) = 2\\
b_{1} = \text{Im}(z_{1}) = 1\\
a_{2} = \text{Re}(z_{2}) = 1\\
b_{2} = \text{Im}(z_{2}) = 2
\end{split} \]
Cartesian representation of a complex number
The Cartesian representation of a complex number is just the pair of real numbers which gives the real and imaginary part of the complex number.
The general form of Cartesian representation of a complex number is:
\[\bbox[#FFFF9D]{z=(a, b)}\]which is equivalent with:
\[z=(\text{Re}(z), \text{Im}(z))\]The Cartesian representation of the z1 and z2 complex numbers is:
\[ \begin{split}z_{1} = (2,1)\\
z_{2} = (1,2)
\end{split} \]
Using Cartesian representation we can plot a complex number in a 2-D plane, having the real part as the x-axis coordinate and the imaginary part as the y-axis coordinate.
Image: Complex numbers Cartesian representation
The complex number represents a point (a,b) on the Re-Im plane, which is the endpoint of a line starting from the origin O(0,0).
For a better understanding, we’ll plot our z1 and z2 complex numbers on the Re-Im plane.
Image: Complex numbers Cartesian representation example
Polar representation of a complex number
The polar representation of the complex number is using a radius and an angle. Similar to the Cartesian representation, instead of providing the coordinates of a point on the Re-Im plane, we can provide the length (radius) of the line and its angle with the x-axis (Re-axis).
Image: Complex numbers representation
The general form of polar representation of a complex number is:
\[\bbox[#FFFF9D]{z=r \angle \varphi}\]The radius r can be calculated using the Pythagorean theorem:
\[r^{2}=a^{2}+b^{2}\]which gives:
\[r = \sqrt{a^{2}+b^{2}}\]The angle φ can be calculated either from the sine or cosine function:
\[ \begin{split}\cos(\varphi) = \frac{a}{r}\\
\sin(\varphi) = \frac{b}{r}
\end{split} \]
which can be written as:
\[ \begin{split}\varphi = \arccos \left ( \frac{a}{r} \right )\\
\varphi = \arcsin \left ( \frac{b}{r} \right )
\end{split} \]
The real and imaginary parts of the complex number, a and b, can be written function of the radius r and angle φ:
\[ \begin{split}a = r \cdot \cos(\varphi)\\
b = r \cdot \sin(\varphi)
\end{split} \]
Replacing these in the polynomial form of the complex number, gives:
\[z = r \cdot \left ( \cos(\varphi) + i \cdot \sin(\varphi) \right )\]The angle φ can also be calculated function of a and b, using the tangent function:
\[\tan(\varphi) = \frac{b}{a}\]which gives the angle φ in radians as:
\[\varphi = \arctan \left ( \frac{b}{a} \right )\]Let’s compute the radius and the angle for our z1 and z2 complex numbers:
\[ \begin{split}r_{1}&=\sqrt{2^{2}+1^{2}}=\sqrt{5} \approx 2.24\\
r_{2}&=\sqrt{1^{2}+2^{2}}=\sqrt{5} \approx 2.24\\
\varphi_{1} &= \arctan \left ( \frac{1}{2} \right ) = 0.4636476 \text{ rad} \approx 26.57^{\circ}\\
\varphi_{2} &= \arctan \left ( \frac{2}{1} \right ) = 1.1071487 \text{ rad} \approx 63.43^{\circ}
\end{split} \]
Observation: To get the degrees from radians we need to multiply the radians with 180° and divide by π. For example:
\[0.4636476 \text{ rad} = 0.4636476 \cdot \frac{180}{\pi} = 26.565051^{\circ}\]If we check the above plot of the complex numbers z1 and z2 we can observe that the calculated angles and radius matches the graphical representation. Written in polar form, the complex numbers are:
\[ \begin{split}z_{1}=2.24 \angle 0.4636\\
z_{2}=2.24 \angle 1.1071
\end{split} \]
Exponential representation of a complex number
In exponential form, a complex number z is represented as:
\[\bbox[#FFFF9D]{z=r \cdot e^{i \cdot \varphi}}\]Where the exponential component eiφ can be written function of the sine and cosine functions (Euler’s formula):
\[e^{i \cdot \varphi} = \cos(\varphi) + i \cdot \sin(\varphi)\]which gives z equal with:
\[z=r \cdot \left ( \cos(\varphi) + i \cdot \sin(\varphi) \right )\]Written in exponential form, the complex numbers z1 and z2 are:
\[ \begin{split}z_{1}&=2.24 \cdot e^{i \cdot 0.4636}\\
z_{2}&=2.24 \cdot e^{i \cdot 1.1071}
\end{split} \]
Moivre’s formula for power of complex numbers
If we need to compute the power of a complex number, zn, where n is a natural number, we can use Moivre’s formula:
\[\bbox[#FFFF9D]{z^{n}=r^{n} \cdot \left ( \cos(n \cdot \varphi) + i \cdot \sin(n \cdot \varphi) \right )}\]For example, let’s compute the 5th power of the complex number z1 defined above:
\[z_{1}^{5}=2.24^{5} \cdot \left ( \cos(5 \cdot 0.4636) + i \cdot \sin(5 \cdot 0.4636) \right ) \approx -38 + 41 \cdot i\]Synthesis of complex numbers representation
In the table below we’ll summarise the different methods of representing a complex number and the conversion formulas from one method to another.
| Method | Formula | Conversion |
| Polynomial | \[z = a + b \cdot i\] | \[ \begin{split} a &= r \cdot \cos(\varphi)\\ b &= r \cdot \sin(\varphi) \end{split} \] |
| Cartesian | \[z=(a, b)\] | None |
| Polar | \[z=r \angle \varphi\] | \[ \begin{split} r &= \sqrt{a^{2}+b^{2}}\\ \varphi &= \arctan \left ( \frac{b}{a} \right ) \end{split} \] |
| Exponential | \[z=r \cdot e^{i \cdot \varphi}\] | \[e^{i \cdot \varphi} = \cos(\varphi) + i \cdot \sin(\varphi)\] |
| Moivre | \[z^{n}=r^{n} \cdot \left ( \cos(n \cdot \varphi) + i \cdot \sin(n \cdot \varphi) \right )\] | None |
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