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:

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, *z _{1}* and

*z*, in the following polynomial form:

_{2}z_{1} &= 2 + i\\

z_{2} &= 1 + 2 \cdot i

\end{split} \]

From *z _{1}* and

*z*we can extract the real and imaginary parts as:

_{2}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:

which is equivalent with:

\[z=(\text{Re}(z), \text{Im}(z))\]The Cartesian representation of the *z _{1}* and

*z*complex numbers is:

_{2}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.

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 *z _{1}* and

*z*complex numbers on the Re-Im plane.

_{2}### 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).

The general form of **polar representation** of a complex number is:

The radius *r* can be calculated using the **Pythagorean theorem**:

which gives:

\[r = \sqrt{a^{2}+b^{2}}\]The angle *φ* can be calculated either from the sine or cosine function:

\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*:

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:

which gives the angle *φ* in radians as:

Let’s compute the radius and the angle for our *z _{1}* and

*z*complex numbers:

_{2}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:

If we check the above plot of the complex numbers *z _{1}* and

*z*we can observe that the calculated angles and radius matches the graphical representation. Written in polar form, the complex numbers are:

_{2}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:

Where the exponential component *e ^{iφ}* can be written function of the sine and cosine functions (

**Euler’s formula**):

which gives z equal with:

\[z=r \cdot \left ( \cos(\varphi) + i \cdot \sin(\varphi) \right )\]Written in exponential form, the complex numbers *z _{1}* and

*z*are:

_{2}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, *z ^{n}*, where

*n*is a natural number, we can use

**Moivre’s formula**:

For example, let’s compute the 5^{th} power of the complex number *z _{1}* defined above:

### 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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