Complex numbers are the numbers that are expressed in the form *a+bi*, where *a* and *b* are real numbers and “*i*” is the imaginary unit. The imaginary unit value is the square root of negative one, *i* = (√-1). For example, 5 + 3*i* is a complex number, where 5 is a real number (Re) and 3*i* is an imaginary number (Im).

## Examples of complex numbers

Complex numbers have the general form $latex a+bi$, where *a* is the real part of the number and *bi* is the imaginary part:

### EXAMPLES

Complex numbers are represented by the letter *z*:

- In the complex number $latex z=3+7i$, 3 is the real part and
*7**i*is the imaginary part. - In the complex number $latex z=-6+\frac{1}{2}i$, -6 is the real part and
*$latex \frac{1}{2}i$*is the imaginary part. - In the complex number $latex z=\frac{3}{4}-4i$, $latex \frac{3}{4}$ is the real part and -4
*i*is the imaginary part. - In the complex number $latex z=-\sqrt{3}+\sqrt{7}i$, $latex -\sqrt{3}$ is the real part and $latex \sqrt{7}i$ is the imaginary part.

## Properties of complex numbers

The following are some of the most important properties of complex numbers:

- Complex numbers obey the commutative property of addition and multiplication:

$latex z_{1}+z_{2}=z_{2}+z_{1}$

$latex z_{1}\times z_{2}=z_{2}\times z_{1}$

- Complex numbers obey the associative property of addition and multiplication:

$latex (z_{1}+z_{2})+z_{3}=z_{1}+(z_{2}+z_{3})$

$latex (z_{1}\times z_{2}) \times z_{3}=z_{1}( z_{2} \times z_{3})$

- Complex numbers obey the distributive property:

$latex z_{1}(z_{2})+z_{3})=z_{1}\times z_{2}+z_{1}\times z_{3}$

- Adding two complex conjugate numbers will result in a real number.
- Multiplying two complex conjugate numbers will also result in a real number.
- If
*x*and*y*are real numbers and we have $latex x+yi=0$, therefore, $latex x=0$ and $latex y=0$.

Start now: Explore our additional Mathematics resources

## How to graph complex numbers?

We can graph complex numbers using the complex plane. In this plane, the *x*-axis represents the real part and the *y*-axis represents the imaginary part:

Graphing complex numbers is very easy, we just have to locate each coordinate separately. For example, if we have to graph the number $latex z=3+2i$, we move 3 units on the real axis (the *x*-axis) and 2 units on the imaginary axis (the *y*-axis):

## What are complex numbers used for?

Complex numbers have many applications in real life, especially in engineering fields, such as electricity. Complex numbers are also used to find solutions to quadratic equations. When the equations do not touch the *x*-axis, the solutions of the equation are represented by complex numbers.

Complex numbers are particularly useful in electricity, especially in alternating current (AC) electronics. Alternating current electricity produces a sine wave because it switches between positive and negative.

Calculating this type of current could be too complicated as the waves do not fit together properly. However, the calculation of alternating currents is greatly simplified with the use of complex numbers.

Another application of complex numbers is in signal processing, which is used in cell phone technology, wireless technologies, radars, and even brain wave neurology. Basically, complex numbers are used to facilitate the calculation of processes or applications that use sine or cosine waves.

## What is the origin of complex numbers?

It is difficult to find the exact origin of complex numbers since these numbers were being used by mathematicians long before they had a proper definition.

It is known that the first reference to complex numbers was by Cardan in 1545 while he was studying the roots of polynomials. During this period, the number notation $latex \sqrt{-1}$ was used.

However, this notation was only used for the purpose of categorizing the properties of some polynomials. Back then, $latex \sqrt{-1}$ was only considered a useful notation for categorizing polynomials and was not seen as a real mathematical object.

The notation *i* was introduced by Euler in 1777. Euler defined *i* and –*i* as the two square roots of -1. It was with Euler that the notation “*i*” for imaginary numbers appeared and was formalized. However, because the existence of these numbers was not fully understood, the numbers *i* and –*i* were called “imaginary”.

In 1797 and 1799, Wessel and Gauss respectively, gave a geometric interpretation to complex numbers by representing them as points on a plane. With this, it was achieved that complex numbers are considered as something more concrete and less mysterious.

It was in 1833 that Hamilton formally defined complex numbers by showing that these numbers are composed of pairs of real numbers and the numbers that Euler defined as “*i*“. This is where the modern formulation of complex numbers began.

## See also

Interested in learning more about imaginary and complex numbers? Take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**