By definition, a complex number is a number in the from a+bi where and and b are real and i is the imaginary constant where i^{2 }= -1. But, what actually is a complex number and why are mathematicians and physicists so obsessed with them?

## The square root of a negative

The first time you will probably meet complex numbers is when you solve quadratic or cubic equations with no real root. This simply means that the solution to the equation does not fit on the number line we know. For example:

x^{2 }-2x+17=0

(x-1)^{2 }-1+17=0

(x-1)^{2 } +16=0

(x-1)^{2 }= -16

x-1= ±√(-16)

x = 1 ±√(-16)

So what happens now? We know the square root of a negative number doesn’t exist. When mathematicians discovered this problem, they did just as they always have done when something doesn’t work – make up something new. This is how complex numbers came about. So the solution of the equation above is:

x = 1 ± 4i

To find the value of b (in the form a+bi), simply square root the number ignoring the minus.

## Representing a complex number

The number line of the real numbers goes from left to right, imaginary numbers are added on an axis perpendicular to this, the two axis cross at (0,0).

This type of a diagram is called an argand diagram and is a useful tool to visualize the imaginary and real parts of a complex number.

## Applications

Complex numbers are fundamental to our understanding of mathematics and physics, with applications in fluid dynamics, quantum mechanics and electromagnetism to list a few.