Complex Numbers Explained in 5 Minutes

Complex Numbers Explained in 5 Minutes

It's not a great name - complex numbers are not complex! To see how they occur think about polynomials, which are equations made from powers, for example..

x^2-9=0 which has solutions x=3 and x=-3

It turns out that although polynomials are easy to write down they are not so easy to solve.

Using integers {..,-3,-2,-1,0,1,2,3, ..} it's possible to solve some of these equations, but not all.

Using rational numbers or "fractions", a/b where a and b are integers, it's possible to solve many more but still not all.

A major breakthrough occurred when mathematicians introduced a third type of number called irrational numbers which are not integers and not rational numbers. This allowed many more polynomial equations to be solved.. but still not all!

In fact, the situation was a bit embarrassing, because one of the simplest polynomial equations could still not be solved even with three types on numbers available. It was this..


This incredibly simple equation cannot be solved by an integer, it cannot be solved by a rational number, and it cannot be solved by an irrational number!

So what did mathematicians do? No problem, they just invented a new number. They called it i for "imaginary", another poor name choice.

i=the square root of -1

So i solves the original equation, that is..


Of course i can be used to define an infinite amount of new numbers, just take any number and multiply it by i. Mathematicians called the new class of numbers "complex numbers". Not a great name.

So, it turns out that with 4 types of numbers available (integer, rational number, irrational numbers and complex numbers) it was possible to solve all polynomial equations.

There was a big bonus. In addition to helping solve all polynomial equations, the number i turned out to be amazingly useful in other areas of mathematics and it's also used in many areas of physics. So the number i, defined as "that number which when multiplied by itself gives -1", turned out to be an amazing invention!

Content written and posted by Ken Abbott