Application of complex numbers

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from 'The Scientific Harrovian - Issue 6-1, December 2020'

Josh Nash And His Contribution...

Real Life Applications of Complex Numbers

Josiah Wu

Let’s start with the basics. At a young age, we were taught how to count with positive numbers, such as one, two or three. Later in primary school, we were also introduced to negative numbers: for example, -19 is a negative number. I’m also going to assume that you are familiar with square roots (if not, you should revise). It is commonly taught to students that one cannot take the square root of negative numbers. But what if we could?

You may be wondering, “How is it possible to take the square root of a negative number?” In fact, mathematicians before the 16th century would’ve thought so as well. This was until Italian mathematician Gerolamo Cardano broke the convention by inventing imaginary numbers, in a desperate attempt to solve cubic equations. Throughout history, mathematicians have always loved to break their own rules: apart from taking the square root of a negative number, Ramanujan once proved that 1 + 2 + 3 + 4… all the way up to infinity is equal to -1/12. Another mathematician, Georg Cantor, proved that there are as many even numbers as positive integers. Therefore, what Cardano did was not uncommon (at least in historical records).

So what is an imaginary number? An imaginary number is a multiple of i = √-1. For example, √-25 is an imaginary number because it can be rewritten as √-25 = √25 × -√1 =5i. Furthermore, one can add a real number to an imaginary number to form a complex number. To demonstrate this, one can add 3, a real number, to 3i, an imaginary number, to form the complex number 3+3i.