How To Really Calculate The Square Root Of i?

If you are the typical person, you are probably not looking to calculate the square root of i, the unit imaginary number. If you are a math enthusiast, however, things are different. Such a question intrigues you, and you see possibilities that you can work with.

The concept of the imaginary number ‘i’ is tricky on its own. But when we stack the question of the square root of i (√i) on top of it, things get even tricker.

At least, that is what it looks like on the surface. This essay is aimed to show how such a seemingly complex question could be answered rather simply. And whilst doing so, we will also be strengthening certain intuitions surrounding the basics of complex numbers.

Let’s start with the concept of imaginary numbers and how they are used in combination with real numbers.

This essay is supported by Generatebg

Imaginary Numbers and Complex Numbers

A Brief Imaginary History

Imaginary numbers were born out of a specific need in mathematics. You see, mathematicians of the past kept running into certain contradictory situations. They were challenged with computing a number whose square value leads to a negative real number. But no such (real) number existed back then.

So, eventually, the mathematicians decided to “invent” such numbers, whose unit is ‘i’, and is defined as the square root of -1 (i = √(-1)).

An imaginary number is hence defined as a real number multiplied by the imaginary unit (i).

It was Rene Descartes who first coined the term “imaginary numbers” as a derogatory term for these fictitious and (seemingly) useless numbers.

But later on, mathematicians such as Leonhard Euler, Augustin-Lois Cauchy, and Carl Friedrich Gauss proved how useful these “imaginary” numbers were. Consequently, imaginary numbers gained immense mathematical popularity and the terminology stuck.

Fast forward to today, the concept of imaginary numbers is used in all sorts of applications and has become indispensable to mathematics.

A Complex Present

The mathematicians of the past figured out that imaginary numbers are particularly useful if they are combined with real numbers in the form: [x+(i*y)], where x and y are real numbers, and i is the imaginary unit.

Such complex numbers are usually visualized on a cartesian plane with the real numbers on the X-axis and the imaginary numbers on the Y-axis. Such a cartesian plane is known as the complex plane.

With this, we have the basic toolkit that we need to answer our main question.

We will now proceed to calculate √i using a mathematical approach first, and then an intuitive approach next.

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A Mathematical Approach to Calculate the Square Root of i

The first step we need to take is to recognise the fact that any real or imaginary number can be expressed as a complex number on the complex plane. For example, the imaginary unit i can be represented as a complex number as follows:

i = x + iy

where x = 0 and y = 1

Therefore, i = 0 + (1*i)

If any real or imaginary number can be expressed as a complex number, it should also be possible to express √i as a complex number:

Let us now see what happens if we square this equation.

We end up with the following equation system that could be solved using algebra:

When we plug these values back into the equation for √i, we get the following result:

This is the answer (solution) to our original question. But what does this mean exactly? To answer that, let us proceed with an intuitive approach to our problem.

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An Intuitive Approach to Calculate the Square Root of i

Let us now consider the complex plane once again. If we further suppose a vector from 0 to i, multiplying this vector by i rotates it by π/2 radians (or 90°) in the anticlockwise direction to arrive at -1. If we multiply this vector by i, we rotate it once again by π/2 radians to arrive at -i. Similarly, further multiplication leads to rotations that arrive at 1 and i respectively.

If multiplying by i rotates our unit imaginary vector by π/2 radians (anticlockwise), then multiplying by √i would rotate it by π/4 radians (or 45°). This would mean that √i would have two root points along the unit circle: one at 45° from 1 towards i, and another at 135° from -1 towards -i.

There we go. The weird-looking x and y values we got from the mathematical approach could be intuitively understood as the outcomes of (angularly) slicing the rotation performed by (1*i) or (-1*-i) in half.

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Final Thoughts

Wejust showed how such a seemingly ‘complex’ problem (pun totally intended) could be solved relatively easily.

The key is to not be fazed by the challenge but to build incrementally on our fundamental understanding of the problem.

Complex number theory is quite often used in wave equations and phase calculations. So, most modern digital and analogue technologies make use of such mathematics.

Problems such as calculating √i really just touch the surface of such applications and give us a sneak peek into the complex architecture of our present-day technological devices and applications!

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Further reading that might interest you: How To Intuitively Understand Euler’s Identity? and Logarithms: The Long Forgotten Story Of Scientific Progress.