IMAGINARY NUMBERS?!

As I said in my previous post, mathematics is the dictionary to the scientific guidebook of the universe. But this dictionary has not always been as complete as it is today, for hundreds of years mathematicians worked their whole lives to solve various theoretical problems to lay down the groundwork to solve even more difficult problems. One of these conundrums that had been puzzling mathematicians for quite a long time, is the equation

When making an attempt to solve this equation, we are left stumped with 

This confronting equation has tormented many renowned mathematicians of the past, as they tried and failed to tie it to a real value, as they do with any other equation. Some of you may know that the square of any real number is a positive value, so to find the square root of a negative number, is unfeasible. This is how ‘imaginative’ numbers came into play, and this article is an attempt at explaining the basics of imaginary numbers, how ‘imaginary’ isn’t a great name, and how they have applications in the real world. 

The above equation is most commonly recognised with 𝑖 rather than 𝑥, the 𝑖 being known as iota, denoted by Leonhard Euler, a well-known mathematician. So the equation rather looks like this:

𝑖 has some curious properties when raised to different powers; it cycles with four different variations continuously: 𝑖, -1, -𝑖, 1. Even when raised to very large numbers, such as 𝑖^3460, it is still either 𝑖, -1, -𝑖 or 1.

Now grab a pen and some paper, and try to find what 𝑖⁵ is. It should look something like this:

To save space, you can take me at my word that 𝑖⁶ is equal to - 1. Using this knowledge of this pattern, to find 𝑖^3460 you divide the power by the lowest number possible, 2 for example. Once having done this, you can solve 𝑖² which is -1, and then put -1 to the number that came from the division of the 2. Like this:

I don’t like the usage of the word imaginary, as there is very thin line between 𝑖 and a number with a physical value, as you can see by the cycling from ‘imaginary’ to real. So to use a different label, I’ll introduce complex numbers. A complex number is in the form 

Where ‘a’ is the ‘real’ number and ‘b’ is imaginary. Now, a complex number does not have to include non-zero numbers. I will explain this with examples of different complex numbers and what they look like when applied to the above form.

These are all complex numbers, even without a non-zero counterpart. 

There is plenty of advanced applications in the world with 𝑖, many being in quantum mechanics and engineering. The usage of complex numbers introduces a two-dimensional cartesian plane, with the x-axis being the ‘real’ numbers and the y-axis being the ‘imaginary’ numbers. Using this plane, you can plot complex numbers, which opens into more opportunities. I am still learning these different applications, and I hope to one day present them on this blog with confidence, but for now, the basics of 𝑖 with suffice. I hope you could easily follow along with my way of explanation, and if not then I implore you to seek your own explanation by researching it yourself. I greatly recommend Khan Academy, a free online learning source in which I am learning about complex and imaginary numbers. Thank you for reading this post, and continuing to support this blog.