Tuesday, February 18, 2014

Getting mathematical - Understanding complex numbers intuitively

Hello everybody!

I will be posting a series sharing some intuitive ideas about the concept of number systems in mathematics - natural numbers, rational numbers, irrational numbers and complex numbers.Essentially, what led to their discovery and their relevance in the real-world.  How to explain the notion behind them to a kid.This post is focused on complex numbers. The question that I am raising in this context is:

What does bi denote in the form a+bi? We call it imaginary part? But it has to have some real world significance? What is it? 

As simple as it sounds but understanding number system in mathematics does require quite some thought. Things which we (at least I did) took for granted and never questioned when they were taught. I remember the multiple sessions that we had in high school, solving problems involving  complex numbers - a+bi. Remember? But leaving aside the details about how to multiple them, find its conjugate etc, I wonder if I understood their real-world relevance until few days ago. I recognized a gaping hole in my education seeing that I was not aware of the things that I am going to talk about in this post earlier - the time when I actually studied these concepts in school.

Although I would love to have said that the concepts in this post are from my own understanding and research but that is not true! Please know that everything that I write in this post comes from what I learned from the book [1] and the blog [2] that I followed to solidify my understanding.

Getting started...

Here is one word answer to the question above: Rotation. 

Details: So, we have learned in high school that this imaginary number i (pronounced as iota) is a number whose square is -1. This raises the question that what number can it be whose square is a negative number. Lets try to understand it by using the concepts that we do understand.

In the equation square(x)=9. What is x? It would be 3 and -3. 
Another way to read this is: scale the number 1 by 3, then scale the result further by 3 => giving us 9. Do you see that? 

In other words, I am saying that you can observe this square operation as two steps:
1) 1 * 3 = 3 (This pertains to scaling the number by 3)
2) 3 * 3 = 9 (Scaling the result 3 by 3 times again)

Now, consider solving the following equation:

                                   1.x.x = -1

The question arises : What transformation (just like scaling in the example above) does x represent?

Answer: Start with point (1,0) on x-axis. Now, imagine rotating the point by 90 degrees. What point will you get?

(0,1).. Right?

Rotate it again by 90 degrees. Where will you end up?

It will be (-1,0). Right? Which is what we wanted. We wanted to see how we can transform 1 to -1! 

See the figure below for a pictorial representation of what I said above:

[2] Understanding complex numbers

So, the number i represents rotation. Complex numbers are a way to measure a number in terms of rotation.

They are mostly employed in solving differential equations or physics equations.

This was all that I wanted to share with you all. I really enjoyed learning about them. I hope you also found it interesting?



[1] "College Trigonometry", Richman, F., Walker, C., and E. Walker, 1970.
[2]  "A Visual, Intuitive Gide to Imaginary Numbers", betterexplained.com, K. Azad.

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