Hello everybody!

I will be posting a series sharing some

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.

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

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This was all that I wanted to share with you all. I

Dippy

[1] "College Trigonometry", Richman, F., Walker, C., and E. Walker, 1970.

[2] "A Visual, Intuitive Gide to Imaginary Numbers", betterexplained.com, K. Azad.

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

*(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.*

**i**
In the equation square(x)=9. What is

**? It would be 3 and -3.***x*__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?

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

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

*represents rotation. Complex numbers are a way to measure a number in terms of rotation.***i**They are mostly employed in solving differential equations or physics equations.

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This was all that I wanted to share with you all. I

**really**enjoyed learning about them. I hope you also found it interesting?

Dippy

**References**

[1] "College Trigonometry", Richman, F., Walker, C., and E. Walker, 1970.

[2] "A Visual, Intuitive Gide to Imaginary Numbers", betterexplained.com, K. Azad.