Reality of Imaginary Numbers

Before we get to know about imaginary numbers, I would like you to take back to the history of numbers development. When we were learning about countable numbers (Natural Numbers), suddenly talking about zero was not easy. It took years to understand zero in the real sense. Now comes the negative integers. 



Assume you are in the ancient era, having 5 cows and 4 dogs as your pet, and your neighbor is borrowing 7 cows from you. Is it practically possible? What will you do? 

Probably, you will start scolding your neighbor. How can you give him 7 cows when you have only 5. Practically you are trying to subtract 7 from 5, which you think is impossible.

Can you now subtract 7 from 5? Yes of course you can, and the answer is -2. This tiny negative symbol is replicating debt.

The concept of a negative number was not easy to digest, but now our whole banking system runs with this negative symbol. Whenever you take a loan from a bank, your account balance shows negative. We are now very much comfortable with these negative numbers.

Imaginary numbers too have a similar story, we began with finding a solution of equation x2 + 1 = 0.

As any real number can not be the solution to such an equation (x being real x square will be position and when added t o1 can never become 0). So we need something beyond real numbers.

We defined a symbol i (iota) to solve this problem. Symbol iota (i) = sqrt(-1) or i^2=-1

Here iota is one imaginary unit, to be located along vertical axes (Imaginary axis).

 Imaginary Number



The square root of a negative real number is an imaginary number while solving the equation 

x2 + 1 = 0 we 

get x =± i  which is imaginary, so the quantity is denoted by ‘i’ called ‘iota’.

 


Term “imaginary” is a bit misleading, In fact Imaginary numbers are not man made, they appear everywhere in nature. Without imaginary numbers it's be almost impossible to explain waves, the motion of fluids, or Quantum mechanics.

Wave Motion



Properties of iota (i):

i = sqrt(-1)  so i2 = –1, i3 = – 1 and i4 = 1.

 Hence n Î N, in = i, – 1, – i, 1 attains four values according to the value of n, so

i4n + 1 = i,         i4n + 2 = -1      i4n + 3 = – i,         i4n + 4 = 1.

 

Introduction To Complex Numbers

A complex number z can be written in the following standard form:


z = a + bi where a, b are real numbers 

 


  		•

    a is known as a real part of z

    i.e.

    a = Re (z)


    b is known as the imaginary part of z

    i.e.

    b = Im (z)


    We can write z = Re (z) + i Im (z)

     

     
        The modulus of a complex number is denoted by | z |

 

                                                                  Representation of Complex Numbers (see applet below)



Drag the pointer to move to a new complex Number and observe the real and imaginary part of the complex numbers here.


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Happy Intuitive Learning!