Imaginary Number

We know there are natural numbers, real numbers etc. Like them, there is a imaginary number. It is denoted by 'i'. How does it come?

We know that square root of a positive number is always positive. For example: 4²=16 and √16=4. Here 4 is positive number. So, it satisfies the equation X² =16.

So square of any positive or negative number is a positive number.

But look at the equation:     x²+4=0 that implies x² = − 4

What value of x will satisfy this equation? There is no real number whose square is negative and that is what we have been told that we can't take the square root of a negative number.

To solve this problem, an Imaginary Number 'i' was proposed( or invented, discovered whatever it is) where  

i² = -1

and 

i=√-1

So, we can write √−4 as  √−1.√4 = i2 and √−7 as  √−1.√7 = i7.

Imaginary number 'i'  has interesting property. Higher integral powers of 'i' cycles through i,−1,− i and 1. 

i²     =  −1
i³     =  i².i  =  −1· i  =  − i
i⁴     =  i³.i  =  − i· i  =  − i²  =  −(−1)  =  1

i⁵     =  i⁴.i  =  1· i  =  i

Similarly,
i^-1   =  1/i   =   i²/i   =  − i
i^-2   =  1/i²   =   1/-1   =  − 1
i^-3   =  1/i³   =   1/-i   =  (−1)(i^-1) = (−1)(− i) = i
i^-4   =  1/i⁴   =   1/i².i²   =  (−1)(−1) = 1
and
i⁰ =1

 

i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis
Photo courtesy: wikipedia

The concept of an imaginary number may be intuitively more difficult to grasp than that of a real number. It is widely used in mathematics and  physics. In electrical and electronic engineering, it is denoted by 'j' because 'i' is used as symbol of current.