Question

Can there exist a set of numbers which does not contain real numbers? If yes, how do we represent it .

Answer 1

Yes ,they are called imaginary numbers.
A non-real, or imaginary, number is any number that, when multiplied by itself, produces a negative number.
Mathematicians use the letter "i" to symbolize the square root of -1. An imaginary number is any real number multiplied by i. For example, 5i is imaginary; the square of 5i is -25.

They are represented with an i next to them.

Hope this helps :)