What are nonreal numbers?

Answer 1

The square of 0 is 0. For any other number, whether it is positive or negative, the square is positive. Thus, the square of any real number is non-negative. This means that only non-negative number can have square roots.

So what happens if you are trying to solve the equation x^2 +4 = 0. It is not a particularly complicated equation.

Simplify to x^2 = -4 and so x = sqrt(-4). But then what? 2*2 = 4 and (-2)*(-2) is also 4 so neither can be an answer.

To get around that, mathematicians introduced imaginary numbers. Then, the answer to the above question is that x = 2i (2j if you are a physicist). And they added the special rule for multiplication: i*i = -1. These are the simplest form of non-real numbers.

The next level up are complex numbers and these are numbers of the form a + bi where a and b are real and i is the imaginary number representing the square root of -1.

Yet another level up are the quaternions which are numbers of the form a + bi + cj + dk where a, b, c and d are real, and i,j k are similar to the i of the complex numbers but with further multiplication rules for i, j and k. See the link below for more on quaternions.

http://en.wikipedia.org/wiki/Quaternion

Answer 2

The set of imaginary numbers contains the square root of negative one and its multiples.