Why is square root of the number -9 not a real number?

For the same reason that the square root of ANY negative number is not a real number.Real numbers are positive, negative, or zero.

* The square of a positive number is a positive number.

* The square of a negative number is a positive number.

* The square of zero is zero.

In other Words, in no case will you get a REAL number whose square is a negative number. The square roots of negative numbers are said to be "imaginary" - a name given for historical reasons. They are just as "real" or "unreal" as the so-called real numbers, but the point is that they are a different kind of numbers.