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What are positive square roots called?
They are called real numbers. Negative square roots must be complex numbers.
Is there a number that doesn't have a square root?
Any negative number, and infinity. However, a number (called "i") was created to represent the square roots of negative numbers.
Why don't negative numbers have square roots?
They can, but they are called imaginary numbers. This is because a "square" of something is the number that, when multiplied by itself, will equal the first number. A negative multiplied by a negative equals a positive, and positives can't turn into negatives on their own, so there are no square roots of negative numbers.
What kind of number system is a square root?
A square root is not a number system. Square roots of non-negative numbers may be rational or irrational, but they all belong to the set of real numbers. The square roots of negative numbers do not. To include them, the number system needs to be extended to the complex numbers.
What has integers as its square roots?
Such a number is called a "perfect square".
