Question

If I get the square root of x equals a negative integer, is that a real number?

Answer 1

This is a weird question. First, you should never get that square root of x is equal to a negative number.

In short. Logically if \(\sqrt{x}=-1\) then YES \(\sqrt{x}\) is a real number. This is because in an 'a implies b' statement, if the antecedent a is false then b is \(\textit{vacuously}\) true.

So I think your question should be stated as follows: Is it possible that the square root of a real number is equal to a negative number.

This is not true by by the order axioms of the real numbers. A direct result of one of them states that if a>0 and b>0 then ab>0 which gives a<0 and b<0 implies ab>0.

Answer 2

\[ \sqrt{4} = -2 \text{ or } +2\]
i.e. -2 * -2= 4
to get an imaginary number you have to take the square root of a negative number. Example
\[ \sqrt{-4} = -2i \text{ or } +2i \]

Answer 3

\[\sqrt{4}\] means the positive square root of 4. If you want the negative square root you must write:
\[-\sqrt{4}\]