When you square a negative number, you still get a positive number: –2 times –2 also equals 4. When you square zero, you get zero. Positive numbers, negative numbers, and zero all give you nonnegative squares, and those three possibilities cover the whole number line. This means that there is no number on the number line that gives you a negative number when you square it. The square root of a negative number seemed like a ridiculous concept.
