Hello, I am seeking some aid in proving that the square of a number is always non-negative. Here is some of my proof: A number, call it a, is either positive, negative, or zero. A number squared is produced when you take the number and multiply it by itself. So, we have three cases to consider. When a < 0: If a < 0, then 0 - a > 0; and so, 0 - a = -a must be in the set of positive numbers. If -a is in the set of positive numbers, then, by property P12 (Closure under multiplication: If a and b are in P, then a • b is in P), (-a)(-a) must be in the set of positive numbers, implying that it is a positive number, and the negative signs may vanish. (-a)(-a) = a x a = a^2 > 0. Initially, I thought that this proof would be valid; but as I went along further along, I developed an inkling of a feeling that it was not so. What is wrong with the proof of the case a > 0?