Why Does a Negative Number Times a Negative Number Equal a Positive?

The result follows from keeping the distributive property valid for positive and negative numbers. Consider 0 = 0 × (−1) = (1 + (−1)) × (−1). Distributing gives 0 = (−1) + (−1 × −1), so (−1 × −1) = 1. Applying the same reasoning to positive numbers a and b gives (−a)(−b) = ab.

Think of it as reversing direction twice

On a number line, multiplying by a negative reverses direction once. Multiplying by another negative reverses direction again, so the result points in the positive direction. Combined with the distributive property, this picture explains the rule more clearly than memorizing the signs alone.

This is a property of the integers and real numbers, not a special exception for negative values.