Product of Additive Inverses

Summary

Product of Additive Inverses By Susam Pal on 29 May 2025 A negative number multiplied by another negative number results in a positive number. Most of us learnt this rule during our primary or secondary school years. 'Negative times negative equals positive' was a phrase drummed into us during mathematics lessons. In this article, we will prove this rule, not just for numbers but for any algebraic structure that, in a general sense, behaves somewhat like numbers. Contents Illustration Let us begin with a quick illustration that shows why the product of two negative numbers must be positive for arithmetic to make sense. Consider \[ 7 \times 8 = 56. \] The above equation can also be written as \[ (10 - 3) \times (10 - 2) = 56. \] Using the distributive property of multiplication over subtraction, we get \[ (10 - 3) \times 10 + (10 - 3) \times (-2) = 56. \] Using the distributive property again, we have \[ 10 \times 10 + (-3) \times 10 + 10 \times (-2) + (-3) \times (-2) = 56. \] Now, we will take it for granted that a positive times a negative is negative. We will prove all of this rigorously later, but for now, we are just working through an illustration, so we will accept that rule and see where it leads. The equation becomes: \[ 100 + (-30) + (-20) + (-3) \times (-2) = 56. \] Adding the first three terms gives \[ 50 + (-3) \times (-2) = 56. \] Subtracting \( 50 \) from both sides, we get \[ (-3) \times (-2) = 6. \] What we have seen here is that if we accept \( 7 \times 8 = 56, \) and that positive times negative gives a negative result, then we must also accept that \( (-3) \times (-2) = 6. \) Ring Axioms From this section onwards, we take a rigorous approach. We want to show that the rule 'negative times negative equals positive' holds, in a general sense, for any set of elements that share certain properties with numbers. As it turns out, these elements do not need to possess all the properties of complex numbers, real numbers, or even rational numbers. In fact,...