Product of Additive Inverses

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, if they satisfy a small and specific set of properties held by the integers, then the rule still holds. These properties are known as the ring axioms.

A ring is an algebraic structure consisting of a set \( R \) with two binary operations \( + \) and \( \cdot, \) called addition and multiplication respectively, satisfying the following axioms:

Associativity of addition: For all \( a, b, c \in R, \) we have \( a + (b + c) = (a + b) + c. \) Commutativity of addition: For all \( a, b \in R, \) we have \( a + b = b + a. \) Additive identity: There exists an element \( 0 \in R \) such that for all \( a \in R, \) we have \( a + 0 = a = 0 + a. \) Additive inverse: For each \( a \in R, \) there exists an element \( -a \in R \) such that \( a + (-a) = 0 = (-a) + a. \) Associativity of multiplication: For all \( a, b, c \in R, \) we have \( a \cdot (b \cdot c) = (a \cdot b) \cdot c. \) Left distributivity of multiplication over addition: For all \( a, b, c \in R, \) we have \( a \cdot (b + c) = (a \cdot b) + (a \cdot c). \) Right distributivity of multiplication over addition: For all \( a, b, c \in R, \) we have \( (b + c) \cdot a = (b \cdot a) + (c \cdot a). \)

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