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 \( (-3) \times (-2) = 6. \)

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:

1. Associativity of addition: For all \( a, b, c \in R, \) we have \( a + (b + c) = (a + b) + c. \)

2. Commutativity of addition: For all \( a, b \in R, \) we have \( a + b = b + a. \)

3. Additive identity: There exists an element \( 0 \in R \) such that for all \( a \in R, \) we have \( a + 0 = a = 0 + a. \)

4. Additive inverse: For each \( a \in R, \) there exists an element \( -a \in R \) such that \( a + (-a) = 0 = (-a) + a. \)

5. Associativity of multiplication: For all \( a, b, c \in R, \) we have \( a \cdot (b \cdot c) = (a \cdot b) \cdot c. \)

6. Left distributivity of multiplication over addition: For all \( a, b, c \in R, \) we have \( a \cdot (b + c) = (a \cdot b) + (a \cdot c). \)

7. Right distributivity of multiplication over addition: For all \( a, b, c \in R, \) we have \( (b + c) \cdot a = (b \cdot a) + (c \cdot a). \)

Note that we do not assume that the ring contains multiplicative identity, nor do we assume that multiplication is commutative. Many familiar types of numbers form rings. For example, the set of integers forms a ring with the usual addition and multiplication operations. The sets of rational numbers, real numbers, and complex numbers satisfy the ring axioms too.

Rings need not consist of numbers; they may contain elements of any type. As long as a set of elements, together with suitable addition and multiplication operations, satisfies the seven axioms above, it forms a ring. For example, the set of all polynomials in the indeterminate \( t \) with coefficients in some ring \( R \) forms a ring under the usual addition and multiplication of polynomials. Such a ring is called a polynomial ring and it is denoted \( R[t]. \)

Some texts include the following additional axioms for the closure properties of a ring:

1. Closure under addition: For all \( a, b \in R, \) we have \( a + b \in R. \)

2. Closure under multiplication: For all \( a, b \in R, \) we have \( a \cdot b \in R. \)

However, stating these axioms explicitly is usually considered redundant because a binary operation is closed by definition. A binary operation \( \circ \) on a set \( M \) is defined to be a function \[ \circ : M \times M \to M; \quad (a, b) \mapsto a \circ b. \] This definition automatically implies the closure property, since the domain and codomain are the same. The addition and multiplication operations on a ring \( R \) may be defined as \begin{align*} + &: R \times R \to R; \quad (a, b) \mapsto a + b, \\ \cdot &: R \times R \to R; \quad (a, b) \mapsto a \cdot b. \end{align*} These definitions imply that a ring is closed under addition and multiplication. In practice, while deciding if some set \( R \) forms a ring, we should always verify that the addition and multiplication operations indeed have \( R \) as the codomain to confirm that the closure property holds.

Theorem 1. Let \( R \) be a ring with \( + \) and \( \cdot \) operations. Then for all \( a \in R, \) we have \[ -(-a) = a. \]

Proof. This result follows directly from the additive inverse axiom. First, observe that \[ a + (-a) = 0. \] Therefore \( a \) is an additive inverse of \( -a, \) i.e., \[ -(-a) = a. \] This completes the proof.

Notice that this proof does not involve the multiplication operation of a ring at all. In fact, it holds true in a more general algebraic structure known as a group, which requires only a binary operation with associativity, an identity element, and inverses. A ring, under addition, is also a group. Since the proof relies solely on these additive group properties, this theorem holds for all groups. However, for brevity, and to avoid introducing group axioms separately, I have stated and proved this theorem in the context of rings.

It is also worth noting that the additive inverse is unique in a ring (as well as in any group), but since this fact is not needed for later results, its proof has been omitted. Even if, hypothetically, there were two distinct additive identities, \( 0_1 \) and \( 0_2, \) in a ring (there are not, of course), the arguments below would still hold if we simply focussed on either one of them and labelled it \( 0. \)

Theorem 2. Let \( R \) be a ring with \( + \) and \( \cdot \) operations. Then for all \( a \in R, \) we have \[ a \cdot 0 = 0 \cdot a = 0. \]

Proof. Using the additive identity axiom, we get \[ 0 + 0 = 0. \] Multiplying both sides on the left by \( a, \) we get \[ a \cdot (0 + 0) = a \cdot 0. \] Using the left distributivity axiom, we get \[ a \cdot 0 + a \cdot 0 = a \cdot 0. \] Let \( b = a \cdot 0. \) Then \[ b + b = b. \] Since a ring is closed under multiplication, \( b \in R. \) By the additive inverse axiom, there exists \( -b \in R \) such that \( b + (-b) = 0. \) Adding \( -b \) to both sides of the above equation, we get \[ (b + b) + (-b) = b + (-b). \] By associativity of addition in a ring, we get \[ b + (b + (-b)) = b + (-b). \] Since \( b + (-b) = 0, \) the above equation becomes \[ b + 0 = 0. \] By the additive identity axiom, we get \[ b = 0. \] Since \( b = a \cdot 0, \) the above equation may be written as \[ a \cdot 0 = 0. \] A similar argument shows that \[ 0 \cdot a = 0. \] This completes the proof.

Theorem 3. Let \( R \) be a ring with \( + \) and \( \cdot \) operations. Then for all \( a, b \in R, \) we have \[ a \cdot (-b) = (-a) \cdot b = -(a \cdot b). \]

Theorem 4. Let \( R \) be a ring with \( + \) and \( \cdot \) operations. Then for all \( a, b \in R, \) we have \[ (-a) \cdot (-b) = a \cdot b. \]