Multiplication in the Real Numbers

Multiplication as an Operation on the Real Numbers

The previous tutorial studied addition in \(\mathbb{R}\), including additive inverses and the reversibility of adding a fixed number. Multiplication is another operation on the real numbers, governed by the field axioms introduced earlier in this course. These axioms let us group and reorder factors, distribute multiplication over addition, and use the number \(1\) as a multiplicative identity.

For \(x,y\in\mathbb{R}\), their product \(xy\) is again a real number; this is the closure property of multiplication. The associative law says that \((xy)z=x(yz)\), and the commutative law says that \(xy=yx\). Thus a finite product can be regrouped and reordered without changing its value. The distributive law connects multiplication to addition:

By commutativity, distribution also works when the sum is on the left: \((x+y)z=xz+yz\). These are field axioms, not consequences of the corresponding addition rules. They are what justify expanding a product across a sum. The multiplicative identity is \(1\), so \(1x=x1=x\) for every real \(x\). The number \(0\), by contrast, makes a product equal to zero, as we will prove below.

A distinction between \(0\) and \(1\) is important. Multiplying by \(1\) leaves a number unchanged, whereas multiplying by \(0\) gives zero. Neither statement depends on whether the other factor is positive, negative, or zero. We first establish the zero rule from the field axioms.

Zero and Products with Negative Numbers

Proof. Since \(0+0=0\), distributivity gives

Add the additive inverse \(-(x0)\) to both sides. The left side becomes \(0\), and the right side becomes \(x0+0=x0\), by associativity, the additive-inverse property, and the additive identity property. Hence \(0=x0\). Commutativity of multiplication gives \(0x=x0=0\). \(\square\)

This proof matters because the zero rule is not a separate multiplication axiom in the field description being used here; it follows from distributivity and the additive rules. It also allows us to derive how multiplication interacts with additive inverses. Recall that \(-x\) denotes the unique additive inverse of \(x\).

Proof. By distributivity and the additive-inverse property,

Therefore \((-x)y\) is an additive inverse of \(xy\). The uniqueness of additive inverses gives \((-x)y=-(xy)\). By commutativity, \(x(-y)=(-y)x=-(yx)=-(xy)\), using the first identity and \(yx=xy\).

For the last identity, apply the first identity with \(y\) replaced by \(-y\), and then use the second identity and the fact that the additive inverse of an additive inverse is the original number:

This proves all three formulas. \(\square\)

The sign of a product is therefore determined by the signs of its factors: changing the sign of exactly one factor changes the product to its additive inverse, while changing the signs of both factors leaves the product unchanged. These conclusions come from distributivity and the additive-inverse rules, rather than from a separate sign convention.

Distributivity in Calculation and Simplification

Distributivity is the rule that expands a product involving a sum. It also works in reverse: a sum of terms with a common factor can be written as one product. The reversal is justified by the same equality, read from right to left. Associativity and commutativity allow the terms to be arranged so that a common factor is visible, but do not by themselves permit a factor to be distributed over addition.

For expressions with variables, the same laws apply because variables stand for real numbers. For example, distributing \(4\) over \(a+b\) gives \(4a+4b\). If the sum is multiplied by a negative number, the negative factor applies to every term. One way to check that the signs are handled correctly is to first use distributivity and then evaluate each product using the product-with-inverses theorem.

Multiplication by a Fixed Number

Fixing one factor defines a function on the real numbers. Its behavior depends on whether that fixed factor is zero. Multiplication by zero sends every real number to zero, so it cannot be undone. Multiplication by a nonzero number, however, can be reversed by multiplying by its multiplicative inverse. The field axioms ensure that every nonzero real number has such an inverse.

Proof. Closure of multiplication ensures that \(M_a(x)=ax\) is real for every \(x\in\mathbb{R}\), so \(M_a\) has the stated codomain. We compute the two compositions. For each \(x\in\mathbb{R}\), associativity and the inverse and identity properties give

For each \(y\in\mathbb{R}\), the other order gives

Thus \(M_{a^{-1}}\circ M_a=\operatorname{id}_{\mathbb{R}}\) and \(M_a\circ M_{a^{-1}}=\operatorname{id}_{\mathbb{R}}\). By the inverse-function result established earlier in the course, \(M_a\) is bijective and its inverse is \(M_{a^{-1}}\). \(\square\)

The nonzero hypothesis is essential. If \(a=0\), then \(M_0(x)=0x=0\) for every real \(x\), by the zero-product theorem. Since distinct real numbers have the same output under \(M_0\), that function is not injective and therefore is not bijective. This is why division by a real number is permitted only when the divisor is nonzero.

Using Multiplication Rules Carefully

Each multiplication manipulation should be tied to the rule that justifies it. Associativity changes grouping, commutativity changes the order of factors, and distributivity connects products with sums. The identity \(1\) leaves a factor unchanged, while the zero-product rule shows that a zero factor makes the entire product zero. The product-with-inverses theorem determines how negative factors affect signs.

A common error is to confuse multiplication by zero with multiplication by one. For example, \(0x=x\) is not a multiplication identity; the zero-product theorem gives \(0x=0\). Another common error is to distribute across multiplication rather than addition. The field axioms give \(x(y+z)=xy+xz\), but do not give \(x(yz)=xy+xz\). The latter generally fails: with \(x=2\), \(y=3\), and \(z=4\), its left side is \(2(3\cdot4)=24\), while its right side is \(2\cdot3+2\cdot4=14\).

The earlier course result called the Cancellation and Zero-Product Properties gives another important warning: cancellation of a common factor requires that factor to be nonzero. For instance, \(0\cdot2=0\cdot5\), but \(2\ne5\). Multiplication by zero is not reversible, just as the multiplication-map theorem predicts. When solving an equation by undoing multiplication, first confirm that the factor being undone is nonzero.