Distributivity¶

Definition¶

Let $\star$ and $\circ$ be binary operations on a set $A$ . The operation $\star$ is left distributive over $\circ$ if

a\star(b\circ c)=(a\star b)\circ(a\star c)

for all $a,b,c\in A$ $A$ , and right distributive over $\circ$ if

(a\circ b)\star c=(a\star c)\circ(b\star c)

for all $a,b,c\in A$ $A$ . It is distributive over $\circ$ if it is both left and right distributive.

Interpretation¶

Distributivity is the rule for multiplying out brackets, and read backwards, for factoring out a common factor. It is the law that connects two operations, whereas associativity and commutativity each concern a single operation.

Area model of 3(4 + 5): a 3-by-9 grid rectangle split into a blue 3-by-4 part of area 12 and a red 3-by-5 part of area 15, which together give 27.

The figure reads the first example, $3(4+5)=3\cdot4+3\cdot5=27$ $27$ , as areas. A rectangle of height $3$ and width $4+5=9$ has area $3(4+5)$ $5)$ ; cutting it where the width $4$ ends splits it into rectangles of areas $3\cdot4=12$ and $3\cdot5=15$ $15$ . Multiplying out the brackets is this cut, and factoring out the common factor $3$ is gluing the two rectangles back together.

Drag $\boldsymbol{x}$ and $\boldsymbol{y}$ and change the scalars. Adding first and then scaling lands on the same point as scaling first and then adding: these are the two distributive laws of a vector space.

If $\star$ is commutative, left and right distributivity are equivalent. For a non-commutative operation such as the matrix product of Week 3, both laws must be stated.

The order of the two operations matters: " $\star$ distributes over $\circ$ " is a different statement from " $\circ$ distributes over $\star$ ". Multiplication of real numbers distributes over addition, but addition does not distribute over multiplication. Many algebra slips come from applying a distributive law that does not hold, such as splitting $c/(a+b)$ into $c/a+c/b$ $c / b$ .

Key Results¶

A ring, and therefore a field, requires multiplication to be distributive over addition. Distributivity is the only ring axiom that links the two operations.
In any ring, $a\cdot0=0$ $0$ : distributivity gives $a\cdot0=a\cdot(0+0)=a\cdot0+a\cdot0$ $0$ , and adding $-(a\cdot0)$ to both sides leaves $0=a\cdot0$ $0$ . The same argument on the right gives $0\cdot a=0$ $0$ .
In any ring, $(-a)b=-(ab)$ $- (ab)$ : by distributivity $ab+(-a)b=(a+(-a))b=0\cdot b=0$ $0$ , so $(-a)b$ is the additive inverse of $ab$ . In particular $(-a)(-b)=ab$ $ab$ .
Repeated use gives the general laws $a(b_1+\cdots+b_n)=ab_1+\cdots+ab_n$ and $(a+b)(c+d)=ac+ad+bc+bd$ $b d$ ; the second needs both the left and the right law.
The vector space axioms of Week 2 contain two distributive laws for scalar multiplication: $\alpha(\boldsymbol{x}+\boldsymbol{y})=\alpha\boldsymbol{x}+\alpha\boldsymbol{y}$ and $(\alpha+\beta)\boldsymbol{x}=\alpha\boldsymbol{x}+\beta\boldsymbol{x}$ $β x$ . Here the operations combine elements of two different sets, scalars and vectors.

Examples¶

Multiplication of real numbers distributes over addition: $3(4+5)=3\cdot4+3\cdot5=27$ $27$ .
Complex multiplication distributes over complex addition, which is how products are expanded: $(1+\mathrm{i})(2+3\mathrm{i})=2+3\mathrm{i}+2\mathrm{i}+3\mathrm{i}^2=-1+5\mathrm{i}$ $5 i$ .
For sets, intersection distributes over union and union distributes over intersection: $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$ and $A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$ $C)$ .
Addition of real numbers does not distribute over multiplication: $1+(2\cdot3)=7$ $7$ , but $(1+2)\cdot(1+3)=12$ $12$ .
A one-sided law: on $\mathbb{R}$ define $a\star b=a$ $a$ . Then $(a+b)\star c=a+b=(a\star c)+(b\star c)$ $c)$ , so $\star$ is right distributive over $+$ ; but $1\star(1+1)=1$ while $(1\star1)+(1\star1)=2$ $2$ , so it is not left distributive.
Division is right distributive over addition, $(a+b)/c=a/c+b/c$ for $c\neq0$ $0$ , but not left distributive: $1/(1+1)=\tfrac{1}{2}$ $\frac{1}{2}$ , whereas $1/1+1/1=2$ $2$ .
For powers of positive reals, $(ab)^c=a^cb^c$ $a^{c} b^{c}$ , so exponentiation is right distributive over multiplication; but $2^{1\cdot1}=2$ $2$ , whereas $2^1\cdot2^1=4$ $4$ .
For matrices of compatible sizes, the matrix product distributes over matrix addition on both sides: $\boldsymbol{A}(\boldsymbol{B}+\boldsymbol{C})=\boldsymbol{A}\boldsymbol{B}+\boldsymbol{A}\boldsymbol{C}$ and $(\boldsymbol{B}+\boldsymbol{C})\boldsymbol{D}=\boldsymbol{B}\boldsymbol{D}+\boldsymbol{C}\boldsymbol{D}$ $C D$ .

Prerequisites¶

Binary Operation

Sources¶

Math II: Linear Algebra · Week 1

Aliases: distributive law, distributive property, distributive Keywords: left distributivity, right distributivity, distributes over, multiplying out brackets, factoring out Symbols: alpha, beta, element of, intersection, real numbers, union

Last updated: 17 October 2026 21:53