## Commutative, Associative, and Distributive Laws

Table of Contents

## Commutative Laws

Commutative Laws say we can swap numbers over and still get the same answer.

### Commutative Laws for Addition

**a + b = b + a**

4 + 7 = 7 + 4

### Commutative Laws for Multiplication

**a × b = b × a**

3 × 8 = 8 × 3

From these laws it follows that any finite sum or product is unaltered by reordering its terms or factors.

Commutativity holds for many systems, for examples: the real or complex numbers. In system of n × n matrices or the system of quaternions, commutativity of multiplication is invalid.

Scalar multiplication of 2 vectors is commutative

**a·b = b·a **

But, vector multiplication is not commutative

**a × b = −b × a. **

The commutative law does not necessarily hold for multiplication of conditionally convergent series.

### Commutative Law for subtraction or division

The Commutative Law does **not work** for subtraction or division:

#### when we subtract

**a – b ≠ b – a**

10 – 7 = 3, but 7 – 10 = -3

#### when we Divide

**a ÷ b ≠ b ÷ a**

15 ÷ 5 = 3, but 5 ÷ 15 = ⅓

### Commutative Percentages!

We know,

**a × b = b × a**

So it is true that

**a% of b = b% of a**

10% of 60 = 60% of 10, which is 6

## Associative Laws

Associative Laws say that it doesn’t matter how we group the numbers or which we calculate first.

### Associative Laws for Addition

**(a + b) + c = a + (b + c)**

(1 + 2) + 3 = 1 + (2 + 3)

### Associative Laws for Multiplication

**(a × b) × c = a × (b × c)**

(4 × 5) × 6 = 4 × (5 × 6)

### Associative Law for subtraction or division

The Associative Law does **not work** for subtraction or division:

#### when we subtract

**(a – b) – c ≠ a – (b – c)**

(7 – 4) – 1 = 3 – 1 = 2, but 7 – (4 – 1) = 7 – 3 = 4

#### when we divide

**(a ÷ b) ÷ c ≠ a ÷ (b ÷ c)**

(8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4

The terms or factors may be associated in any way desired. While associativity holds for ordinary arithmetic with real or imaginary numbers, there are certain applications such as nonassociative algebras in which it does not hold

## Distributive Law

Distributive Law says we’ll get the same answer when we:

- multiply a number by a group of numbers added together, or
- do each multiply separately then add them

Distributive Law needs careful attention. And we write it like this:

**a × (b + c) = a × b + a × c**

7 × (8 + 9) = 7 × 8 + 7 × 9

### Distributive Law for Division

The Distributive Law does **not work **for division:

12 ÷ (4 + 2) = 12 / 6 = 2, but 12 ÷ 4 + 12 ÷ 2 = 3 + 6 = 9

The monomial factor a is distributed, or separately applied to each term of the binomial factor b + c, resulting in the product ab + ac.

It is easy to understand that the result of:

**adding several numbers first, then multiplying the sum by some number**

is the same as

**multiplying each separately by the number first, then adding the products**

## Summary

Commutative Laws | a + b = b + a a × b = b × a |

Associative Laws | (a + b) + c = a + (b + c) (a × b) × c = a × (b × c) |

Distributive Law | a × (b + c) = a × b + a × c |