---
tags:
  - prealgebra
---

# The set of whole numbers

We recall the set of whole numbers:

$$ \mathbb{W} = {0, 1, 2, 3, ...} $$

# The properties of $\mathbb{W}$

> In mathematics, a **property** is any characteristic that applies to a given
> set.

## The commutative property

### Addition

When **adding** whole numbers, the placement of the addends does not affect the
sum.

Let **a**, **b** represent whole numbers, then:

$$ a + b = b + a $$

### Multiplication

When **multiplying** whole numbers the placement of the
[multiplicands](static/Symbols-and-formal-conventions-80aeaf1872f94a0d97a2e8d07e3855bd)
does not affect the
[product](static/Symbols-and-formal-conventions-80aeaf1872f94a0d97a2e8d07e3855bd).

Let **a, b** represent whole numbers, then:

$$ a \cdot b = b \cdot a $$

### Subtraction

**Subtraction** is not commutative, viz:

$$ a - b \neq b - a $$

### Division

Division is not commutative, viz:

$$ a \div b \neq b \div a $$

## The associative property

### Addition

When grouping symbols (parentheses, brackets, braces) are used with addition,
the particular placement of the grouping symbols relative to each of the addends
does not change the sum.

Let **a**, **b, c** represent whole numbers, then:

$$ (a + b) + c = a + (b + c) $$

### Multiplication

Let **a, b, c** represent whole numbers, then:

$$ a \cdot (b \cdot c) = (a \cdot b) \cdot c $$

### Subtraction

Subtraction is not associative, viz:

$$ (a - b) - c \neq a - (b - c) $$

### Division

Division is not associative

$$ (a \div b) \div c \neq a \div (b \div c) $$

## The property of additive identity

If **a** is any whole number, then:

$$ a + 0 = a $$

We therefore call zero the additive identity: whenever we add zero to a whole
number, the sum is equal to the whole number itself.

## The property of multiplicative identity

If **a** is any whole number, then:

$$ (a \cdot 1 = a) = (1 \cdot a = a) $$

## Multiplication by zero

If **a** is any whole number, then:

$$ (a \cdot 0 = 0) = (0 \cdot a = 0) $$

## Division by zero

Division by zero is **undefined** but zero divided is zero.