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 Symbols-and-formal-conventions-80aeaf1872f94a0d97a2e8d07e3855bd does not affect the 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.