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](https://www.notion.so/Symbols-and-formal-conventions-80aeaf1872f94a0d97a2e8d07e3855bd) does not affect the [product](https://www.notion.so/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.
