Integers

Properties of Division of Integers

1. The integers are not closed under division. Look at the following table.

Expression	Value	Inference
(– 8) ÷ (– 4)	= 2	An integer
(– 4) ÷ (– 8)	= 1/2	Not an integer
(–250) ÷ 750	= – 1/3	Not an integer
100 ÷ (– 1000)	= –1/10	Not an integer

2. The division is not commutative for integers. Look at the following table.

Expression 1	Expression 2	Inference
(– 8) ÷ (– 4) = 2	(– 4) ÷ (– 8) = 1/2	(– 8) ÷ (– 4) ≠ (– 4) ÷ (– 8)
(– 9) ÷ 3	3 ÷ (– 9)	(– 9) ÷ 3 ≠ 3 ÷ (– 9)
30 ÷ (– 6)	(– 6) ÷ 30 =	30 ÷ (– 6) ≠ (– 6) ÷ 30

3. Like whole numbers, any integer divided by zero is meaningless. In other words, for any integer a, a ÷ 0 is not defined.

Expression	Inference
8 ÷ 0	Not defined
(– 9) ÷ 0	Not defined
(– 1632) ÷ 0	Not defined

4. Zero divided by an integer other than zero is equal to zero. In other words, for any integer a, 0 ÷ a = 0 for a ≠ 0.

Expression	Value
0 ÷ 3	0
0 ÷ (– 9)	0
0 ÷ (– 1632)	0

5. When we divide a whole number by 1 it gives the same whole number. Similarly, any integer divided by 1 gives the same integer. In general, for any integer a, we have a ÷ 1 = a.

Expression	Value
(– 8) ÷ 1	= (– 8)
(–11) ÷ 1	= –11
(–13) ÷ 1	= –13

6. When we divide an integer by (–1) it gives the same integer with changed sign. In general, for any integer a, we have a ÷ (–1) = –a.

Expression	Value
8 ÷ (–1)	= – 8
(–25) ÷ (–1)	= 25
– 48 ÷ (–1)	= 48