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 |
Excellent post as usual sir.
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The post become most attractive due to additional exercise…or skill development…super sir!!
Thank you
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Thank you so much.
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Sir , I am from class 8
Maths was really easy
I solved all in first attempt
Plz upload more interesting maths for us , as I have shared the post with my friends
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Thank you so much. More things will come up shortly.
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