Properties of Division of Integers
Drag and drop the given division expressions into the correct classification.
What do you observe?
We observe that integers are not closed under division.
We know that division is not commutative for whole numbers. Let us check it for integers also.
Is (– 9) ÷ 3 the same as 3 ÷ (– 9) ?
Is (– 30) ÷ (– 6) the same as (– 6) ÷ (– 30) ?
Can we say that division is commutative for integers?
Like whole numbers, any integer divided by zero is meaningless and zero divided by an integer other than zero is equal to zero. Thus, for any integer a, a ÷ 0 is not defined but 0 ÷ a = 0 for a ≠ 0.
When we divide a whole number by 1 it gives the same whole number. But is it true for negative integers also?
Observe the following :
− 8 ÷ 1 = − 8 − 11 ÷ 1 = − 11 − 13 ÷ 1 = − 13 − 25 ÷ 1 = − 25 − 37 ÷ 1 = − 37 − 48 ÷ 1 = − 48 This shows that negative integer divided by 1 gives the same negative integer.
In general, for any integer a,
a ÷ 1 =
What happens when we divide any integer by (–1)?
Try completing the following table
What do we observe? We see that if any integer is divided by (–1), it does not give the same integer.
But, does the division of integers follow the associative rule? Let's find out.
Take for example: [(–16) ÷ 4] ÷ (–2) and (–16) ÷ [4 ÷ (–2)]
- [(–16) ÷ 4] ÷ (–2) =
÷ (–2) = - Now, (–16) ÷ [4 ÷ (–2)] = (–16) ÷
= - We see that the result is different for the individual expressions.
- Thus, associative property is not valid for division of integers.
Can you say that division is associative for integers?
Is for any integer a.Take different values of a and check.
(i) 1 ÷ a = 1 ?
Hint:Substitute a with 1,2,-3
a = 1 ÷ (1) =
a = 1 ÷ (2)
a = 1 ÷ (-3)
(ii) a ÷ (–1) = –a?
Substitute a with 2,-3,0
a = (2) ÷ (-1)
a = (-3) ÷ (-1)
a = (0) ÷ (-1)
Example 2 : In a test (+5) marks are given for every correct answer and (–2) marks are given for every incorrect answer.
(i) Radhika answered all the questions and scored 30 marks though she got 10 correct answers.
(ii) Jay also answered all the questions and scored (–12) marks though he got 4 correct answers. How many incorrect answers had they attempted?
Solution:
(i)Marks given for one correct answer = 5
So, marks given for 10 correct answers = 5 × 10 = 50
Radhika’s score = 30
Marks obtained for incorrect answers = 30 – 50 = –
Marks given for one incorrect answer = (–2)
Therefore, number of incorrect answers = (–20) ÷ (–2) =
(ii)Marks given for 4 correct answers = 5 × 4 = 20
Jay’s score = –12
Marks obtained for incorrect answers = –12 – 20 = –
Marks given for one incorrect answer = (–2)
Therefore number of incorrect answers = (–32) ÷ (–2) = 16
Example 3 : A shopkeeper earns a profit of ` 1 by selling one pen and incurs a loss of 40 paise per pencil while selling pencils of her old stock.
(i)In a particular month she incurs a loss of ` 5. In this period, she sold 45 pens. How many pencils did she sell in this period?
(ii)In the next month she earns neither profit nor loss. If she sold 70 pens, how many pencils did she sell?
Solution:
(i) Profit earned by selling one pen = 1
Profit earned by selling 45 pens = 45, which we denote by + ` 45
Total loss given = 5, which we denote by – 5
Profit earned + Loss incurred = Total loss
Therefore, Loss incurred = Total Loss – Profit earned.
= (– 5 – 45) = (–50) =
Loss incurred by selling one pencil = 40 paise which we write as – 40 paise
So, number of pencils sold = (–5000) ÷ (– 40) =
(ii) In the next month there is neither profit nor loss.
So, Profit earned + Loss incurred = 0
i.e., Profit earned = – Loss incurred.
Now, profit earned by selling 70 pens = 70
Hence, loss incurred by selling pencils = 70 which we indicate by 70 or – 7,000 paise.
Total number of pencils sold = (–7000) ÷ (– 40) =