Integers

Addition and Subtraction of Integers

Closure under Addition

We have learnt that sum of two whole numbers is again a whole number, which is known as the closure property for addition of the whole numbers. Since addition of integers gives integers too, we say integers are closed under addition.

In other words, for any two integers a and b, a + b is an integer.

Look at the following table.

Statement	Value	Observation
17 + 23	= 40	Result is an integer
(–10) + 3	= –7	Result is an integer
19 + (– 25)	= – 6	Result is an integer
27 + (– 27)	= 0	Result is an integer
(– 20) + 0	= – 20	Result is an integer
(– 35) + (– 10)	= – 45	Result is an integer

Closure under Subtraction

Since the difference of integers gives integers too, we say integers are closed under subtraction.

In other words, if a and b are two integers then a – b is also an integer. Do the whole numbers satisfy this property?

Look at the following table.

Statement	Value	Observation
7 – 9	= – 2	Result is an integer
17 – (– 21)	= 38	Result is an integer
(– 8) – (–14)	= 6	Result is an integer
(– 18) – (– 18)	= 0	Result is an integer
(– 29) – 0	= – 29	Result is an integer

Commutative Property

We know that the whole numbers can be added in any order. In other words, addition is commutative for whole numbers. In the same way, we say that addition is commutative for integers.

In general, for any two integers a and b, we can say a + b = b + a.

Look at the following table.

Operation 1	Operation 2	Observation
5 + (– 6) = –1	(– 6) + 5 = –1	5 + (– 6) = (– 6) + 5
(– 45) + 0 = – 45	0 + (– 45) = – 45	(– 45) + 0 = 0 + (– 45)
(– 8) + (– 9) = – 17	(– 9) + (– 8) = – 17	(– 8) + (– 9) = (– 9) + (– 8)
(– 23) + 32 = 10	32 + (– 23) = 10	(– 23) + 32 = 32 + (– 23)
(– 52) + 52 = 0	52 + (– 52) = 0	(– 52) + 52 = 52 + (– 52)

We know that subtraction is not commutative for whole numbers. Subtraction is not commutative for integers too. Consider the following table.

Operation 1	Operation 2	Observation
5 – ( –3) = 5 + 3 = 8	(–3) – 5 = – 3 – 5 = – 8	5 – ( –3) ≠ (–3) – 5
(– 45) – 0 = – 45	0 – (– 45) = 45	(– 45) – 0 ≠ 0 – (– 45)
(– 8) – (– 9) = 1	(– 9) – (– 8) = – 1	(– 8) – (– 9) ≠ (– 9) – (– 8)

Associative Property

Consider any three whole numbers. The sum of last two added with the first one is always equal to the sum of first two added with the last one. This is the associative property. In the same way, we say that addition is associative for integers.

In general, for any integers a, b and c, we can say a + (b + c) = (a + b) + c.

Look at the following table.

Operation 1	Operation 2	Observation
(–5) + [(–3) + (–2)] = (–5) + (–5) = –5 –5 = –10	[(–5) + (–3)] + (–2) = (–8) + (–2) = –8 –2 = –10	(–5) + [(–3) + (–2)] = [(–5) + (–3)] + (–2)
(–3) + [1 + (–7)] = (–3) + (–6) = –3 + –6 = –9	[(–3) + 1] + (–7) = (–2) + (–7) = –2 –7 = –9	(–3) + [1 + (–7)] = [(–3) + 1] + (–7)

Additive Identity

When we add zero to any whole number, we get the same whole number. Zero is an additive identity for whole numbers. In the same way, zero is an additive identity for integers.

In general, for any integer a, we can say a + 0 = a = 0 + a.

Look at the following table.

Operation 1	Operation 2	Observation
(– 8) + 0 = – 8	0 + (– 8) = – 8	0 is the additive identity
0 + (–37) = –37	(–37) + 0 = –37	0 is the additive identity
0 + 0 = 0	0 + 0 = 0	0 is the additive identity
[– (– 23)] + 0 = 23	0 + [– (– 23)] = 23	0 is the additive identity

Example. Write down two pairs of integers whose sum is 0.

Solution. Here, the solution is not unique. There are many such pairs. Two pairs are given by

(–110) + 110 = 0 and 5 + (–5) = 0.

Example. Write down two pairs of integers whose sum is –3.

Solution. Here, the solution is not unique. There are many such pairs. Two pairs are given by

(–1) + (–2) = –3 and (–5) + 2 = –3.

Example. Write down two pairs of integers whose difference is 2.

Solution. Here, the solution is not unique. There are many such pairs. Two pairs are given by

(–7) – (–9) = 2 and 1 – (–1) = 2.

Example. Write down two pairs of integers whose difference is –5.

Solution. Here, the solution is not unique. There are many such pairs. Two pairs are given by

(–9) – (– 4) = –5 and (–2) – 3 = –5.

Exercise 1. Write a pair of integers whose sum gives

• (a) a negative integer;
• (b) zero;
• (c) an integer smaller than both the integers;
• (d) an integer smaller than only one of the integers;
• (e) an integer greater than both the integers.
• (f) sum is –7

Exercise 2. Write a pair of integers whose difference gives

• (a) a negative integer;
• (b) zero;
• (c) an integer smaller than both the integers;
• (d) an integer greater than only one of the integers;
• (e) an integer greater than both the integers.
• (f) difference is –10

Exercise 3. Solve as instructed.

• (a) Write a pair of negative integers whose difference gives 8.
• (b) Write a negative integer and a positive integer whose sum is –5.
• (c) Write a negative integer and a positive integer whose difference is –3.

Exercise 4. In a quiz, team A scored – 40, 10, 0 and team B scored 10, 0, – 40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order?

Exercise 5. Fill in the blanks to make the following statements true:

• (i) (–5) + (– 8) = (– 8) + (…………)
• (ii) –53 + ………… = –53
• (iii) 17 + ………… = 0
• (iv) [13 + (– 12)] + (…………) = 13 + [(–12) + (–7)]
• (v) (– 4) + [15 + (–3)] = [– 4 + 15] + …………