Closure under Addition
Closure property under addition states that the sum of any two integers will always be an integer.
Let’s consider the following pairs of integers.
23 + 12 = 35 (Result is an integer)
5 + (-6) = -1 (Result is an integer)
-12 + 0 = -12 (Result is an integer)
Since addition of integers gives integers, we say integers are closed under addition.
In general, for any two integers a and b, a + b is an integer.
Closure under Subtraction
Closure property under subtraction states that the difference of any two integers will always be an integer.
Let’s consider the following pairs of integers.
6 – 8 = -2 (Result is an integer)
15 – (-6) = 21 (Result is an integer)
-19 – 0 = -19 (Result is an integer)
Since subtraction of integers gives integers, we say integers are closed under subtraction.
Thus, if a and b are two integers then a – b is also an integer.
Commutative property of addition states that swapping of terms will not change the sum.
Let’s consider the following examples.
(-7) + (-5) = (-5) + (-7)
(-8) + 12 = 12 + (-8)
(-29) + 0 = 0 + (-29)
So, we can say that addition is commutative for integers.
In general, for any two integers a and b, we can say a + b = b + a
But, subtraction is not commutative for integers.
Let’s consider the following example.
7 – (–2) ≠ (–2) –7 [Because 7 – (–2) = 9 whereas (–2) –7 = (-9)]
We conclude that subtraction is not commutative for integers.
Associative property of addition states that the way of grouping of numbers will not change the result.
Let’s consider the following example.
(–6) + [(–4) + (–3)] and [(–6) + (–4)] + (–3)
In the first sum (–4) and (–3) are grouped together and in the second (–6) and (–4) are grouped together.
(–6) + [(–4) + (–3)] = (-6) + (-7) = (-13)
[(–6) + (–4)] + (–3) = (-10) + (-3) = (-13)
In both the cases, we get –13.
So, Addition is associative for integers.
In general for any integers a, b and c, we can say a + (b + c) = (a + b) + c
Subtraction of integers is not associative in nature.
Let’s consider the following example.
(–5) - [(–3) - (–2)] and [(–5) - (–3)] - (–2)
In the first difference (–3) and (–2) are grouped together and in the second (–5) and (–3) are grouped together.
(–5) - [(–3) - (–2)] = (-5) - (-1) = -4
[(–5) - (–3)] - (–2) = (-2) + 2 = 0
In both the cases, we get different results.
So, Subtraction is not associative for integers.
Additive identity property states that when we add zero to any integer, we get the same integer.
Let’s observe the following examples:
(– 8) + 0 = – 8
0 + (– 17) = – 17
0 + (–50) = -50
The above examples show that zero is an additive identity for integers.
In general, for any integer a
a + 0 = a = 0 + a
Example 1: Write a pair of integers whose sum is zero (0) but difference is 12.
Solution: Since sum of two integers is zero, one integer is the additive inverse of other integer (such as – 3, 3; – 4, 4 etc.). But the difference has to be 12. So, the integers are 6 and – 6 as 6–(–6) is 12.
Example 2: Which of the following shows the maximum rise in temperature?
(a) 12° to 30° (b) – 8° to + 7°
Solution: 12° to 30° = 30° - 12° = 18° (rise in temperature)
– 8° to + 7° = 7° - (-8)° = 15° (rise in temperature)
(a) 12° to 30° shows the maximum rise in temperature
Example 3: Encircle the odd one of the following
(a) (–9, 9) (b) (–5, 5) (c) (–5, 3) (d) (–11, 11)
Solution: (a) (–9) + 9 = 0
(b) (–5) + 5 = 0
(c) (–5) + (3) = (-2)
(d) (–11) + 11 = 0
In all the case we get the same answer except in case (c)
Example 4: Fill in the blanks to make the statements true.
(a) [(–8) + ______] + ________ = ________ + [(–3) + ________] = –3
(b) (– 23) + _____ = – 23
Solution: (a) [(–8) + (-3)] + 8 = (-8) + [(–3) + 8] = –3
(b) (– 23) + 0 = – 23
Example 5: Write down a pair of integers whose
(a) sum is –4 (b) difference is –5
(c) difference is 1 (d) sum is 0
Solution:
(a) (–2) + (–2) = –4
(b) (–8) – (– 3) = –5
(c) (–7) – (–8) = 1
(d) (–16) + 16 = 0
(We can write more pairs for the above examples)