Addition of integers
Integers 7th Math
Addition of integers
Addition of integers having similar sign
Rule to Add integers with similar sign
To add integers having similar sign; add them without taking their sign into consideration and put a common sign before the result.
This means if two or more positive integers are to be added, then add all of them regardless of their sign and put a positive sign before the result.
And if two or more negative integers are to be added, then add them regardless of their sign and put a negative sign before the result
Steps to add two integers having common sign
Step: (i) Add the given integers simply without taking their sign into consideration.
Step: (ii) Put a common sign to their sum. That is suppose both of the integers are positive, then put a positive sing before their sum. Or if both of the integers are negative, then put a negative sign before their sum.
Example:
Addition of two positive integers
Example Question:(1)Add 25 and 45
Solution:
Given, 25 + 45
Here, both of the given integers have positive sign. This means both of the integers are positive.
Thus, Add them simply without taking their sign into consideration.
25 + 45 = 70
Now, since both of the given integers are positive, thus put a positive sign before their sum.
= + 70
Since a positive integer is written without putting any sign before it.
Thus, `+ 70' or '70' both are considered as positive.
Hence, 25 + 45 = 70 Answer
Example Question:(2) Add 55 and 27
Solution:
Given, 55 + 27
Here, both of the given integers have positive sign. This means both of the integers are positive.
Thus, Add them simply without taking their sign into consideration.
55 + 27 = 82
Now, since both of the given integers are positive, thus put a positive sign before their sum.
Thus, the sum 82 is written as + 82
Since, it is not necessary to write a positive sign before a positive integers.
Thus, 55 + 27 = 82 Answer
Example Question:(3)Add 37 and 23
Solution:
Given, 37 + 23
Here, both of the given integers have positive sign. This means both of the integers are positive.
Thus, Add them simply without taking their sign into consideration.
37 + 23 = 60
Since, both of the given integers are positive, thus their sum will be positive.
Thus, 37 + 23 = 60 Answer
Addition of more than two positive integers
Same Rule is applied to add more than two positive integers
Example:
Question:(i) Add 25, 35, and 10
Solution:
Given 25 + 35 + 10
Here all the given three integers have positive sign. So they are added regardless of their sign and a positive sign will be given to their sum. This means sum of all positive integers is a positive integer.
Thus, 25 + 35 + 10 = 70Answer
Question:(ii) Add 25, 35, and 10, 40
Solution:
Given 25 + 35 + 10 + 40
Here all the given integers have positive sign. So they are added regardless of their sign and a positive sign will be given to their sum. This means sum of all positive integers is a positive integer.
Thus, 25 + 35 + 10 + 40 = 110Answer
Addition of two negative integers
Example Question:(1) Add – 12 and (– 16)
Solution:
Given, Add – 12 and (– 16)
Step: (i) Since, both of the two integers have negative sign, thus first of all add them regardless of their sign
12 + 16 = 28
Step (2): Now, put a negative sign before the value of their sum. [Because negative is the common sign of given integers]
= – 28
Thus – 12 + (–16) = – 28 Answer.
Question:(2)Add – 15 and –25
Solution:
Given, – 15 and – 25
Step: (i) Add the given integers regardless of their sign
15 + 25 = 40
Step: (ii) Put a common sign (negative sign) before their sum.
= – 40
Thus, – 15 + (– 25)= – 40 Answer.
Question:(3) Add –10 and –35
Solution:
Given, – 10 + (–35)
Step: (i) Add the given integers without taking their sign into consideration
i.e. 10 + 35 = 45
Step (ii) Put common sing (negative) sing before the sum
= – 45
Thus, – 10 + (–35) = – 45
Question:(4) Add – 22 and – 115
Solution:
Given, – 22 and – 115
Add them regardless of their sign and put a common sign (negative sign)before their sum
– 22 + (– 115)
= -–137 Answer
Addition of more than three negative integers :
Two or more than two negative integers are added in similar ways.
Question:(5) Add –10, –15 and –25
Solution:
Given, –10, –15 and –25
Add all the given integers without taking their sign into consideration
10 + 15 + 25 = 50
Now, put the common sing (negative sign) of the given integers to their sum
Thus, = – 50
Thus, – 10 + (– 15)+(–25)
= – 50 Answer
Question:(6) Add –20, –25, –40 and –10
Solution:
Given, –20, –25, –40 and –10
Add all the given integers without taking their sign into consideration
= 20 + 25 + 40 + 10 = 95
Now put the common sign (negative sing) of the given integer to their sum
=– 95
Thus, (–20) + (–25) + (–40) + (–10)
= – 95 Answer
Addition of two integers having different sign
Addition of one positive and one negative integers.
Rule to add one positive and one negative integer
To add one positive and one negative integers, find the difference between them regardless of their sign. And give the sign of the greater integer to their difference.
Example:
Example Question (1) Add 10 and – 15
Solution:
Given, 10 and – 15 are to be added.
Step: (i) Find the difference between the given integers regardless of their sign
15 – 10 = 5
Step: (ii) Put the sign of greater integer to the difference calculated
Here, since 15 is greater (without taking the sign of given integers into consideration). And 15 has negative sing. Thus, negative sing will be put before the result calculated.
Here, result = 5. Thus a negative sing will be assigned before 5
That is, 5 will become –5
Thus, 10 + (–15) = – 5
Question:(2)Add 20 and –15
Solution:
Given, 20 and –15 are to be added
Step (i) Forget sign of given integers.
Thus, integers are, 20 and 15
Step (ii) Find the difference
20 – 15 = 5
Step: (iii) Assing the sing of greater integer to the result obtained.
Here, 20 is greater and 20 had positive sign
Thus, 5 will become +5
Thus, 20 – 15 = 5 Answer
Question:(3)Add –30 and 40
Solution:
Given, –30 and 40
Step (i) Forget sign of given integers.
Thus, we have, 30 and 40
Step (ii) Find the difference
40 – 30 = 10
Step: (iii) Assing the sing of greater integer to the result obtained.
Here, 40 is greater and 40 had positive sign
Thus, 10 will remains + 10
Thus, –30 + 40 = 10 Answer
Question:(4)Add –50 and 30
Solution:
Given, –50 and 30
Step (i) Forget sign of given integers.
Thus, we have, 50 and 30
Step (ii) Find the difference
50 – 30 = 20
Step: (iii) Assing the sing of greater integer to the result obtained.
Here, 50 is greater and 40 had negative sign
Thus, 20 will become – 20
Thus, –50 + 30 = – 20 Answer
Properties of addition oof integers
Closure property of Addition of integers
The sum of two integers is always an integer.
Example:
(a) 5 + 9 = 14
Here 5 and 9 are integers and their sum 14 is also an integers
(b) 5 + (–9)= –4
Here, 5 and –9 are integers and their sum –4 is also an integer.
(c) – 5 + 9 =4
Here –5 and 9 are integers and their sum 4 is also an integer.
Commutative law of Addition of Integers
Let a and b are two integers
∴ a + b = b + a
Example:
(1) 5+4 = 9
and 4 + 5 = 9
i.e. 5 + 4 = 4 + 5
(2) –5 + 1 = – 4
and 1 + (–5) = –4
i.e – 5 + 1 = 1 + (–5)
(3) 16 + (–15)= 1
And –15 + 16 = 1
i.e. 16 + (–15) = –15 + 16
Thus addition of integers is commutative
Associative Law of Addition of Integers
If a, b and c are any three integers therefore, ( a + b )+ c = a +( b + c )
Example:
(a) (3+2) + 5 = 5 + 5 = 10
and 3 + (2 + 5) = 3 + 7 = 10
∴(3 + 2) + 5 = 3 + (2 + 5)
(b) Let –2, 3 and –7 are three integers
∴( –2 + 3)+ (–7)
= 1 + (–7) = –6
And –2 + {3 + (–7)}
= –2 + (–4) = –6
∴ (–2 + 3) + (–7)= –2 + {3+ (–7)}
Additive Identity
Let `a` is an integer
∴ a + 0 = a
or, 0 + a = a
i.e. a + 0 = 0 + a = a
This `0` is called the additive identity for integers
Example:
(a)2 + 0 = 0 + 2 = 2
(b) –5 + 0 = 0 + (–5) = –5
Additive inverse:
For any integer a, –a is an addition inverse
That is a + (–a) = – a + a = 0
Thus, opposite of an integers is called additive inverse. The sum of an integer and its additive inverse is always equal to '0'
Similarly, Additive inverse of `–a` is `a`
because, –a + a = 0
Similarly, Additive inverse of 5 is –5
Because 5 + (–5)= 5 – 5 = 0
Similarly, additive inverse of –4 is 4
Because, – 4 + 4 = 0
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