CBSE Class 7 Mathematics Integers Notes

Integers

 The collection of ….–3, –2, –1, 0, 1, 2, 3, 4,….. is known as intergers. 

Integers on number line

Note i)0 is greater than all negative integers but it is lesser than all positive integers.

ii) 1 is the smallest positive integer.

iii)  –1 is the largest negative integer.

iv) 0 is neither +ve nor –ve

Operations on integers

We are going to learn the following operations on integers.

i) Addition   ii)   Subtraction   iii)  Multiplication and   iv)  Division

Addition of two integers

Rules to add two integers

Rule 1: If a and b are positive integers, then add a and b and put + sign to the result.

Rule 2:  If a is positive integer and b is negative integer, and subtract the smaller one from bigger one  and put the sign of bigger number to the result

Rule 3: If  a is a negative integer and b is positive integer, then also subtract the smaller one from bigger one and put the sign of bigger number to the result.

Rule 4: If a and b are negative integers, then add ‘a’ and ‘b’ and put ‘–’ sign to the result.

Example 1: Find the value of 8 + 3

Solution: Both 8 and 3 are positive integers.

8 + 3 = 11 (using rule 1)

Example 2: Find the value of 8 + (–13)

Solution: 8 is positive integer and –13 is negative integer. Also 13 > 8

Following rule 2, 8 + (–13) = – [13 – 8] = – 5

Example 3:Find the value of (–8) + 3

Solution: –8 is negative integer and 3 is positive integer by rule 3, (–8) + 3 = – (8 – 3) = –5

Example 4:  Find the value of (–8) + (–3)

Solution:(–8) and (–3) are negative integers. By rule 4

(–8) + (–3) = –(8 + 3) = –11

Addition of three or more integers

Hint: This can be done by method of grouping. Group first two integers the next two integers and so on.

Example 5: Find the value of (–5) + (–6) + (+7)

Solution:  (–5) + (–6) + 7 = {(–5) + (–6)} + 7 (Group first two integers)

=    (– 11) + 7 {Using Rule 4}

 =    –4   {Using Rule 3}

Example 6: Find the value of (–5) + (–4) + (–3) + (–2) + (–1)

Solution: (–5) + (–4) + (–3) + (–2) + (–1)

=    {(–5) + (–4)} + {(–3) + (–2)} + (–1) (Group first two and next two integers)

=    {(–9) + (–5)} + (–1) (Use rule 4 and then group first two integers)

=    (–14) + (–1)(Using rule 4)

=    –15 (Using rule 4)

Example 7:  Find the value of (–25) + (13) + (–49)

Solution: (–25) + (13) + (–49) = {(–25) + (13)} + (–49)  (By grouping)

 =    (–12) + (–49)   (Using rule 2)

 =    –61   (Use rule 4)

Example 8:  Find the value of 1000 + (–999) + (–1)

Solution: 1000 + (–999) + (–1) = {1000 + (–999)} + (–1)     (By grouping)

= 1 + (–1) (Using rule 2)

= 0  (Using rule 2)

Subtraction of two integers

Rules to subtract two integers.

Rule 1:  If a and b are positive integers, then to get a – b, subtract the smaller number from bigger number and put the sign of bigger number to the result.

Rule 2: If a is positive integer and b is negative integer, then to obtain a – b, a and b and put + sign to the result

Rule 3: If a is negative integer and b is positive integer, then to obtain a – b, add a and b and put – sign to the result.

Rule 4:  If a and b are negative integer, then to obtain a – b, subtract the smaller number from bigger number and put (i) ‘+’ sign if b is bigger. (ii) ‘–’ sign if a is bigger.

Example 9:Find the value of 8 – 3.

Solution: 8 – 3 = 5  (Using rule 1)

Example 10: Find the value of 8 – (13)

Solution: 8 and 13 are positive integer and 13 > 8

∴ 8 – 13 = – (13 – 8) = – 5                             (Use rule 1)

Example 11: Find the value of 8 – (–3)

Solution: 8 – (–3) = + (8 + 3) = + 11            (Use rule 2)

Example 12: Find the value of (–8) – (3)

Solution: (–8) – (3) = –(8 + 3) = –11              (Use rule 3)

Example 13: Find the value of (–8) – (–13)

Solution: (–8) – (– 13) = + (13 – 8) = 5           (Rule 4 : 13 – 8 = 5) (13 > 8) \ (–8) – (–3) = + 5

Example 14: Find the value of (–8) – (–3)

Solution: (–8) – (–3) = – (8 – 3) = –5               (Rule 4 : 8 – 3 = 5) (8 > 3) \ (–8) – (–3) = – 5

Subtraction of two (or) more integers

Method : Grouping first two integers only.

Example 15: Find the value of (–5) – (6) – (7)

Solution: (–5) – (–6) – (7) = {(–5) – (–6)} – 7    (Grouping first two terms)

                = 1 – 7                                                (Using rule 4)

                = –6                                                    (Using rule 2)

Example 16: Find the value of (–5) – (–4) – (–3) – (–2) – (–1)

Solution: (–5) – (–4) – (–3) – (–2) – (–1) = {(–5) – (–4)} – (–3) – (–2) – (–1) {Grouping first two}

= [(–1) – (–3)] – (–2) – (–1)                   {Using rule 4}

= [(+2) – (–2)] – (–1)                            {Using rule 4}

= 4 – (–1)                                              {Using rule 2}

= 5                                                         {Using rule 2}

Example 17: Find the value of 100 – 1 – 99

Solution: 100 – 1 – 99 = (100 –1) – 99       {Grouping first two}

                 = 99 – 99

                 = 0

Properties of addition and subtraction of integers

1. Closure property : If a and b are integers, then (i) a + b is also an integer (ii) a – b is also an integer.

Hence, closure property holds for both addition and subtraction of integers.

2. Associative Property : If a, b, and c are integers, then

i) a + (b + c) = (a + b) + c

ii) a – (b – c) ¹(a – b) – c

Hence associative property holds for addition but not for subtraction.

3. Commutative property : If a and b are integers, then

i) a + b = b + a

ii) a – b b – a

Hence commutative property holds for addition but not for subtraction.

4. Inverse : If a is an integer, then

(i) a + (–a) = 0

(ii) a – a = 0

‘–a’ is called additive inverse of a (or) negative of ‘a’

5. Role of Zero : If a is an integer, then

(i) a + 0 = 0 + a = a Þ 0 is an additive identity

(ii) a – 0 = a but 0 – a ¹ a [as 0 – a = –a]

Multiplication of integers

i) Positive × Positive = Positive

ii) Positive × Negative = Negative

iii) Negative × Positive = Negative

iv) Negative × Negative = Positive

v) If there are odd number of negative integers in multiplication, then the result will be negative integer.

vi) If there are even number of negative integers in multiplication, then the result will be in positive integer.

Example 18: Find the value of (–8) × 5

Division of two integers: Rules to remember

i) Positive / Positive = Positive

ii) Negative / Positive = Negative

iii) Positive / Negative = Negative

iv) Negative / Negative = Positive

Example 23: Find the value of (140) ÷ (–20)

Solution: 140/-20 = 14/-2 = -7

Example 24: Find the value of (–140) / (–20)

Solution: -140/-20 = 14/2 = 7

Example 25: The value of (–100) ÷ 20

Solution: -100/20 = -10/25 = -5

Properties of Multiplication and Division of integers

1. Closure property : If a and b are integers, then

i) a × b is an integer

ii) a ÷ b need not be an integer

Example-26 2 × 3 = 6

                    2 ÷ 3 = 2/3 is a fraction

                    ∴ Closure property is true for multiplication but not for division

2. Commutative property : If a and b are integers, then

i) a × b = b × a

ii) a ÷ b ≠ b ÷ a

Hence commutative property holds for multiplication but not for division.

3. Assosciative property : If a, b and c are integers, then

i) (a × b) × c = a × (b × c)

ii) (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

Hence multiplication integers is associative but not for division of integers.

4. Role of 1 : If a is an integer, then

i) a × 1 = 1 × a = a                      [1 is called multiplicative identity]

ii) a/1 = a but 1/a ¹ a

5. Inverse : If a is an integer, then

i) a*1/a = 1/a*a =1                    [1/a is called multiplicative inverse of a]

ii) a/a = 1; i.e. a÷a = 1

6. Distributive property of multiplication over addition :

If a, b, c are integers, then a × (b + c) = a × b + a × c

Example :

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