Integers Class 7 Notes Maths Chapter 1

Students can go through AP 7th Class Maths Notes Chapter 1 Integers to understand and remember the concepts easily.

Class 7 Maths Chapter 1 Notes Integers

→ Numbers like 1, 2, 3, ………….. are called Natural Numbers and is denoted by N.
N = {1, 2, 3, ………}
Note:

  1. We can’t say the biggest Natural number.
  2. The smallest Natural number is 1.

→ Numbers like 0, 1, 2, 3, ………. are called whole numbers and is denoted by W.
W = {0,1, 2, 3, ………}
i.e. set of Natural numbers including zero are called whole numbers.
Note:

  1. N U { 0 } = W
  2. W – { 0} = N
  3. The smallest whole number is ‘0’
  4. We can’t say the largest whole number.

Integers Class 7 Notes Maths Chapter 1

→ The greatest two digit number is 99 where as three digits is 999, four digits is 9999 – and so on.

→ The smallest two digit number is 10 whereas three digits is 100, four digits is 1000 and so on.

→ Numbers like -1, -2, -3 are called as negative number.
Note:

  1. The smallest negative number can’t be determined.
  2. The greatest negative number is -1.
  3. ‘0’ is neither positive nor negative.

→ Integers : Numbers like ………. -3, -2, -1,
0, 1, 2, 3, ………….. are called Integers, denoted by Z (or) I.
Z = {…….. -3, -2, -1,0, 1, 2, 3, …..}
Note :

  1. The smallest integers can’t be determined.
  2. The greatest integers can’t be determined.
  3. We can locate all integers on a number line.

→ Number line : The representation of integers on a number line is given below.
Integers Class 7 Notes Maths Chapter 1 1
Note:

  1. On a number line all integers are marked in ascending order.
  2. ‘0’ is located in the middle place, negative number are taken as left’ side and positive number are taken on right side if ‘0’ on the number line.

→ Even numbers : Numbers like 2, 4, 6, ……… are called Even numbers.
Note:

  1. All even number are divisible by 2
  2. If you add one to any even number, you will get an odd number.

Integers Class 7 Notes Maths Chapter 1

→ Odd numbers: Numbers like 1, 3, 5, 7, ………. are called Odd numbers.
Note:

  1. All odd numbers are not exactly divisible by 2 ie., gives remainder 1 in each case.
  2. If you subtract 1 from any odd number you will get even number.

→ Properties of Addition and Subtraction of Integers
1) Closure under Addition : We know that the sum of two whole numbers is again a whole number.
Ex: 1) 7 + 8 = 15; 2) 0 + 19 = 19
In case of integers:
3) (- 2) + 5 = 3 ; 4) 8 + (- 7) = 1
Note:

  1. Whole numbers and Integers are closed under addition.
  2. In general for any two integers a and b, a + b is an integer.

2) Closure under Subtraction :
Whole Number:
1) 3 – 0 = 3; 2) 10 – 11 = -4
Whole number are not Closed under subtraction.
Integers : 1) 8 – 1= 7 ; 2) 7 – 17 = – 10
Integers are closed under subtraction.
Note : In general, if a and b are two integers then a-b is also an integer.

3) Commutative Property : Whole number can be added in any order will give the same result.
Ex: 1) 7 + 4 = 11; 4 + 7 = 11
2) 8 + 1 = 9 ; 1 + 8 = 9
In case of Integers : (+ )
1) 7 + (- 3) = 4 ; – 3 + 7 = 4
2) – 3 + 2 = -1 ; 2 + (- 3) = -1
Are the following equal ?
i) (- 8) +(- 9) and (- 9) + (- 8)
Solution:
(- 8) + (- 9) = – 8 – 9 = -17
and, (- 9) +(- 8) = – 9 – 8 = -17
⇒ (- 8) + (- 9) = (- 9) +(- 8)

ii) (- 23) + 32 and 32 + (- 23)
Solution:
(- 23) + 32 = – 23 + 32 = 9
and, 32 + (-23) = 32 – 23 = 9
⇒ (- 23) + 32 = 32 + (-23)

iii) (-45) + 0 and 0 + (-45)
Solution:
(- 45) + 0 = – 45 and, 0 + (- 45) = -45
⇒ (-45)+ 0 = 0 + (-45)
Note:

  1. Addition of integers is commutative.
  2. In general if a, b are integers, a + b = b + a

Subtraction (-)
1) 7 – 3 = 4
3 – 7 = -4 But 4 ≠ -4
2) – 2 – 3 = -5
– 3 – (- 2) = – 3 + 2 = – 1
5 ≠ -1.
Note:

  1. Subtraction of integers is not commutative.
  2. In general if a, bare integers, a – b ≠ b – a

3) Associative property : Consider three integers a, b and c.
For addition (+) :
1) Let a = 3, b = 2, c = 1
a + (b + c) = 3 + (2 + 1) = 3 + 3 = 6
(a + b) + c = (3 + 2) + 1 = 5 + 1 = 6
2) Let a = – 3, b = – 2, c = 4
a + (b + c) = -3 + (-2 + 4) = -3 + 2 = -1
(a + b) + c = (- 3 – 2) + 4 = -5 + 4 = -1
a + (b + c) = (a + b) + c
Note:

  1. In general, for any three integers a, b and c, a + (b + c) = (a + b) + c
  2. Integers satisfy associative property under addition of integers.

For Subtraction (-) :
1) Let a = 3, b = 2, e =1
(a – b) – c = (3 – 2) – 1 = 1 – 1 – 0
a – (b – c) = 3 – (2 – 1) = 3 – 1 = 2
(a – b) – c ≠ a – (b – c)
Note : Integers does not satisfy associative property under subtraction,
i.e., (a – b) – c ≠ (a – b) – c

Integers Class 7 Notes Maths Chapter 1

4) Additive Identity:
Whole numbers : If we add ‘0’ to any whole number, we get the number itself as a result.
Ex: 1) 7 + 0 = 0 + 7 = 7
2) a + 0 = 0 + a = a
∴ ‘0’ is the additive identity in whole number.
Integer : Take any integer and add ‘0’ to it we get the same number as result, then we call ‘0’ as the additive identity in Integers also.
Ex:
1) – 31 + 0 = 0 – 31 = -31
2) b + 0 = 0 + b = b
Note:

  1. In general for any integer
    a, a + 0 = 0 + a = a
  2. ‘0’ is called the additive identity (or) zero.

→ Multiplication of integers :
Multiplication of a positive and a negative integers : We have from the number line.
5 + 5 + 5 = 15 = 3 × 5
– 3 – 3 – 3 = -9 = 3 × -3
Example (1) Number line
Integers Class 7 Notes Maths Chapter 1 2
Example (2)
-4 – 4 – 4 – 4 = – 4 × 4 = – 16
Integers Class 7 Notes Maths Chapter 1 3
Note:

  1. When we multiply a positive integer and a negative integer we get a negative integer as a product.
  2. When we multiply a negative integer and a positive integer we get a negative integer as a product.
  3. In general if a and b are two integers such that a × (- b) = (- a) × b = – (a × b)
  4. Multiplication of an integer with ‘0’ gives us the result ‘0’. Ex: – 7 × 0 = 0
  5. Multiplication of an integer with “1′ gives us the same result.
    Ex: – 13 × 1 = -13

→ Multiplication of two negative integers:
Observe the following :
– 3 × 4 = -12
– 3 × 3 = – 9 = – 12 – (- 3)
– 3 × 2 = – 6 = – 9 – (- 3)
– 3 × 1 = -3 = – 6 – 0 (- 3)
– 3 × 0 = 0 = – 3 – (- 3)
– 3 × – 1 = 0 – (- 3) = 0 + 3 = 3
– 3 × – 2 = 3 – (- 3) = 3 + 3 = 6
Note:

  1. The product of two negative integers is a positive integer, we multiply the two negative integers as whole numbers and put the positive sign before the product.
  2. In general, for any two integers a and b.
    (-a) × (-b) = a × b
    Ex: 1) (-3) × (-4) = 3 × 4 = 12
    2) (-1) × (-9) = 1 × 9 = 9

→ Product of three or more Negative Integers:
Observe the following examples:
a) (-4) × (-3) = 12

b) (-4) × (-3) × (-2)
= [(-4) × (-3) ] × (-2)
= 12 × (-2) = -24

c) (-4) × (-3) × (-2) × (-1)
= [(-4) × (-3) × (-2)] × (-1)
= (-24) × (-1)

d) (-5) × [(-4) × (-3) × (-2) × (-1)]
= (-5) × 24 = – 120

From the above products we observe that
a) the product of two negative integers is a positive integer.
b) the product of three negative integers is a negative integer.
c) product of four negative integers is a positive integer.

Note:

  1. The product of even number of negative integers is positive integer.
  2. The product of odd number of negative integers is negative integers.

Examples:

  1. – 2 × – 2 × – 2 × – 2 = 16
  2. – 3 × – 3 × – 3 × – 3 × – 3 = -243.
  3. – 1 × – 1 × – 1 ……. 2022 times = 1
  4. – 1 × – 1 × – 1 ………… 2023 times = -1

→ Properties of Multiplication of Integers : Closure under Multiplication : The
product of two integers is always an integer.
Ex: – 3 × 4 = – 12, an integer

Note : a × b is an integer, for all integers a and b.
Commutativity of Multiplication:
Consider:

  1. – 4 × 9 = – 36
    9 × – 4 = – 36
  2. – 15 × (- 10) = 150
    – 10 + (- 15) = 150
  3. 7 × 2 = 14
    2 × 7 = 14

From above we can conclude that multiplication of integers is commutative.
Note: a × b = b × a. For all integers a & b.
Multiplication by Zero :
Consider:

  1. – 71 × 0 = 0
  2. 8 × 0 = 0

From above we can conclude that multiplication of an integer with zero gives ‘0’ as product.
Note : a × 0 = 0 × a. For all integers a., Multiplicative Identity:
Consider:

  1. – 3 × 1= – 3
  2. 2 × 1 = 2

From above we can say that

  1. 1 is the multiplicative identity.
  2. a × 1 = 1 × a = a, a is an integer.

Associativity for Multiplication:
Consider: 1) (1 × – 2) × 3 = – 2 × 3 = – 6
1 × (- 2 × 3) = 1 × -6 = -6
From above we can say that

  1. The product of three integers does not depend upon the grouping of integers and this called associative property for multiplication of integers.
  2. a × (b × c) = (a × b) × c, for any three integers a, b and c.

Integers Class 7 Notes Maths Chapter 1

Distributive Property:
Consider:

  1. 7(2 + 3) = 7 × 2 + 7 × 3 = 14 + 21 = 35
  2. (8 + 1)2 = 8 × 2 + 1 × 2 = 16 + 2 = 18

From above we can say that

  1. a × (b + c) = a × b + a × c, for any three integers a, b & c.
  2. (a + b) × c = a × c + b × c

Also

  1. 8 × (5 – 2) = 8 × 5 – 8 × 2 = 40 – 16 = 24
  2. (3 – 2) × 4 = 3 × 4 – 2 × 4 = 12 – 8 = 4

From above we can say that

  1. a × (b – c) = a × b – a × c
  2. (a – b) × c = a × c-b × c
  3. a × (b – b) = a × 0 = 0
  4. (a – a) b = 0 × b = 0

Division of Integers:
Consider:

  1. 10 ÷ 2 = 5
  2. – 10 ÷ 2 = -5
  3. 10 ÷ (- 2) = – 5
  4. – 10 ÷ (- 2) = 5

From above we can conclude that
i) When we divide a negative integer by positive integers, we divide them as whole positive integer, we divide them as whole numbers and then put a minus sign (-) before the quotient.

  1. a ÷ (- b) = (- a) ÷ b, where b ≠ 0
  2. (- a) ÷ (- b) = a ÷ b, where b ≠ 0
  3. a ÷ 0 is not defined, a ≠ 0
  4. – a ÷ 0 is not defined, a ≠ 0
  5. (- a) ÷ (- a) = 1, a ≠ 0
  6. – a ÷ a = – 1, a ≠ 0
  7. 0 ÷ 0 is not defined.
  8. a ÷ a = 1, a ≠ 0
  9. a ÷ 1 = a, a ≠ 0
  10. a ÷ (- 1) = – a, a ≠ 0

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