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:
- We can’t say the biggest Natural number.
- 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:
- N U { 0 } = W
- W – { 0} = N
- The smallest whole number is ‘0’
- We can’t say the largest whole number.
→ 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:
- The smallest negative number can’t be determined.
- The greatest negative number is -1.
- ‘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 :
- The smallest integers can’t be determined.
- The greatest integers can’t be determined.
- We can locate all integers on a number line.
→ Number line : The representation of integers on a number line is given below.
Note:
- On a number line all integers are marked in ascending order.
- ‘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:
- All even number are divisible by 2
- If you add one to any even number, you will get an odd number.
→ Odd numbers: Numbers like 1, 3, 5, 7, ………. are called Odd numbers.
Note:
- All odd numbers are not exactly divisible by 2 ie., gives remainder 1 in each case.
- 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:
- Whole numbers and Integers are closed under addition.
- 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:
- Addition of integers is commutative.
- 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:
- Subtraction of integers is not commutative.
- 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:
- In general, for any three integers a, b and c, a + (b + c) = (a + b) + c
- 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
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:
- In general for any integer
a, a + 0 = 0 + a = a - ‘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
Example (2)
-4 – 4 – 4 – 4 = – 4 × 4 = – 16
Note:
- When we multiply a positive integer and a negative integer we get a negative integer as a product.
- When we multiply a negative integer and a positive integer we get a negative integer as a product.
- In general if a and b are two integers such that a × (- b) = (- a) × b = – (a × b)
- Multiplication of an integer with ‘0’ gives us the result ‘0’. Ex: – 7 × 0 = 0
- 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:
- 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.
- 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:
- The product of even number of negative integers is positive integer.
- The product of odd number of negative integers is negative integers.
Examples:
- – 2 × – 2 × – 2 × – 2 = 16
- – 3 × – 3 × – 3 × – 3 × – 3 = -243.
- – 1 × – 1 × – 1 ……. 2022 times = 1
- – 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:
- – 4 × 9 = – 36
9 × – 4 = – 36 - – 15 × (- 10) = 150
– 10 + (- 15) = 150 - 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:
- – 71 × 0 = 0
- 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:
- – 3 × 1= – 3
- 2 × 1 = 2
From above we can say that
- 1 is the multiplicative identity.
- 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
- The product of three integers does not depend upon the grouping of integers and this called associative property for multiplication of integers.
- a × (b × c) = (a × b) × c, for any three integers a, b and c.
Distributive Property:
Consider:
- 7(2 + 3) = 7 × 2 + 7 × 3 = 14 + 21 = 35
- (8 + 1)2 = 8 × 2 + 1 × 2 = 16 + 2 = 18
From above we can say that
- a × (b + c) = a × b + a × c, for any three integers a, b & c.
- (a + b) × c = a × c + b × c
Also
- 8 × (5 – 2) = 8 × 5 – 8 × 2 = 40 – 16 = 24
- (3 – 2) × 4 = 3 × 4 – 2 × 4 = 12 – 8 = 4
From above we can say that
- a × (b – c) = a × b – a × c
- (a – b) × c = a × c-b × c
- a × (b – b) = a × 0 = 0
- (a – a) b = 0 × b = 0
Division of Integers:
Consider:
- 10 ÷ 2 = 5
- – 10 ÷ 2 = -5
- 10 ÷ (- 2) = – 5
- – 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.
- a ÷ (- b) = (- a) ÷ b, where b ≠ 0
- (- a) ÷ (- b) = a ÷ b, where b ≠ 0
- a ÷ 0 is not defined, a ≠ 0
- – a ÷ 0 is not defined, a ≠ 0
- (- a) ÷ (- a) = 1, a ≠ 0
- – a ÷ a = – 1, a ≠ 0
- 0 ÷ 0 is not defined.
- a ÷ a = 1, a ≠ 0
- a ÷ 1 = a, a ≠ 0
- a ÷ (- 1) = – a, a ≠ 0