RationalNumbers.pptx ppt ppt ppt ppt ppt

B.J.P.S Samiti’s
M V HERWADKAR ENGLISH MEDIUM
HIGH SCHOOL
CLASS : VII SUBJECT : MATHEMATICS
TOPIC : RATIONAL NUMBERS
BY : SHRAMMI CHAUHAN

Introduction
 Do you remember Natural Numbers, Whole Numbers & Integers?
 What are Rational Numbers?

Rational Numbers
o Rational Numbers : Numbers that can be written in the form p/q, where p & q are
integers & q ≠ 0 is called a Rational Number.
• eg : 4/7 , 11/13 , -2/9 , 9/-5 etc.
o Properties for Rational
• Closure
• Commutative
• Associative.

Properties for Rational Numbers
o Closure
NUMBERS CLOSED FOR (CLOSURE)
ADDITION SUBTRACTION MULTIPLICATION DIVISION
WHOLE NUMBER
REMARK
OPERATION
EXAMPLE
YES
a + b = c
2 + 7 = 9
NO
a - b = c
4 - 7 = -3
YES
a x b = c
2 x 4 = 8
NO
a ÷ b = c
4 ÷ 7 = 4 / 7
INTEGER REMARK
OPERATION
EXAMPLE
YES
a + b = c
-7 + 4 = -3
YES
a - b = c
-11 - 2 = -13
YES
a x b = c
-4 x 3 = -12
NO
a ÷ b = c
-5 ÷ 6 = -5 / 6
RATIONAL NUMBER
REMARK
OPERATION
EXAMPLE
YES
a + b = c
1/5 + 2/5 = 3/5
YES
a - b = c
4/7 – 3/7 = 1/7
YES
a x b = c
-5/6 x 1/5 = -5/30
NO
a ÷ b = c
4/5 ÷ 0 = NOT DEFINED

o Commutative
NUMBERS COMMUTATIVE FOR
WHOLE NO. REMARK
OPERATION
EXAMPLE
YES
a + b = b + a
2 + 7 = 7+2
NO
a - b ≠ b – a
4 - 7 ≠ 7 - 4
YES
a x b = b x a
2 x 4 = 4 x 2
NO
a ÷ b ≠ b ÷ a
4 ÷ 7 ≠ 7 ÷ 4
INTEGER REMARK
OPERATION
EXAMPLE
YES
a + b = b + a
7 + 4 = 4 + 7
NO
a - b ≠ b – a
11 - 2 ≠ 2 - 11
YES
a x b = b x a
-4 x 3 =3 x (-4)
NO
a ÷ b ≠ b ÷ a
-5 ÷ 6 ≠ 6 ÷ (-5)
RATIONAL NO.
REMARK
OPERATION
EXAMPLE
YES
a + b = b + a
1/5 + 2/5 = 2/5 + 1/5
NO
a - b ≠ b – a
4/7 – 3/7 ≠ 3/7 – 4/7
YES
a x b = b x a
-5/6 x 1/5 = 1/5 x (-5/6)
NO
a ÷ b ≠ b ÷ a
4/5 ÷ 6/7 ≠ 6/7 ÷ 4/5
NATURAL NO. REMARK
OPERATION
EXAMPLE
YES
a + b = b + a
1 + 5 = 5 + 1
NO
a - b ≠ b – a
4 – 3 ≠ 3 - 4
YES
a x b = b x a
3 x 4 = 4 x 3
NO
a ÷ b ≠ b ÷ a
8 ÷ 4 ≠ 4 ÷ 8

o Associative
NUMBERS ASSOCIATIVE FOR
WHOLE NO. REMARK
OPERATION
EXAMPLE
YES
a + (b + c)= (a + b) + c
5 + (2 + 7) = (5 + 2) + 7
NO
a - (b - c) ≠ (a - b) - c
5-(4 – 7) ≠ (5 – 4) - 7
YES
a x (b x c) = (a x b) x c
7x(2 x 4) = (7x2)x4
NO
a ÷ (b ÷ c) ≠ (a÷ b) ÷ c
6÷ (4 ÷ 7) ≠ (6÷ 4) ÷ 7
INTEGER REMARK
OPERATION
EXAMPLE
YES
a + (b + c)=(a + b) + c
-5 + (2 + 7)=(-5 + 2) + 7
NO
a - (b - c) ≠ (a - b) - c
8-(11 - 2) ≠ (8 - 11)-2
YES
6 x(-4 x 3) = (6 x -4) x
3
NO
a ÷ (b ÷ c) ≠ (a÷ b) ÷ c
3÷ (-5 ÷ 6) ≠ (3÷ -5) ÷ 6
RATIONAL NO.
REMARK
OPERATION
EXAMPLE
YES
a + (b + c)= (a + b) + c
5/7+(1/5 + 2/5) =
(5/7+1/5) + 2/5
NO
a - (b - c) ≠ (a - b) - c
6/8-(4/7 – 3/7) ≠
(6/8 – 4/7) - 3/7
YES
4/9 x (-5/6 x 1/5) =
(4/9 x -5/6) x 1/5
NO
a ÷ (b ÷ c) ≠ (a÷ b) ÷ c
2/3 ÷ (4/5 ÷ 6/7) ≠
(2/3 ÷ 4/5) ÷ 6/7
NATURAL NO. REMARK
OPERATION
EXAMPLE
YES
a + (b + c)= (a + b) + c
15 +(2 + 5)=(15 + 2) + 5
NO
a - (b - c) ≠ (a - b) - c
10 - (7 – 3) ≠ (10 - 7) -
3
YES
9x(6 x 5) = (9x6)x5
NO
a ÷ (b ÷ c) ≠ (a÷ b) ÷ c
2÷ (8 ÷ 4) ≠ (2÷ 8 )÷ 4

The Role of Zero (0)
 Additive Identity : Zero is called the identity for rational numbers under addition
because adding zero to any rational number, the result is same rational number.
This is known as Additive identity property.
• It is additive identity for integers and whole numbers as well.
• a + 0 = 0 + a = a ; [ a = whole number ]
example : 8 + 0 = 0 + 8 = 8
• b + 0 = 0 + b = b ; [ b = integer ]
example : -5 + 0 = 0 + (-5) = -5
• c + 0 = 0 + c = c ; [ c = rational number ]
example : 2/7 + 0 = 0 + 2/7 = 2/7

The Role of One (1)
 Multiplicative Identity : When 1 is multiplied to any number, the is the same number.
Therefore, 1 is the multiplicative identity for whole numbers, integers and rational
numbers.
 Multiplicative Inverse : A rational number c/d is called the reciprocal or multiplicative
inverse of another non-zero rational number a/b.

Additive Inverse of a Number
Additive Inverse or Negative of a number is defined as the value which on adding with
the original number results in zero.
 For integer a,
a + (-a) = a – a = 0
Example : 9 + (-9) = 9 – 9 = 0
 For integer – a,
-a + a = -a + a = 0
Example : -9 + 9 = 0
 For rational number a/b, the additive inverse is –a/b
a/b + (-a/b) = a/b – a/b = 0
Example : 9/11 + (-9/11) = 9/11 – 9/11 = 0
 For rational number -a/b, the additive inverse is a/b
-a/b + a/b = -a/b + a/b = 0
Example : -9/17 + 9/17 = 0

Distributive Property of multiplication
over Addition
According to the distributive property of multiplication over addition or
subtraction of two numbers together then resultant product will be same
as when we take the product of that number indivisually and then do
addition or subtraction.
a(b + c) = ab + ac
a(b – c) = ab – ac
Q. Find using distributivity :
a) {7/5 x (-3/12 )} + {7/12 x }
b) 2/5 x 3/7 - 1/6 x 3/2 + 1/14 x 2/5

Representation of Rational Number
on a Number Line
A number line us a visual representation of numbers on a straight line.
Usually, 0 is in the middle, positive numbers on the right side of zero and negative
numbers on the left side of zero.

Steps to Represent Rational Number
on a Number Line
STEPS:
1. Draw a horizontal line with arrows onboth ends, making the origin as (0) as the center.
2. Mark integers ensuring equal spacing between them.
3. If rational number is a fraction, divide the interval between consecutive integers into
equal.
The number of equal parts should be equal to the denominator of the fraction.
4. Count the number of parts from zero , movinfg to the right for positive fractions and to
the left for negative fraction . The number of parts should be equal to the numerator
of the fractions.
5. Mark the point on the number line that represents the rational number.

Rational Number between two
Rational Number
A. When Denominators are same
i) Check the values on the numerator of the rational numbers.
ii) Find by how many values the numerators differ from each other.
iii) If the difference between two numerators is more, then we can write
the rational numbers between the given rational numbers in the
increasing order of numerator.
Example : Find rational numbers between 2/13 and 11/13.
Now, the denominators are same and difference of numerator is more so,
rational numbers are : 3/13, 4/13, 5/13, 6/13, 7/13, 8/13, 9/13 and 10/13

Rational Number
A. When Denominators are same
iv) If the difference between two numerators is less or we need to find
more rational numbers, then we multiply the numerator and denominator
by multiples of 10.
Example : Find rational numbers between 2/7 and 5/7.
Now, the denominators are same and difference of numerator is less so,
we multiply both numerator and denominator by 10/10.
2/7 x 10/10 = 20/70 and 5/7 x 10/10 = 50/70
so, the rational numbers between 20/70 and 50/70 are: 21/70, 22/70,
23/70, 24/70 …….49/70

Rational Number
B. When Denominators are different
i) Find the LCM of two rational numbers.
ii) Multiply and divide the two rational numbers by the value that results in
the denominators equal to the obtained LCM .
iii) Once the denominators become the same follow the same rules with
the same denominators.
Example : Find rational numbers between ½ and 7/3
LCM of 2 and 3 = 6
½ x 3/3 = 3/6 and 7/3 x 2/2 = 14/6
Now multiply both rational numbers by 10/10 , we get 30/60 and 140/60
So, rational numbers between ½ and 7/3 are 31/60, 32/60, 33/60, 34/60,
….. 139/60

Rational Number
C. Finding the average
i. Let the two rational numbers be ‘a’ and ‘b’
Add them : (a+b)
ii. Divide the sum by 2. i.e. (a+b)/2
iii. Continue this process to find rational numbers between original
numbers and the new rational number.
Example : Find rational number between ¼ and 2/3.

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