Revision Notes on Rational Numbers
Learning Goal
Rational Numbers and it’s types
Equivalent Rational Numbers
Rational Numbers on number line
Operations of rational numbers (Add, Subtract, Multiply and Divide)
Comparison of Rational Numbers
Simplification of Rational Numbers
Numbers and Their Types in Mathematics
Natural Numbers
Counting numbers are called natural numbers. Ex- 1, 2, 3, 4……..
Whole Numbers
All natural numbers including 0 are whole numbers.
Integers
All negative and positive numbers including 0 are called Integers.
Rational Numbers
The numbers that can be expressed in the form p/q where p and q are integers but (q ≠ 0) are called Rational Numbers.
Rational Number includes all natural, whole numbers, fractions and integers.
Equivalent Rational Numbers
To find it multiplying or dividing the numerator and denominator by the same integer.
Numbers are said to be equivalent if they are proportionate to each other.
Example
Therefore 1/2, 2/4, 4/8 are equivalent to each other as they are equal to each other.
Positive and Negative Rational Numbers
1. Positive Rational Numbers are the numbers whose both the numerator and denominator are positive.
Example: 3/4, 12/24 etc.
2. Negative Rational Numbers are the numbers whose one of the numerators or denominators is negative.
Example: (-2)/6, 36/(-3) etc.
Remark: The number 0 is neither a positive nor a negative rational number.
Rational Numbers on the Number Line
Rational Numbers can also be represented on a number line like integers i.e. positive rational numbers are on the right to 0 and negative rational numbers are on the left of 0.
Representation of rational numbers can be done on a number line as follows
Rational Numbers in Standard Form
Conditions-
Method to Write Rational Numbers in Standard Form
Reduce it to its standard form or the lowest form by dividing its numerator and denominator by their HCF.
Example
Find the standard form of 12/18
Solution
2/3 is the standard or simplest form of 12/18
Comparison of Rational Numbers
Case-1. Positive rational numbers
Convert it into like fraction by finding LCM.
Example
Compare 4/5 and 3/8 and tell which one is greater.
Solution
Find LCM of the denominator of both the numbers.
LCM (5, 8) = 40
Case-2. Negative rational numbers
Example
Compare – (2/5) and – (3/7) and tell which one is greater.
Solution
LCM (5, 7) = 35
Case-3. Positive and Negative rational numbers
Rational Numbers between Rational Numbers
Example
Find the rational numbers between 3/5 and 3/7.
Solution
LCM (5 and 7) = 35
Hence the rational numbers between 3/5 and 3/7 are
These are not the only rational numbers between 3/5 and 3/7.
If we find the equivalent rational numbers of both 3/5 and 3/7 then we can find more rational numbers between them.
Hence we can find more rational numbers between 3/5 and 3/7.
Remark: There are “n” numbers of rational numbers between any two rational numbers.
Operations on Rational Numbers
1. Addition
a. Addition of two rational numbers with the same denominator
i. We can add it using a number line.
Example:
Add 1/5 and 2/5
Solution:
On the number line we have to move right from 0 to 1/5 units and then move 2/5 units more to the right.
ii. If we have to add two rational numbers whose denominators are same then we simply add their numerators and the denominator remains the same.
Example
Add 3/11 and 7/11.
Solution
As the denominator is the same, we can simply add their numerator.
b. Addition of two Rational Numbers with different denominator
If we have to add two rational numbers with different denominators then we have to take the LCM of denominators and find their equivalent rational numbers with the LCM as the denominator, and then add them.
Example
Add 2/5 and 3/7.
Solution
To add the two rational numbers, first, we need to take the LCM of denominators the find the equivalent rational numbers.
LCM of 5 and 7 is 35.
c. Additive Inverse
If the sum of two numbers is zero than they are called additive inverse of each other.
To fine additive inverse change (+)Ve sign into (-)Ve and vice-versa.
This shows that the additive inverse of 3/7 is – (3/7)
This shows that
2. Subtraction
If we have to subtract two rational numbers then we have to add the additive inverse of the rational number that is being subtracted to the other rational number.
a – b = a + (-b)
Example
Subtract 4/21 from 8/21.
Solution
i. In the first method, we will simply subtract the numerator and the denominator remains the same.
ii. In the second method, we will add the additive inverse of the second number to the first number.
3. Multiplication
a. Multiplication of a Rational Number with a Positive Integer.
Simply multiply the integer with the numerator and the denominator remains the same.
Example
b. Multiply of a Rational Number with a Negative Integer
Simply multiply the integer with the numerator and the denominator remains the same and the resultant rational number will be a negative rational number.
Example
c. Multiply of a Rational Number with another Rational Number
Multiply the numerator of two rational numbers and multiply the denominator of the two rational numbers.
Example
Multiply 3/7 and 5/11.
Solution
4. Division
a. Reciprocal
Reciprocal is the multiplier of the given rational number which gives the product of 1.
Reciprocal of a/b is b/a
Product of Reciprocal
If we multiply the reciprocal of the rational number with that rational number then the product will always be 1.
Example
b. Division of a Rational Number with another Rational Number
Multiply the reciprocal of the rational number with the other rational number.
Example
Divide
Solution