[Maths Class Notes] on Rational Numbers Pdf for Exam

We work with basic math in our daily life like counting your fingers and toes, calculating your age, finding temperatures of a cold winter day in Michigan which would be below 0 degrees, dividing a birthday cake among twelve of your friends, and many more. In each of these circumstances, we use different types of numbers. The types of numbers that we already know are natural numbers, whole numbers and integers.

When you were studying fractions, you were just learning the way of writing a numerical quantity, with a numerator and a denominator.  You also learned about using arithmetic operations with the fractions and about representing them as decimals. Now you are growing older, and are ready to dig mathematics a little deeper.  You are going to learn about the numbers themselves and not just the way you write them on a piece of paper. Let us walk through a new type of number called rational numbers.

Rational Numbers 

After integers, rational numbers are one of the most prevalent types of numbers we learn in math. “Ratio” is how it’s referred to. As a result, rational numbers have a strong connection to the concept of ratio.

A rational number has the form [;frac{p}{q}], where p and q are both integers and q is not equal to 0. Q is the abbreviation used for rational numbers.

The p/q form of numbers makes it difficult to distinguish between fractions and rational numbers. Fractions are made up of whole numbers, whereas the numerator and denominator of rational numbers are made up of integers.

Standard Form of Rational Numbers  

A rational number is not just a fraction. It means that any number, be it a natural number or integers, can be written in the form of p/q such that q is never equal to 0. But we cannot write every number in that way.

A rational number is said to be in a standard form if and only if

a) The denominator is an integer greater than 0. 

b) The only common divisor between the numerator and the denominator is 1.

Steps to Express a Rational Number in the Standard Form:

a) Check if the denominator is positive or negative. If the denominator is negative, change it to positive by multiplying both numerator and denominator by -1. 

b) Take the absolute values of the numerator and denominator and find their greatest common divisor.

c) Perform the division of each of the numerator and denominator by the obtained greatest common divisor. The resulting rational number is the standard form.

Positive and Negative Rational Numbers

A positive rational number is one that has the same signs (either positive or negative) in both its numerator and denominator. Eg: −5/−7,10/13.

A negative rational number has opposite signs in both its numerator and denominator.

Eg: −5/7,10/−13

Rational Numbers – Properties, Methods, Calculation and Examples

Properties

The properties of rational numbers are:

Addition of Rational Numbers:

As per the closure property, a + b is also a rational number for any two rational numbers a and b.

Subtraction of Rational Numbers:

As per the closure property, a – b is also a rational number for any two rational numbers a and b.

Multiplication of Rational Numbers:

As per the closure property, a x b is also a rational number for any two rational numbers a and b.

Division of Rational Numbers:

As per the closure property, a ÷ b is also a rational number for any two rational numbers a and b. However, it should be noted that rational numbers are not closed when divided but if we eliminate 0 from the equation, all rational numbers are divided.

Addition of Rational Numbers:

Let’s take any two rational numbers a and b 

So, a + b = b+ a

The addition will always be commutative for rational numbers.

Subtraction of Rational Numbers:

Let’s take any two rational numbers a and b

so, a – b ≠ b –  a

The subtraction will never be commutative for rational numbers.

Multiplication of Rational Numbers:

Let’s take any two rational numbers a and b

So, a*b = b*a

The multiplication will always be commutative for rational numbers.

Division of Rational Numbers:

Let’s take two rational numbers a and b

So, a ÷ b ≠ b ÷ a

The division will never be commutative for rational numbers.

• Let’s take 3 rational numbers a,b, and c

• Add a and b and later add c to it i.e (a+b)+c

• Add b and c and later add a to the sum of it i.e a+(b+c)

• Both the above sums will be the same

Hence, the associative property works addition and multiplication of rational numbers. However, it won’t work for subtraction and division of rational numbers.

Eg. Rational numbers can be added or multiplied regardless of how they are grouped.

The property states that for any three numbers a, b and c:

a*(b+c) =  (a*b) + (a*c)

Methods, Calculations, and Examples

Like fractions, rational numbers can be added, subtracted, multiplied, and divided. On rational numbers, below are the four basic arithmetic operations:

Rational Number – Addition and Subtraction

In the same manner that fractions may be added and subtracted, rational numbers can be added and subtracted. To add or subtract two rational numbers, first, equalize their denominators, then add their numerators.

Example : 

1/2 – (-2/3)

= 1/2 + 2/3 

= 1/2 × 3/3 + 2/3 × 2/2 

= 2/6 + 4/6 

= 6/6 

= 1

Rational Number – Multiplication and Division

Rational numbers can be multiplied and divided in the same way as fractions. We multiply the numerators and denominators of any two rational integers independently before simplifying the resultant fraction.

Example: 

3/5 × -2/7 

= (3 × -2)/(5 × 7)

= -6/35

To divide any two fractions, multiply the first fraction (dividend) by the reciprocal of the second fraction (which is the divisor).

Example: 

3/5 ÷ 2/7

=3/5 × 7/2 

= 21/10 

Rational Numbers Examples

– Both the numerator 2 and the denominator 7 are integers.

, wherein 5 is the quotient when we divide the integer 5 by the integer 1.

– As the square root of 49 can be simplified to 7, which is also the quotient obtained when the integer 7 is divided by 1

Thus, all terminating decimal
s are rational numbers.