Rational Numbers, Representation, Standard Form of Rational Numbers

# Rational Numbers, Representation, Standard Form of Rational Numbers

## Rational Numbers

A number of the form p/q, where p and q are integers and q ≠ 0, is called a rational number. All the natural numbers, whole numbers and integers can be written in the form of p/q by taking 1 as the denominator. For example, 2 = 2/1, 0 = 0/1, -6 = -6/1. Thus, they are rational numbers. Fractions are also the examples of rational numbers.

## Rational Numbers Symbol

Rational numbers are the superset of the set of natural numbers, whole numbers, integers and fractions. It is denoted by the symbol Q. It can written as:
Q = {2, -6,1/4 , -4/7, 0.43, -0.27, -185, -2/5, 431, 5/11, 0.416, -0.425, 1/7}

## Rational Numbers Examples

A fraction is always a rational number where as a rational number may or may not be a fraction. All fractions are rational number because in fraction p, q are natural numbers and all natural numbers are integers.
What about decimal numbers say 0.3, 0.4, 6.5, etc.? Each of the decimals can be represented as fraction say 0.3 = 3/10, 0.4 = 4/10, 6.5 = 65/10. Hence, all decimal numbers are rational numbers. A rational number may be positive, zero or negative.
Thus, rational numbers examples are:
5, -8,1/2 , -5/7, 0.49, -0.25, -385, -2/3, 435, 5/9, 0.316, -0.225, 1/9, -12/31, 0, -492654, etc.
All natural numbers, whole numbers, integers, fractions, terminating decimals and their negatives are rational numbers.

## What are Rational Numbers?

As you have seen above that a number that can be written in the form p/q, where and are integers and ≠ 0, is called a rational number.

### Positive rational numbers

If both the numerator and the denominator of a rational number are either positive or negative integers, then the rational number is called a positive rational number.
For example, 5/9, 4/15, –3/–5, etc.

### Negative rational numbers

A rational number is said to be negative, if the numerator
and the denominator have opposite signs.
For example, –2/5 , 4/–7, –5/7, 1/–3, etc., are negative rational numbers.

## Rational numbers are defined as: A number which can be written in the form p/q, where p and q are integers and q ≠ 0, is called a rational number.

For example, 7 = 7/1 is a rational number.
0 = 0/5 is a rational number.
-3 = -3/1 is a rational number.

## Representation of Rational Numbers on the Number Line

We know how to represent integers and fractions on the number line. In the same way, rational number can also be represented on the number line.

Draw a line and choose any point 0 on it to represent zero. The positive rational numbers are represented on the part of line lying to the right of zero and the negative rational numbers are represented on the part of the line lying to the left of zero.

As in the case of integers the pairs 1 and –1, 2 and –2, 3 and –3, etc., are equidistant from 0, in the same way the rational numbers 1/4 and – 1/4 , 3/4 and – 3/4 , 5/4 and      –5/4, etc., are equidistant from 0.

## Standard Form of Rational Numbers

A rational number of the form p/q is said to be in standard form if p and q have no common factor other than 1 and q is a positive integer.
For example, consider the rational number 15/–35.
(15 ÷ –5)/(–35 ÷ –5) = –3/7
Thus, 15/–35 when written as –3/7 is said to be a rational number in the standard form.

## Comparing and Ordering Rational Numbers

To compare and order the rational numbers, follow these steps:
1. Write the rational numbers in the standard form, i.e., with positive denominator.
2. Find the LCM of the denominators.
3. Write the equivalent rational number of each rational number with common
denominator as the LCM obtained in step 2.
4. Arrange the rational numbers according to the numerators in ascending or descending order.

Example: Arrange the following rational numbers in ascending order:
4/–9, –7/12, 11/–8, –2/3

Solution: 1. The given rational numbers can be written in standard form as:
–4/9 , –7/12, –11/18 , –2/3
2. LCM of 9, 12, 18 and 3 is 36.
3. Write the equivalent rational number of each rational number with common
denominator as the LCM obtained.

## Properties of Rational Numbers

1. Closure Property: Rational numbers are closed under addition, subtraction and
multiplication. For two rational numbers a and b, a + b, a – b and a × b are also rational numbers.

2. Commutative Property: Rational numbers are commutative under addition and
multiplication. For two rational numbers a and b, a + b = b + a and a × b = b × a.

3. Associative Property: Rational numbers are associative under addition and
multiplication. For three rational numbers a, b and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c

4. Additive Identity: The rational number 0 is the additive identity, i.e., if a is a rational
number, then a + 0 = 0 + a = a.

5. Multiplicative Identity: The rational number 1 is the multiplicative identity, i.e., If a
is a rational number, then a × 1 = 1 × a = a.

6. Additive Inverse: If p/q is a rational number, then – p/q is its additive inverse, i.e.,
p/q + (– p/q) = 0.

7. Multiplicative Inverse: The multiplicative inverse of p/q is q/p, since p/q × q/p = 1.

8. Distributive Property: If a, b and c are rational number, then a(b + c) = a ×  b + a × c
and a(b c) = a × b a × c.

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