Rational Numbers

In Maths, rational numbers are represented by the p / q form where q is not equal to zero. And that's kind of a real number. Any fraction with non-zero numbers is a logical number. Therefore, we can say '0' and it is a logical number, as we can represent it in many ways such as 0/1, 0/2, 0/3, etc. However, 1/0, 2/0, 3/0, etc. are not rational, since they give us infinite values.Also, check the irrational  numbers here and compare them with rational numbers.

In this article, we will learn about the correct number of numbers, the properties of logical numbers and their types, the differences between logical and unreasonable numbers, and the examples solved. It helps to understand ideas better. Also, read various examples of rational numbers and learn how to find logical numbers in a better way. To represent rational numbers in a number line, we need to simplify and write on the decimal form fir

What is a Rational Number?

Rational numbers, in Mathematics, can be defined as any number that can be represented in the form of p / q when q ≠ 0. Also, we can say that any fraction falls under the category of rational numbers, where denominator and numerator are integers  and denominator is not equal to zero . When a rational number (e.g., a fraction) is divided, the result will be in the form of a decimal, which may be the completion of a terminating or a recurring decimal.

How to identify rational numbers?

To identify a rational number check the below conditions

• Represented in the form of p / q, when q ≠ 0.
• The p / q ratio was added simplified and displayed in decimal form.

 Set of numbers that can be said as rational :

1. Enter positive, negative, and zero numbers
2. It can be displayed as a fraction

Examples of Rational Numbers: 

p	q	p/q	Rational
20	2	20/2=10	Yes
2	1000	2/1000 = 0.002	Yes
40	3	40/3=13.333	Yes

Types of Rational Numbers

Rational number makes sense if we can write it as a fraction, where both the number and the number are integers and the denominator is a non-zero number.

The diagram below helps us to understand more about sets of numbers.

ImageSource-Google: Byjus

Real numbers (R) includes all the rational numbers (Q).

Real numbers include all integers (Z).

Integers include natural numbers (N).

Every number is a rational number because every whole number can be displayed as a fraction.

 Standard Form of Rational Numbers

The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive.

For example, 13/39 is a rational number. But it can be made as simple as 1/3; common factors

between a divisor and dividend is only one .

So we can say that the logical number ⅓

is in the Standard form.

Positive and Negative Rational Numbers

Since we know that a rational number is in the form of p / q, where p and q are integers. Also, q should be a non-zero number. A rational number can be positive or negative. If a rational number is positive, both p and q are positive integers. If a sensible number takes the form - (p / q), then p or q takes the negative value. It means

-(p/q) = (-p)/q = p/(-q).

 let’s discuss some of the examples of positive and negative rational numbers.

Positive Rational Numbers	Negative Rational Numbers
If  the numerator and denominator are  the same signs.	If numerator and denominator are of opposite signs.
All are greater than 0	All are less than 0
Example: 12/17, 9/11 and 3/5 are positive rational numbers	Example: -2/17, 9/-11 and -1/5 are negative rational numbers

Arithmetic Operations on Rational Numbers

In Maths, artithmetical operations are the basic functions we perform in integers. Let's discuss here how we can make this work with sensible numbers, say p / q and s / t.

Addition: When we add p / q and s / t, we need to make the denominator the same. So, we get (pt + qs) / qt.

Example: 1/2 + 3/4 = (2 + 3) / 4 = 5/4

Subtraction: Similarly, if we subtract p / q and s / t, and then again, we need to subtract the equal value, first, and then subtract.

Example: 1/2 - 3/4 = (2-3) / 4 = -1/4

Multiplication: In the event of a multiplication, while multiplying by two logical numbers, the number and the number of rational numbers increases, respectively. If p / q is multiplied by s / t, then we get (p × s) / (q × t).

Example: 1/2 × 3/4 = (1 × 3) / (2 × 4) = 3/8

Separation: If p / q is separated by s / t, then it is represented as:

(p / q) ÷ (s / t) = pt / qs

Example: 1/2 ÷ 3/4 = (1 × 4) / (2 × 3) = 4/6 = 2/3

Multiplicative Inverse of Rational Numbers

Since a rational number is represented in the form of p / q, which is partial, then the repetitive multiplication of a logical number is a multiple of the given fraction.

For example, 4/7 is a rational number, and a multiple of a positive number 4/7 is 7/4, which (4/7) x (7/4) = 1

Rational Numbers Properties

Since a rational number is a subset of a real number, a rational number will obey  to all the properties of real number system structures. Some of the key properties of rational numbers are as follows:

• Results are always a logical number when we multiply, add, or subtract any two logical numbers.
• A  rational number remains the same when we divide or multiply both the number and the number by the same factor.
• If we add zero to a rational number we will get the same number itself.
• Rational numbers are closed under addition, subtraction, and multiplication.

Learn more about properties of rational numbers here.

Rational Numbers and Irrational Numbers

There is a difference between Rational and Irrational Numbers. A fraction with non-zero numbers is called a rational number. The number ½ is a rational number because it is read as 1 number divided by the number 2. All  numbers that are not rational are called irrational. See the chart below to distinguish between rational and irrational.

ImageSource-Google: Byjus

Rationals can be positive, negative or zero. When specifying a rational number , a negative symbol is in front of or with numerical value, which is a standard mathematical notation. For example, we define negative 5/2 as -5/2.

An irrational number cannot be written as a simple fraction but can be represented by decimal. It has endless digits that do not repeat after a decimal point. Some of the most irrational common numbers are:

Pi (π) = 3.142857…

Euler's number (e) = 2.7182818284590452 …….

√2 = 1.414213…

How to Find Rational Numbers Between Two Rational Numbers

There are "n" numbers of rational numbers between two rational numbers. Rational numbers between two rational numbers can be easily found using two different methods. Now, let's look at two different ways.

Method 1:

Find the equal number of rational numbers and find the correct numbers between them. Those numbers should be the rational numbers needed.

Method 2:

Find the number that means the number of two rational numbers. The value that means must be a  number required. To find rational numbers, repeat the same process with old and newly acquired rational numbers.

Solved Examples

Example 1:

Mark each of the following as irrational or rational: ¾, 90/12007, 12 and -5.

Solution:

Since the rational number is the one that can be shown as a ratio. This indicates that it can be displayed as a fraction where both numerator and denominator are integers.

¾ is a rational number as it can be displayed as a fraction. 3/4 = 0.75

Fraction 90/12007 makes is rational.

12, can also be expressed as 12/1. And a reasonable number.

The value of √5 = 2.2360679775 …… .. It is a non terminating value and therefore cannot be written as a fraction. It's an irrational number.

Example 2:

Find whether the mixed fraction, 1 1/2 is a rational number.

Solution:

The simplest form of   1 1/2  is  3/2

Numerator = 3, which is a integer 

Denominator = 2, is a integer and is non-zero 

So, yes, 3/2 is a rational number.

Example 3:

Find out if the given numbers are rational or irrational.

(a) 1.75 (b) 0.01 (c) 0.5 (d) 0.09 (d) -3

Solution:

The numbers provided are in decimal form. To determine whether a given number is a decimal or not, we must convert it into a fractional form (e.g., P / q)

If the denominator of the fraction is not equal to zero, then the number is rational , or else, it is irrational .

Decimal Number	Fraction	Rational Number
1.75	7/4	yes
0.01	1/100	yes
0.5	1/2	yes
0.09	1/11	yes
√ 3	?	No

Frequently Asked Questions on Rational Numbers {FAQ}

How to identify the rational numbers?

 If the number is expressed in the form of p / q it is a rational number.

Here p and q are integers, and q is not equal to 0. A rational number should have a numerator and a denominator. Examples: 10/2, 30/3, 100/5.

What is the difference between rational and irrational numbers?

A rational number is a number that is  expressed as a  where the denominator should not be equal to zero, and an odd number cannot be expressed in the form of fractions. Rational numbers are terminating decimals but irrational numbers do not. An example of a rational number is 10/2, and for an irrational number is the popular arithmetic Pi (π) equal to 3.141592653589 …….

Is 0 a rational number?

Yes, 0 is a rational number because it is a whole number, which can be written in any way like 0/1, 0/2, where b is a non-zero number. It can be written as follows: p / q = 0/1. Therefore, we conclude that 0 is a rational number.

How to get a rational number between 3 and 4?

 The rational number between 3 and 4 = 1/2 (3+4)

= 7/2