What are Rational Numbers?

Rational numbers are the subgroup of real numbers that are specifically written in the form of p/q, where both p and q are integers and q can never be 0. Thus, rational numbers include other numbers such as natural numbers, whole numbers, fractions, decimals, and others. Real numbers are the union of Rational numbers and Irrational Numbers

What is a Rational Number?

The numbers which can be expressed as fractions of two integers and can be written as a positive number, negative number, prime, and even zero is called rational numbers.

Rational Number can be expressed as

 p/q, where q ≠0

For example, 2/3 is a rational number which is in the form of p/q, and here 2 and 3 both are integers, and q (2) ≠ 0

Rational numbers can be expressed as fractions, decimals, and even zeros. All the numbers with a non-zero denominator that can be written in p/q form are rational numbers.

Types of Rational Numbers

Various types of numbers can be represented as rational numbers some of which are discussed below: 

Fraction Number

A rational number is a ratio of two integers which can be written in the form of p/q where q is not equal to zero. Hence, any fraction with a non-zero denominator is a rational number.

Example

-2 / 5 is a rational number where -2 is an integer being divided by a non-zero integer 5

Decimal Number

A rational number can be also written in the decimal form if the decimal value is definite or has repeating digits after the decimal point.

Example

0.3 is a rational number. As the value 0.3 can be further expressed in the form of ratio or fraction as p/q

0.3 = 3/10

Also, 1.333333… can be represented as 4/3 hence, 1.33333… is a rational number.

Is 0 a Rational Number?

Yes, 0 is a rational number as it has a non-zero denominator. It can be written in the p / q form as,

0 = 0/1 = p / q

where,
q is not equal to 0

How to Identify Rational Numbers?

Rational Numbers have various properties from which we can identify them some of which are given below:

• Natural numbers, Whole Numbers, Fractions, and Integers all are rational numbers.
• All terminating decimals are Rational Numbers.
• All recurring decimals are Rational Numbers.
• Non-Recurring and Non Terminating decimals are Irrational Numbers.
• All the numbers which can be expressed as p/q where p and q are integers are Rational Numbers.

Example: Check whether 2.69696969… is a rational number or not?

Solution:

Given Number 2.69696969… has repeating decimals, hence it is a rational number.

Positive and Negative Rational Numbers

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

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

Rational Number Standard Form

The standard form of a rational number is,

p/q

where,
p and q are both integers with no common integers,
q can never be zero.

For example, 2/6 is a rational number but it is not in its standard form as 2/6 has a common factor of 2 and it can further be simplified as 1/3. Thus, its standard form is 1/3.

Rational Numbers Arithmetic Operations

We can perform various arithmetic operations on rational numbers which are,

• Addition
• Subtraction
• Multiplication
• Division

Addition

Let’s take two rational numbers p/q and s/t, adding these two using rules of addition we get 

p/q + s/t = (pt+qs)/qt

Example:  Add 3/5 + 2/7

Solution:

3/5 + 2/7 = (3×7 + 2×5) / 5×7
                = (21 + 10) / 35
                = 31 / 35

Subtraction

Let’s take two rational numbers p/q and s/t, subtracting these two using rules of subtraction we get

p/q – s/t = (pt  – qs)/qt 

Example: Subtract 3/5 – 2/7

Solution:

3/5 – 2/7 = (3×7 – 2×5) / 5×7
                = (21 – 10) / 35
                = 11 / 35

Multiplication

Let’s take two rational numbers p/q and s/t, multiplying these two using rules of multiplication we get 

p/q × s/t = (p × s) / (q × t)

Example: Multiply 3/5 × 2/7

Solution:

3/5 × 2/7 = (3 × 2) / (5 × 7)
                = 6 / 35

Division

Let’s take two rational numbers p/q and s/t, we know that divide is the inverse of multiply then dividing these two using rules of division we get 

(p/q) / (s/t) = p/q × t/s = (p × t) / (q × s)

Example: Divide (3/5) / (2/7)

Solution:

(3/5) / (2/7) = 3/5 × 7/2
                    = (3 × 7) / (5 × 2)
                    = 21 / 10

Multiplicative Inverse of Rational Numbers

A Multiplicative Inverse of Rational Numbers is a number that when multiplied by the number results in 1. The general form of a rational number is p/q then its multiplicative inverse is q/p.

For example: For a rational number 2/3, then its multiplicative inverse is 3/2, such that,

2/3 × 3/2 = 1

Properties of Rational Numbers

Various properties of rational numbers are,

• The results are always a rational number if we multiply, add, or subtract any two rational numbers.
• Multiplying or dividing the numerator and denominator of any rational number with the same number does not change the number such that, p/q = ap/aq.
• Multiplication, Division, Addition, and Subtraction of any two rational numbers result in a rational number.
• The additive inverse of the rational number is zero as p/q + 0 = p/q
• The multiplicative inverse of the rational number is 1 as p/q × 1  = p/q

Rational Numbers and Irrational Numbers

Rational Numbers and Irrational Numbers both are subsets of real numbers the basic difference between them is that Rational Numbers can be represented as p/q whereas Irrational Numbers can not be represented as p/q.

All natural numbers, whole numbers, decimals, and others are subsets of rational numbers while irrational numbers are those numbers that are non-repeating and non-terminating numbers.

Examples of Rational Numbers

• 1, 2, 3,…
• 1/2, 2/3, 4/5,…
• 2.3 = 23/10, etc.

Examples of Irrational Numbers

• √2 = 1.414213…
• √3 = 1.7320508…
• Pi (π) = 3.142857…
• Euler’s Number (e) = 2.7182818284590452…….

How to Find Rational Numbers between Two Rational Numbers?

We can find Rational Numbers between Two Rational Numbers by two methods which are,

Method 1

For the given rational numbers find their equivalent rational numbers and then the number between them is found easily.

Example: Find the rational number between 1/2 and 4/3.

Solution:

1/2 = 3/6

4/3 = 8/6

Then rational numbers between 3/6 and 8/6 are 4/6, 5/6, 6/6, 7/6.

Method 2

In the second method, we find the mean of the given two numbers (m) and then find the mean of the first number with m and the mean of the second number with m, and repeated this process to get more numbers.

Example: Find the rational number between 1/2 and 4/3.

Solution:

Mean of 1/2, 4/3 = (1/2 + 4/3) / 2 = 11/12

Mean of 1/2, 11/12 = (1/2 + 11/12) / 2 = 17/24

Mean of 11/12, 4/3 = (11/12 + 4/3) / 2 = 27/24

Then rational numbers between 3/6 and 8/6 are 17/24, 11/12, 27/24.

Solved Examples on Rational Number

Example 1: What are the rational numbers between 3 and 5?

Solution:

 Rational Numbers between 3 and 5 are 31/10, 32/10, 33/10, 34/10, 35/10, 36/10,…………..,49/10

Lets express 3 and 5 as rational numbers as

3 = 3×10/10 = 30/10

5= 5×10/10 = 50/10

Hence, the rational numbers between 3 and 5 are 30/10 and 50/10 are 31/10, 32/10, 33/10, 34/10, 35/10, 36/10, 37/10, 38/10, 39/10, 40/10, …………..49/10.

Example 2: What are the five rational numbers between 0 and 1?

Solution:

Five rational numbers between 0 and 1 are 0.1, 0.2, 0.3, 0.4 and 0.5.

Example 3: Simplify, 1/2 + 2/3 – 4/5

Solution:

1/2 + 2/3 – 4/5

= 7/6 – 4/5

= (35 – 24) / 30 = 9/30

= 3/10

Example 4: Simplify, 1/2 × 2/3 ÷ 4/5

Solution:

1/2 × 2/3 ÷ 4/5

= 1/2 × 2/3 × 5/4

= 5/12

FAQs on Rational Number

Question 1: What are Rational Numbers?

Answer:

Rational Numbers are subsets of real numbers, they can be written in the form of p/q where p and q are integers with no common factors, and q is not equal to 0. Examples of rational numbers are 1/5, 2/7, etc.

Question 2: Is 2.6 a rational number?

Answer: 

Yes, 2.6 is a Rational Number.  As rational numbers can be expressed as decimals values as well as fractions.  The number can also be written as 26/10 which is the ratio of two integers.

The number 2.6 can be represented as shown below,

2.6 = 26/10 = 13/5

The number 13/5 is the ratio of two integers that are 13 integers divided by 5 integers and expressed in fraction form (as p/q where q is not equal to 0).

Question 3: Is 3.14 a rational number?

Answer:

Yes, the number 3.14 is a rational number. Since rational numbers can also be expressed as decimals with repeating digits after the decimal point.

The given number 3.14  can be expressed as

3.14/100 = 22/7

The number 22/7 is the ratio of two integers that are 22 integers divided by integer 7.

Question 4: Is 0 a rational number?

Answer:

 Yes, 0 is a rational number because it has a non-zero denominator. Since the number 0 can also be written as 0/1.

Question 5: Is Pi(π) a rational number?

Answer:

Pi (π) can not be expressed in the form of p/q and hence it is not a rational number. Pi is a non-terminating and non-recurring decimal and its value equals 3.142857…