Rational Numbers

A rational number is a type of real number expressed as p/q, where q ≠ 0. Any fraction with a non-zero denominator qualifies as a rational number. Examples include 1/2, 1/5, 3/4, and so forth. Additionally, the number 0 is considered a rational number as it can be represented in various forms such as 0/1, 0/2, 0/3, etc.

In this article, learn about rational numbers, their properties, examples, and others in detail.

What are Rational Numbers?

A rational number is any number that can be written as a simple fraction. This includes all integers, as any integer z can be written as z/1.

Rational Numbers Definition

Rational Number is a real number written in the form of p/q where p and q are integers and q is not equal to zero.

• 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.

Forms of Rational Numbers

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

Fraction Form of Rational Number

A rational number is a ratio of two integers. Hence, in the fraction number, it 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 Form of Rational 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.

Standard Form of Rational Numbers

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.

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

Learn, Zero is Rational or Not

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.
• All the numbers which can be expressed as p/q where p and q are integers are Rational Numbers.

Note: Non-Recurring and Non Terminating decimals are Irrational Numbers.

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

Solution:

Given Number 2.69696969... has repeating value 69 after decimal hence it is a rational number.

Positive and Negative Rational Numbers

• 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)

Addition of Rational Numbers

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 of Rational Numbers

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 of Rational Numbers

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 of Rational Numbers

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.

An irrational number is a type of real number that cannot be expressed as a ratio of two integers (i.e., it cannot be written in the form p​/q, where p and q are integers and q≠0. These numbers have non-repeating, non-terminating decimal expansions.

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.

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Solved Examples on Rational Numbers

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