Rational Numbers: Definition, Properties, Types, and Solved Examples

Rational Numbers

Rational Numbers: The term "rational number" is derived from the word "ratio." Ratios are typically expressed as fractions, such as 7/5, 8/3, or 9/3; these are all prime examples of rational numbers. People often confuse fractions with rational numbers because both share the p/q representation format. However, the distinction lies in their components: while fractions consist of whole numbers covering a broad category—including integers, terminating decimals, and repeating decimals—a rational number is strictly a ratio of two integers. Consider these examples to understand the fundamental difference between ratios, fractions, and rational numbers.

1. If we talk about 9, writing it in the form of 9/1, then 9 is a rational number.
2. If we consider 8/7, which is a rational number and can also be written in the form of a fraction. 
3. We can write 0.333….. in the form of 1/3. Which is in the form p/q, so, 0.333 is a rational number
4. 2.5 can also be written as the ratio of  5/2. Which is the correct form of a rational number, Hence, 2.5 is also a rational number.

By now, you should have a clearer understanding of the nuances between rational numbers, fractions, and ratios. This article focuses specifically on rational numbers, exploring their formal definition, core properties, various types, and practical applications.

What are Rational Numbers?

A rational number is defined as any number that can be expressed as a ratio p/q, where both p and q are integers and the denominator q is not equal to zero. 

In other words, the standard form of a rational number is p/q, where p and q are integers and q≠0. 

There is an infinite set of rational numbers located between any two given rational numbers. For instance, if you take the numbers 10 and 20, you can derive an infinite sequence of rational numbers by adjusting the denominators—such as 10/1, 10/2, 10/3, 10/4, 10/5, and so on—continuing the process indefinitely.

Rational numbers can be expressed in various forms while adhering to their mathematical definition. Examples include integers like 4, 41, or 332; terminating decimals like 0.25; negative numbers like -158 or -0.11; and fractions like 7/4 or -8/3.

Conversely, an expression like 4/0 does not qualify as a rational number because, in this case, q=0.

How do identify Rational Numbers?

 We can conclude that a number is rational if it meets the following criteria:

1. A rational number can be identified by its form i.e a rational number is written in the form p/q where p is the numerator containing integers, q is the denominator containing integers that is not equal to zero. ½, ⅖, ¼, -⅛, 8/7.
2. An integer like 1, 2, 44, 88, -1, -2, -93 etc are also a rational number as  it can be written as 1/1, 2/1, 44/1, 88/1, -1/1, -2/1, -93/1. Here 1 is a denominator.
3. Decimals can be terminating like 0.25, 0.15, 0.125 or non-terminating with repeating patterns only like 0.3333.., 0.1111…, 0.54444…. Are also rational numbers.

Arithmetic Operation of Rational Numbers

Just as with standard fractions, rational numbers support basic arithmetic operations: they can be added, subtracted, multiplied, and divided.

Let's examine how these arithmetic operations are performed.

1. Addition and Subtraction

To add or subtract rational numbers, you must first ensure their denominators are identical.

 If you have p/q and x/y, you must find a common denominator, expressed as (py ± qx)/qy. 
For example, 7/8 + 1/2 can be written as (7×2 + 8×1)/(8×2), resulting in 22/16.

= 22/16, which simplifies to 11/8.

2. Multiplication and Division

Unlike addition and subtraction, multiplication and division of rational numbers do not require a common denominator; the operation is performed directly.

To multiply p/q and x/y, simply calculate (p×x)/(q×y).

For example, 2/5 × 4/3 is calculated as (2×4)/(5×3).

This equals 8/15.

To divide p/q by x/y, use cross-multiplication: (p×y)/(q×x). Let’s look at an example.

8/2 ÷ 9/3 = (8×3)/(2×9) = 24/18, which simplifies to 4/3.

Properties of Rational Numbers

1. The addition, multiplication and subtraction of rational numbers are always rational numbers.
2. When both the numerator and denominator of a rational number are multiplied or divided by the same factor then the resulting rational number remains the same.
3. If we add or subtract (0)zero to a rational number then there are no changes in the resulting rational numbers, that is we will get the same rational number itself.

Positive and Negative Rational Numbers

Understanding these basics makes it simple to distinguish between positive and negative rational numbers. Both follow the p/q format where q≠0. A positive rational number has a numerator and denominator with the same sign (both positive or both negative). A negative rational number has a numerator and denominator with opposite signs.

Examples of positive rational numbers include 8/3, 11/4, -5/-2, 36/7, and 62/11.

Examples of negative rational numbers include 8/-15, -1/8, -4/7, and 13/-14.

Important points on Rational Numbers

1. 0 (Zero) is also a rational number.
2. The denominator of a rational number will never be zero.
3. Between two rational numbers, there are infinite rational numbers.
4. Not only fractional numbers are rational numbers but any number that can be expressed in a fraction is a rational number.
5. A real number which is not considered to be a rational number is an irrational number.

Rational Numbers- Solved Questions

Q1: In a restaurant, there are 18 chefs and 12 waiters. What fraction of the total staff are waiters?

Answer: Let the number of waiters be ‘p’ = 12. 

The total number of staff is ‘q’ = 18 chefs + 12 waiters = 30.

The fraction of waiters is p/q = 12/30.

 Simplifying 12/30 gives 2/5.

 The fraction of waiters in the restaurant is 2/5.

Thus, the fraction of staff who are waiters is 12/30, which reduces to 2/5.

Q2: List five rational numbers less than 5.

Answer: Integers are also rational numbers. Examples less than 5 include -2, -1, 0, 1, and 2.

Q3: Identify the rational numbers from the following list: √9, -8/10, 8/-9, √5, √5/2, 0.25, 0.333…, 2.41421254…

Answer: Let's evaluate each number:

√9 = 3. Since 3 can be written as 3/1, it is a rational number.

-8/10 and 8/-9 are rational numbers as they are in the p/q form, where q≠0.

√5 and √5/2 result in non-terminating, non-repeating decimals; therefore, they are irrational.

0.25 is a terminating decimal (1/4) and is thus a rational number.

0.333… is a non-terminating, repeating decimal (1/3), making it a rational number.

2.41421254… is non-terminating and non-repeating, so it is not a rational number.

Therefore, √9, -8/10, 8/-9, 0.25, and 0.333… are rational.

Q4: Find the sum of the following rational numbers: 8/9 + 6/5 and 9/4 + 8/2.

Answer: To add two rational numbers, use the formula (py + qx)/qy.

For 8/9 + 6/5: (8×5 + 9×6)/(9×5) = (40 + 54)/45 = 94/45.

For 9/4 + 8/2: (9×2 + 4×8)/(4×2) = (18 + 32)/8 = 50/8 = 25/4.

Q5: Find the difference between the following rational numbers: 3/2 - 5/4 and 6/2 - 7/5.

Answer: Use the formula (py - qx)/qy.

For 3/2 - 5/4: (3×4 - 2×5)/(2×4) = (12 - 10)/8 = 2/8 = 1/4.

For 6/2 - 7/5: (6×5 - 2×7)/(2×5) = (30 - 14)/10 = 16/10 = 1.6.

Q6: Is the mixed fraction 5½ a rational number?

Answer: 5½ converted to an improper fraction is 11/2.

Since it is in p/q form (p=11, q=2) and q≠0, it is a rational number.

Yes, 5½ is a rational number.

Conclusion: Yes, 5½ is a rational number.

Q7: Are 0.50, 0.111.., 0.6, 2.789, and 4.234 rational numbers?

Answer: Yes. Terminating decimals (0.50, 0.6, 2.789, 4.234) and repeating decimals (0.111…) are all rational numbers.

Answer: Since these are either terminating or repeating, they can be expressed as p/q.

All these numbers are either terminating or non-terminating repeating decimals.

Hence, all of these are rational numbers.

Q8: Find four rational numbers between 4 and 5.

Answer: Convert 4 and 5 into equivalent fractions with a larger denominator, such as 8/2 and 10/2. Further expanding gives 16/4 and 20/4.

The rational numbers between 16/4 and 20/4 are 17/4, 18/4, and 19/4. Another is 9/2.

Any number between these values is a rational number.

Hence, four rational numbers between 4 and 5 are 9/2, 17/4, 18/4, and 19/4.

Q9: What is an equivalent rational number to 8/7?

Answer: Multiply the numerator and denominator by 4 to get 32/28.

Q10: Is 3.14 a rational number?

Answer: While 3.14 itself is a terminating decimal (314/100) and therefore rational, it is often used as an approximation for π, which is irrational. Given the specific number 3.14, it is rational.

Also Read:

Composite Numbers	Prime Numbers
Even Numbers	Odd Numbers