Rational Numbers: Definition, Properties, and Examples

What is a Rational Number?

A rational number is a type of real number that can be expressed as the quotient p/q, where p and q are integers, and q is not equal to zero. In other words, a rational number is any number that can be represented as a fraction. For example, 2/3 is a rational number because it can be written in fraction form with integer values for the numerator (2) and denominator (3).

Properties of Rational Numbers

1. Terminating Decimals: In decimal form, rational numbers are either terminating or repeating decimals. For example, 1.5 is a terminating decimal, while 1.23232... is a repeating decimal.

2. Addition and Subtraction: Rational numbers follow the same rules for addition and subtraction as integers. For example, (1/2) + (2/3) = 7/6.

3. Multiplication and Division: Rational numbers also follow the same rules for multiplication and division as integers. For example, (1/2) * (2/3) = 1/3.

Examples of Rational Numbers

Some examples of rational numbers include:

• 1/2
• 2/3
• 5/8
• 9/4
• 10/3

Finding Rational Numbers

To find a rational number between two given numbers, you can convert both numbers to fractions, find the least common multiple of their denominators, and then add the numerators and divide by the least common multiple. For example, to find the rational number between 2/3 and 5/6, you would first convert both numbers to fractions:

• 2/3 = 2/3
• 5/6 = 5/6

The least common multiple of 3 and 6 is 6, so you would add the numerators and divide by the least common multiple:

• (2 + 5) / 6 = 7/6

Rational Numbers vs. Irrational Numbers

A rational number is a type of real number, while an irrational number is a real number that cannot be expressed as the quotient of two integers. For example, sqrt(2) and pi are irrational numbers because they cannot be written as fractions.

Rational Number Examples

Let's look at some examples of rational numbers and their properties:

Example 1: Find the sum of 1/2 and 2/3.

• 1/2 + 2/3 = 7/6

Example 2: Find the difference between 5/8 and 3/4.

• 5/8 - 3/4 = 1/12

Example 3: Find the product of 2/3 and 3/4.

• 2/3 * 3/4 = 1/2

Example 4: Find the quotient of 9/4 and 2/3.

• 9/4 / 2/3 = 9/8

Example 5: Find the difference between 10/3 and 5/8.

• 10/3 - 5/8 = 15/8

Conclusion

Rational numbers are a fundamental concept in mathematics, representing any number that can be expressed as a fraction. They play a crucial role in arithmetic operations and have various applications in real-world problems. Understanding the properties and examples of rational numbers is essential for mastering the fundamentals of mathematics.