Rational Numbers: Definition, Properties, and Operations

Definition

• A rational number is a real number that can be expressed as the ratio of two integers, i.e., a fraction.
• It can be written in the form a/b, where a and b are integers, and b is non-zero.

Properties

• Rational numbers can be added, subtracted, multiplied, and divided (except by zero).
• The result of these operations is always a rational number.
• Rational numbers can be expressed in decimal form, which may be finite or infinite.

Examples

• 3/4, 22/7, 1/2 are all rational numbers.
• Integers, such as 5, can be written as rational numbers (5/1).
• Finite decimals, such as 0.5, are also rational numbers.

Irrational Numbers vs. Rational Numbers

• Irrational numbers cannot be expressed as a finite decimal or fraction.
• Examples of irrational numbers include π, e, and the square root of 2.

Operations with Rational Numbers

• Addition and Subtraction: To add or subtract rational numbers, find a common denominator and then add or subtract the numerators.
• Multiplication: To multiply rational numbers, multiply the numerators and denominators separately.
• Division: To divide rational numbers, invert the second number and then multiply.

Equivalent Ratios

• Two ratios a/b and c/d are equivalent if a*d = b*c.
• Equivalent ratios have the same value, but may not have the same form.

Simplifying Rational Numbers

• A rational number can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD).
• The resulting simplified form is an equivalent ratio.