Rational Numbers Properties and Operations

Definition

A rational number is a real number that can be expressed as the ratio of two integers, i.e., it can be written in the form:

a/b

where a and b are integers, and b is non-zero.

Properties

• A rational number can be expressed as a finite decimal or a repeating decimal.
• The set of rational numbers is denoted by Q.
• Rational numbers can be added, subtracted, multiplied, and divided (except by zero) using the same rules as integers.
• The result of any arithmetic operation on rational numbers is always a rational number.

Examples

• 3/4, 22/7, -1/2 are all rational numbers.
• 0.5, 0.25, 0.333... (repeating) are also rational numbers.

Equivalent Ratios

• Two ratios a/b and c/d are equivalent if ad = bc.
• Equivalent ratios represent the same rational number.

Simplification of Rational Numbers

• A rational number can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• A rational number in its simplest form is said to be in lowest terms.

Operations on Rational Numbers

• Addition and subtraction:
  • To add or subtract rational numbers, find a common denominator and then add or subtract the numerators.
  • Example: 1/4 + 1/6 = (3+2)/12 = 5/12
• Multiplication:
  • To multiply rational numbers, multiply the numerators and denominators separately.
  • Example: 1/2 × 3/4 = (1×3)/(2×4) = 3/8
• Division:
  • To divide rational numbers, invert the second number and then multiply.
  • Example: 2/3 ÷ 3/4 = 2/3 × 4/3 = (2×4)/(3×3) = 8/9

Definition of Rational Numbers

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

Properties of Rational Numbers

• Can be expressed as a finite decimal or a repeating decimal.
• Denoted by the set Q.
• Can be added, subtracted, multiplied, and divided (except by zero) using the same rules as integers.
• The result of any arithmetic operation on rational numbers is always a rational number.

Examples of Rational Numbers

• 3/4, 22/7, -1/2 are all rational numbers.
• 0.5, 0.25, 0.333...(repeating) are also rational numbers.

Equivalent Ratios

• Two ratios a/b and c/d are equivalent if ad = bc.
• Equivalent ratios represent the same rational number.

Simplification of Rational Numbers

• Can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• A rational number in its simplest form is said to be in lowest terms.

Operations on Rational Numbers

Addition and Subtraction

• To add or subtract rational numbers, find a common denominator and then add or subtract the numerators.
• Example: 1/4 + 1/6 = (3+2)/12 = 5/12.

Multiplication

• To multiply rational numbers, multiply the numerators and denominators separately.
• Example: 1/2 × 3/4 = (1×3)/(2×4) = 3/8.

Division

• To divide rational numbers, invert the second number and then multiply.
• Example: 2/3 ÷ 3/4 = 2/3 × 4/3 = (2×4)/(3×3) = 8/9.