Rational Numbers
2 chapters
Rational numbers are numbers that can be written as a fraction p/q, where p and q are integers, and q ≠ 0. This set includes integers, decimal numbers with a finite length (terminating decimals), and infinite repeating decimals. Rational numbers are denoted by ℚ.
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REPRESENTATION OF RATIONAL NUMBERS
Every rational number can be written in various ways. For example:
- 1/2 = 0.5
- 3/4 = 0.75
- 1/3 = 0.3333… (or 0.(3))
Rational numbers can be divided into positive, negative, and zero. All integers are rational because they can be expressed as n/1 (e.g., 5 = 5/1).
BASIC ARITHMETIC OPERATIONS
Rational numbers allow for all four basic arithmetic operations:
- ADDITION: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2
- SUBTRACTION: 5/4 – 3/4 = 2/4 = 1/2
- MULTIPLICATION: (2/3) × (3/4) = 6/12 = 1/2
- DIVISION: (3/5) ÷ (2/7) = (3/5) × (7/2) = 21/10 = 2.1
PROPERTIES OF RATIONAL NUMBERS
- DENSITY: Between any two distinct rational numbers, there is always another rational number.
- CLOSURE: The set ℚ is closed under addition, subtraction, multiplication, and division (except for division by 0).
- RECIPROCAL (MULTIPLICATIVE INVERSE): Every rational number p/q (except 0) has a multiplicative inverse q/p.
CONCLUSION
Rational numbers are a key part of number sets, as they allow for the precise expression of parts of a whole. Their properties and calculations with them are the basis for understanding decimal numbers, fractions, and algebra.