What Are Rational Numbers?
Rational numbers are numbers that can be written as the ratio of two integers, where the denominator is not zero. Mathematically, a rational number is expressed as:
r = a/b where a, b ∈ ℤ, b ≠ 0The set of rational numbers is denoted by the symbol ℚ.
Examples of Rational Numbers
- 5 (can be written as 5/1)
- −3/4
- 0.75 (which equals 3/4)
- 0.333… (a repeating decimal, equals 1/3)
Types of Rational Numbers
Integers
All integers are rational because they can be written with a denominator of 1. For example, −7 = −7/1.
Fractions
Any number expressed as a fraction of integers is rational, such as 2/3 or −11/8.
Terminating Decimals
Decimals that end after a finite number of digits, like 0.5 or 2.125, can be converted to fractions.
Repeating Decimals
Decimals with a repeating pattern, like 0.666… or 1.272727…, are also rational because they can be converted into fractions.
Properties of Rational Numbers
Closure Property
Rational numbers are closed under addition, subtraction, multiplication, and division (excluding division by zero).
Commutative Property
a + b = b + a and a × b = b × a
Associative Property
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
Distributive Property
a × (b + c) = a × b + a × c
Identity Elements
- Additive Identity: 0 (a + 0 = a)
- Multiplicative Identity: 1 (a × 1 = a)
Inverse Elements
- Additive Inverse: For any rational number a, there exists −a such that a + (−a) = 0.
- Multiplicative Inverse: For any a ≠ 0, there exists 1/a such that a × (1/a) = 1.
Rational Numbers on the Number Line
Rational numbers can be precisely placed on the number line. Every point corresponds to a rational number. Between any two rational numbers, another rational number can always be found—this is known as the density property.
Operations with Rational Numbers
Addition and Subtraction
To add or subtract rational numbers, convert them to a common denominator:
(a/b) + (c/d) = (a·d + b·c) / (b·d)Multiplication
Multiply the numerators and denominators:
(a/b) × (c/d) = (a·c) / (b·d)Division
Multiply by the reciprocal of the divisor:
(a/b) ÷ (c/d) = (a/b) × (d/c)Converting Between Forms
Fractions to Decimals
Divide the numerator by the denominator. If the division terminates or repeats, the decimal is rational.
Decimals to Fractions
- Terminating decimals: Count the decimal places and write as a fraction. Example: 0.75 = 75/100 = 3/4.
- Repeating decimals: Use algebraic techniques to convert. Example: Let x = 0.333… then 10x = 3.333…, subtracting gives 9x = 3, so x = 1/3.
Rational vs. Irrational Numbers
| Feature | Rational Numbers | Irrational Numbers |
|---|---|---|
| Definition | Can be written as a/b | Cannot be written as a/b |
| Decimal Representation | Terminating or repeating | Non-terminating, non-repeating |
| Examples | 1/2, −4, 0.333… | √2, π, e |
Real-Life Applications of Rational Numbers
Everyday Life
- Splitting bills
- Measuring ingredients
- Comparing discounts
Science
- Precision in measurements
- Scientific ratios
Finance
- Interest rates
- Currency conversions
- Budgeting
Common Misconceptions
Are All Decimals Irrational?
No. Repeating and terminating decimals are rational.
Is Zero Rational?
Yes. 0 = 0/1, which fits the definition.
Can We Divide by Zero?
No. Division by zero is undefined.
Practice Problems
- Convert 0.625 to a fraction.
- Determine if −3 is a rational number.
- Add: 2/5 + 3/10
- Multiply: 4/7 × 2/3
- Is √5 rational or irrational?
Summary
Rational numbers are essential in both academic math and everyday life. They help us count, measure, divide, and compare. From understanding how money works to making accurate scientific measurements, mastering rational numbers empowers logical thinking and problem-solving.