Rational, Irrational and Real Numbers
Complete guide to rational operations, decimal representations, and real number systems
Rational Numbers
To obtain a set of numbers that includes every result of a division, we need to expand the integers. Rational numbers are used for this purpose. These are formed from fractions of integers. The integers are also included in the rational numbers.
A rational number is any number that can be expressed as a fraction \(\displaystyle \frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). The set of rational numbers is denoted by ℚ. Rational numbers include all integers, fractions, and terminating or repeating decimals.
Rational numbers allow the execution of all four basic operations (addition, subtraction, multiplication, and division) without leaving the set of rational numbers (except division by zero).
Operations with Rational Numbers
Addition of Fractions
When adding fractions, the denominators of the two addends must be brought to a common denominator:
\(\displaystyle \frac{a}{b} + \frac{m}{n} =\) \(\displaystyle \frac{a \cdot n}{b \cdot n} + \frac{m \cdot b}{n \cdot b} = \frac{an + mb}{bn}\)
Example of Addition
Subtraction of Fractions
Subtraction of fractions can be performed using the addition rule with negative numbers:
\(\displaystyle \frac{a}{b} - \frac{m}{n} = \frac{a}{b} + \left(-\frac{m}{n}\right) = \frac{an - mb}{bn}\)
Example of Subtraction
Multiplication of Fractions
In multiplication, the numerator is multiplied by the numerator and the denominator by the denominator:
\(\displaystyle \frac{a}{b} \cdot \frac{m}{n} = \frac{a \cdot m}{b \cdot n} = \frac{am}{bn}\)
Example of Multiplication
Division of Fractions
Division by a fraction is equal to multiplying by its reciprocal (the fraction inverted):
\(\displaystyle \frac{a}{b} \div \frac{m}{n} = \frac{a}{b} \cdot \frac{n}{m} = \frac{a \cdot n}{b \cdot m} = \frac{an}{bm}\)
Example of Division
Decimal Representation of Rational Numbers
Rational numbers can be written as decimal fractions. Decimal fractions have a whole part before the decimal point and decimal places after the decimal point. Rational numbers in decimal notation have the property that they either have a finite number of decimal places or a repeating pattern in the decimal places.
Types of Decimal Representations
Decimals that end after a finite number of places. These occur when the denominator (in lowest terms) has only factors of 2 and 5.
Decimals where digit sequences repeat infinitely. The repeating part is indicated by a bar above the repeating digits. These occur when the denominator has prime factors other than 2 and 5.
Examples of Decimal Representations
Terminating Decimals (Finite)
- \(\displaystyle \frac{1}{2} = 0.5\)
- \(\displaystyle \frac{1}{4} = 0.25\)
- \(\displaystyle \frac{3}{8} = 0.375\)
- \(\displaystyle \frac{888}{100} = 8.88\)
Repeating Decimals (Infinite)
- \(\displaystyle \frac{1}{3} = 0.\overline{3} = 0.333...\)
- \(\displaystyle \frac{5}{12} = 0.41\overline{6} = 0.41666...\)
- \(\displaystyle \frac{7}{1111} = 0.\overline{0063} = 0.00630063...\)
- \(\displaystyle \frac{22}{7} = 3.\overline{142857} \approx 3.142857...\)
Every rational number has either a terminating or repeating decimal representation. Conversely, every terminating or repeating decimal represents a rational number.
Irrational Numbers
Irrational numbers are real numbers that cannot be expressed as fractions of integers. They have an infinite number of non-repeating decimal places.
An irrational number is a real number that cannot be written as \(\displaystyle \frac{a}{b}\) where \(a\) and \(b\) are integers with \(b \neq 0\). Irrational numbers have non-terminating, non-repeating decimal representations.
Common Irrational Numbers
Famous Irrational Numbers
- \(\displaystyle \pi = 3.14159265358979...\) (Pi - ratio of circumference to diameter)
- \(\displaystyle e = 2.71828182845904...\) (Euler's number - base of natural logarithm)
- \(\displaystyle \sqrt{2} = 1.41421356237309...\) (square root of 2)
- \(\displaystyle \phi = \frac{1+\sqrt{5}}{2} = 1.61803398874989...\) (Golden ratio)
- \(\displaystyle \sqrt{3} = 1.73205080756887...\) (square root of 3)
If \(n\) is not a perfect square, then \(\displaystyle \sqrt{n}\) is irrational. Similarly, \(\displaystyle \sqrt[3]{n}\) is irrational if \(n\) is not a perfect cube.
Real Numbers
The real numbers encompass both rational and irrational numbers. They form the complete set of all numbers that can be plotted on a number line. The set is denoted by ℝ.
A real number is any number that can be represented as a point on the number line. Real numbers include all rational numbers (integers and fractions) and all irrational numbers.
\(\displaystyle \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\)
Natural ⊂ Integers ⊂ Rationals ⊂ Reals
Structure of Real Numbers
| Category | Description | Examples |
|---|---|---|
| Rational (ℚ) | Can be expressed as a/b where a, b are integers | \(\frac{1}{2}, 5, 0.75, 0.\overline{3}\) |
| Irrational | Cannot be expressed as a/b; non-repeating decimals | \(\pi, e, \sqrt{2}, \phi\) |
| Algebraic | Roots of polynomial equations with rational coefficients | \(\sqrt{2}, \sqrt[3]{5}, \frac{\sqrt{7}}{2}\) |
| Transcendental | Not algebraic; cannot be roots of polynomial equations | \(\pi, e, \ln(2)\) |
The real numbers form a complete ordered field. This means every non-empty set of real numbers that is bounded above has a least upper bound (supremum), a crucial property for calculus and analysis.
Summary: All Number Systems
| Number Set | Symbol | Description | Closure Properties |
|---|---|---|---|
| Natural | ℕ | 0, 1, 2, 3, ... | +, × only |
| Integer | ℤ | ..., -1, 0, 1, ... | +, −, × only |
| Rational | ℚ | a/b where a,b ∈ ℤ, b≠0 | +, −, ×, ÷ (b≠0) |
| Irrational | ℝ \ ℚ | Non-repeating decimals | Not closed under basic operations |
| Real | ℝ | ℚ ∪ (ℝ \ ℚ) | +, −, ×, ÷ (divisor≠0) |
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