Fractions¶
Fractions as Numbers¶
A fraction \(\frac{a}{b}\) with \(a \in \mathbb{Z}\) and \(b \in \mathbb{Z} \setminus \{0\}\) represents the result of the division \(a \div b\). Here \(a\) is the numerator and \(b\) is the denominator.
Two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equal if \(a \cdot d = b \cdot c\).
Example. \(\frac{2}{3} = \frac{4}{6}\), since \(2 \cdot 6 = 3 \cdot 4 = 12\).
Simplifying and Expanding¶
Simplifying¶
A fraction is simplified by dividing both numerator and denominator by the same factor \(k \neq 0\):
A fraction is fully reduced when \(\gcd(a, b) = 1\).
Example. \(\frac{12}{18} = \frac{12/6}{18/6} = \frac{2}{3}\).
Expanding¶
A fraction is expanded by multiplying both numerator and denominator by the same factor \(k \neq 0\):
Example. \(\frac{2}{3} = \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15}\).
Addition and Subtraction¶
Fractions with the same denominator are added directly:
For different denominators, a common denominator is formed first:
Example. \(\frac{2}{3} + \frac{1}{4} = \frac{2 \cdot 4 + 1 \cdot 3}{3 \cdot 4} = \frac{8 + 3}{12} = \frac{11}{12}\).
Subtraction works analogously: \(\frac{a}{b} - \frac{c}{d} = \frac{a \cdot d - c \cdot b}{b \cdot d}\).
Multiplication¶
Fractions are multiplied by multiplying numerator by numerator and denominator by denominator:
Example. \(\frac{2}{3} \cdot \frac{5}{7} = \frac{10}{21}\).
Division¶
Division by a fraction equals multiplication by its reciprocal:
Requirement: \(c \neq 0\).
Example. \(\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \cdot \frac{5}{2} = \frac{15}{8}\).
Summary¶
| Operation | Formula |
|---|---|
| Simplify | \(\frac{a}{b} = \frac{a/k}{b/k}\) |
| Expand | \(\frac{a}{b} = \frac{ak}{bk}\) |
| Addition | \(\frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd}\) |
| Multiplication | \(\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\) |
| Division | \(\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}\) |
References¶
- Courant, Richard; Robbins, Herbert: What Is Mathematics? Oxford University Press, 2nd edition, 1996. Chapter 1.