Mathematics.

number systems

Fractions

Foundations20 minDifficulty1 out of 10

You should know: integers

Overview

A fraction represents a part of a whole, written as a numerator over a denominator (a/b), where the denominator tells you how many equal parts the whole is split into and the numerator tells you how many of those parts you have. Fractions are the everyday notation for rational numbers, and mastering their arithmetic — common denominators, simplifying, multiplying, dividing — is the practical foundation the rest of algebra builds on.

Intuition

Cut a pizza into 8 equal slices; if you eat 3, you've eaten 3/8 of the pizza. The denominator (8) sets the size of each piece; the numerator (3) counts how many pieces you have. Two fractions can look different but represent the same amount — 1/2 and 4/8 are the same pizza-fraction — which is the idea behind equivalent fractions.

Formal Definition

Definition

A fraction is written and manipulated as:

ab,b≠0\frac{a}{b}, \quad b \neq 0

a is the numerator, b is the denominator

Definition
ab=kakb(k≠0)\frac{a}{b} = \frac{ka}{kb} \quad (k \neq 0)

Equivalent fractions: scaling numerator and denominator by the same factor

ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}

Addition via a common denominator

ab×cd=acbd,ab÷cd=ab×dc\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}, \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Multiplication is straightforward; division flips the second fraction

Notation

NotationMeaning
ab\frac{a}{b}a is the numerator (parts you have), b is the denominator (parts per whole)

Properties

Simplifying (lowest terms)

ab is in lowest terms when gcd⁡(a,b)=1\frac{a}{b} \text{ is in lowest terms when } \gcd(a,b) = 1

Reciprocal

ab×ba=1(a,b≠0)\frac{a}{b} \times \frac{b}{a} = 1 \quad (a, b \neq 0)

Applications

Measurement systems (inches, cooking recipes) rely on fraction arithmetic for precise scaling.

Worked Examples

  1. Divide both by their GCD, 6.

    1218=12÷618÷6=23\frac{12}{18} = \frac{12\div6}{18\div6} = \frac{2}{3}

Answer: 2/3

Practice Problems

Difficulty 2/10

Compute (2/5) ÷ (3/10).

Common Mistakes

Common Mistake

Adding fractions by adding numerators and denominators separately, e.g. 1/2 + 1/3 = 2/5.

Fractions must share a common denominator before adding. 1/2 + 1/3 = 3/6 + 2/6 = 5/6, not 2/5.

Summary

  • A fraction a/b represents a of b equal parts of a whole.
  • Equivalent fractions scale numerator and denominator by the same nonzero factor.
  • Adding/subtracting requires a common denominator; multiplying and dividing do not.
  • Simplify to lowest terms by dividing both parts by their GCD.

References

  1. WebsiteWikipedia — Fraction