Chapter X – Of the Multiplication and Division of Fractions
Summary: Establishes rules for multiplying and dividing fractions — multiply numerators and denominators respectively for multiplication; invert the divisor and multiply for division — and confirms that sign rules extend unchanged from integers.
Sources: chapter-1.1.10
Last updated: 2026-04-24
§101–102: Multiplying a Fraction by an Integer
Multiply the numerator (or divide the denominator when possible — it is shorter):
Special case: (source: chapter-1.1.10).
§103–105: Dividing a Fraction by an Integer
Divide the numerator, or equivalently multiply the denominator:
Example: (source: chapter-1.1.10).
§106: Multiplying Two Fractions
Multiply numerators together and denominators together:
Examples: ; (source: chapter-1.1.10).
§107–110: Dividing Two Fractions — Inversion Rule
Method: invert the divisor and multiply:
Derivation: reduce to common denominator : (source: chapter-1.1.10).
Examples:
Dividing by is the same as multiplying by 2; dividing by is the same as multiplying by 3.
§111: Division by Zero
As the divisor shrinks toward 0, the quotient grows without bound: dividing 1 by gives . Hence . See infinity (source: chapter-1.1.10).
§112–113: Notable Results
- Any fraction divided by itself equals 1: .
- ” of ” means .
§114: Sign Rules
The same rules as for integers apply to fractions:
(source: chapter-1.1.10)