ch1.1.5-division-simple-quantities

Chapter V – Of the Division of Simple Quantities

Summary: Defines division as the inverse of multiplication, introduces dividend/divisor/quotient terminology, handles remainders, and establishes that sign rules for division mirror those for multiplication.

Sources: chapter-1.1.5

Last updated: 2026-04-24

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§45–48: Division Defined

To divide a number is to separate it into a given number of equal parts and determine the size of one part. Terminology:

• Dividend: the number to be divided.
• Divisor: the number of equal parts (or the number one divides by).
• Quotient: the size of one part.

Division is the inverse of multiplication: the quotient times the divisor reproduces the dividend (source: chapter-1.1.5).

§49–51: Algebraic Examples

$\frac{ab}{a}=b,\frac{ab}{b}=a,\frac{abc}{a}=bc,\frac{12mn}{3m}=4n$

Dividing any number by 1 returns the number; dividing by itself returns 1 (source: chapter-1.1.5).

§52–54: Remainders

When the dividend is not an exact multiple of the divisor, there is a remainder:

$24÷7:7\times 3=21,\text{remainder }3$

Proof check: $\text{Divisor}\times \text{Quotient}+\text{Remainder}=\text{Dividend}$.

When exact division is impossible among integers, fractions are needed (source: chapter-1.1.5).

§55–57: Sign Rules for Division

Same rules as multiplication:

Operation	Result
$+ab÷+a$	$+b$
$+ab÷-a$	$-b$
$-ab÷+a$	$-b$
$-ab÷-a$	$+b$

Like signs give +; unlike signs give −.

Examples: $18pq÷(-3p)=-6q$; $-54abc÷(-9b)=+6ac$ (source: chapter-1.1.5).

Related pages

• ch1.1.4-integers-and-factors
• ch1.1.6-properties-integers-divisors
• ch1.1.7-fractions-in-general
• positive-negative-numbers
• ch1.1.3-multiplication-simple-quantities