ch1.1.3-multiplication-simple-quantities

Chapter III – Of the Multiplication of Simple Quantities

Summary: Introduces multiplication as repeated addition, establishes coefficient notation and commutativity, and derives the sign rules (like signs give +, unlike signs give −).

Sources: chapter-1.1.3

Last updated: 2026-04-24

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§23–26: Multiplication as Repeated Addition

Multiplication abbreviates repeated addition: $a+a=2\times a,a+a+a=3\times a,\text{etc.}$

When a letter is multiplied by a number, the number (the coefficient) is written before the letter: $20a$, $30b$. When both factors are letters, they are placed side by side: $ab$, $pq$, $apq$ (source: chapter-1.1.3).

§27–29: Commutativity

The order of factors is immaterial: $ab=ba$, since $3\times 4=4\times 3$. With numeric factors, the $\times$ sign or a dot is used to avoid ambiguity (e.g., $3\cdot 4=12$, not 34). Mixed expressions such as $5\cdot 7\cdot 8\cdot abcd=280abcd$ (source: chapter-1.1.3).

§30: Terminology

• Product: the result of multiplying two or more numbers.
• Factors: the individual numbers or letters being multiplied.

§31–34: Sign Rules for Multiplication

Factors	Product
$(+a)(+b)$	$+ab$
$(+a)(-b)$	$-ab$
$(-a)(+b)$	$-ab$
$(-a)(-b)$	$+ab$

Rule: Like signs give +; unlike signs give −.

The case $(-a)(-b)=+ab$ is proved by contradiction: $(-a)(+b)=-ab$, so $(-a)(-b)$ must produce the opposite sign, $+ab$ (source: chapter-1.1.3).

§35–36: Multiplying Products

To multiply products, collect all numeric coefficients and all letter factors: $5ab\times 3cd=15abcd,12pqr\times 7xy=84pqrxy$

(source: chapter-1.1.3)

Related pages

• positive-negative-numbers
• ch1.1.2-signs-plus-minus
• ch1.1.4-integers-and-factors
• ch1.1.5-division-simple-quantities