Multiplication and Product¶

Definition¶

Multiplication combines two numbers $a$ and $b$ to produce the number $a\times b\text{,}$ also written $ab\text{,}$ called their product. Math I: Calculus takes the usual multiplication operations on $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$ as given and uses their standard arithmetic rules.

Interpretation¶

For natural numbers, multiplication may be understood as repeated addition: $3\times4=4+4+4$$4$. More generally, it describes scaling: multiplying by $2$ doubles a quantity, while multiplying by $-1$ reverses its sign.

On the number line, multiplying by $c$ stretches all distances from $0$ by the factor $c$ when $c>0$$0$; a negative factor also reflects every point through $0$. Two reflections cancel, which is the picture behind $(-1)(-1)=1$$1$. The Key Results force this rule: by absorbing zero and the distributive law,

so $(-1)(-1)=1$$1$. The reflection also explains why multiplying both sides of an inequality by a negative number reverses it: $2<3$$3$, but $-2>-3$$-3$.

Products of many equal factors are powers, and products of the consecutive numbers $1,2,\ldots,n$ are factorials.

Key Results¶

• Closure: if $a$ and $b$ belong to the same one of $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}\text{,}$ then $ab$ belongs to that set.
• Identity: $a\cdot1=1\cdot a=a\text{.}$
• Commutativity: $ab=ba\text{.}$
• Associativity: $(ab)c=a(bc)\text{.}$
• Distributivity: $a(b+c)=ab+ac\text{.}$
• Absorbing zero: $0a=a0=0\text{.}$

Examples¶

• $3\times 4=12\text{.}$
• $n\times 0=0$ and $n\times1=n$ for every number $n\text{.}$
• $3(4+5)=3\cdot4+3\cdot5=27\text{.}$

Prerequisites¶

• Addition and Sum

Related Concepts¶

• Division and Quotient
• Exponentiation and Power
• Reciprocal
• Factorial

Sources¶

• Math I: Calculus · Lecture 1

Symbols: integers, natural numbers, rational numbers, real numbers

Last updated: 20 October 2026 03:02