multiplication

#LaTeX

Multiplication is one of the four fundamental operations of arithmetic (along with addition, subtraction, and division).
The result of a multiplication operation is called a product.
We denote multiplication using various symbols:

Name	Symbol	$L A T E X$	Example
Cross	$\times$	`\times`	$a \times b = c$
Dot	$\cdot$	`\cdot`	$a \cdot b = c$
Asterisk	$*$	`*`	$a * b = c$
juxtaposition			$a b = c$

The multiplication of whole numbers can be thought of as repeated addition:
In a product between a multiplicand and its multiplier, both called factors of the product, the multiplicand is repeatedly added to itself a total number of times indicated by the multiplier:

a \times b = \underset{a times}{\underset{⏟}{b + \dots + b}}

Which factor is the multiplier or the multiplicand may be ambiguous or depend on context. One of multiplication's main properties is commutativity, i.e. that 3 copies of 4 makes the same product as 4 copies of 3; therefore the designation of multiplier/multiplicand is not strictly necessary.

Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers), or more generally the area of a rectangle, with side lengths of the factors.

For example, we might represent the expression $3 \times 4 = 12$ using the following diagram:

\documentclass[tikz,margin={2cm 0cm}]{standalone}
\usetikzlibrary{decorations.pathreplacing}
%
\begin{document}
\begin{tikzpicture}
	\def\length{4}
	\def\width{3}
	\foreach \x in {1,...,\length} {
		\foreach \y in {1,...,\width} {
			\node[fill,circle] at (\x,\y) {};
		}
	}
	\draw[decorate,decoration={brace, raise=0.5ex}] (0.5,0.5) -- ++(0,\width) node[midway,left=1ex] {$\width$};
	\draw[decorate,decoration={brace, mirror, raise=0.5ex}] (0.5,0.5) -- ++(\length,0) node[midway,below=1ex] {$\length$};
	\draw[xshift=0.5cm,yshift=0.5cm,help lines] (0,0) grid ++(\length,\width);
\end{tikzpicture}
\end{document}

The inverse operation of multiplication is division ( $\div$ ). For example, since 4 times 3 equals 12, 12 divided by 3 equals 4. In general, multiplying by 3 then dividing by 3 yields the original number. Most generally, dividing a number by itself equals 1 (the multiplicative identity).