= Multiplication
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Multiplication combines two factors into their <product of numbers>. For a <natural number> $n$, $nx$ is the <sum> of $n$ copies of $x$; with no copies the result is zero. This repeated-<addition> interpretation directly explains integer grouping, while <multiplication> of arbitrary <real numbers> also describes scaling by noninteger factors.
A <rectangle> of side <lengths> $3$ and $6$ has <area> $18$. Split the second side into <lengths> $2$ and $4$. The two <areas> are $6$ and $12$, giving the visible identity
$$
3(2+4)=3\cdot2+3\cdot4=18.
$$
Rotating the same <rectangle> interchanges the sides without changing its <area>, explaining $3\cdot6=6\cdot3$. These pictures illustrate <distributivity> and <commutativity> for ordinary positive <numbers>; <matrix multiplication> and other operations require their own rules.
\Image[/multiplication-area-partition.png]
{title=Partitioning a <rectangle> explains <distributivity>; partitioning a unit <square> gives an exact product of fractions}
For a fractional example, take $3/2$ of a quantity and then $2/3$ of that result: the combined scale is $(2/3)(3/2)=1$. The two <fractions> are <multiplicative inverses>. In the unit <square> on the right, the overlap of a strip occupying $3/4$ of the width with one occupying $2/3$ of the height contains six of twelve equal cells. Its <area> is $(3/4)(2/3)=1/2$. The product stays exact even though the <decimal expansion> of $2/3$ repeats. https://openstax.org/books/prealgebra-2e/pages/4-1-visualize-fractions[OpenStax, Prealgebra 2e, equal-part models of fractions]{ref}
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