= Arithmetic
{wiki}
<Arithmetic> studies operations on <numbers>: combining quantities by <addition>, finding differences by <subtraction>, scaling through <multiplication>, and reversing a nonzero scale through <division>. Its usefulness goes beyond producing an answer. A calculation can express how resources are shared, how a measurement changes with scale, or why two different descriptions of a quantity agree.
The allowed <numbers> matter. Start with the <natural numbers> $0,1,2,\ldots$. Counting collections gives $2+3=5$, but removing three from two calls for the <integer> $-1$. Sharing three units equally between two recipients gives the <rational number> $3/2$. These operations motivate larger number systems: <integers> supply <additive inverses>, and <rational numbers> supply <division> by nonzero <integers>. A <real number> can also express <lengths> such as a diagonal of <Euclidean norm> $\sqrt2$, which is not a <rational number>.
The beauty is that bookkeeping, geometry, and algebra can check one another. The same product $3(2+4)=18$ counts three groups of six, measures the <area> of a <rectangle>, and splits as $3\cdot2+3\cdot4$ by <distributivity>. A <number line> makes negative <integers> visible as positions; a <fraction> records an exact share even when its <decimal representation> does not terminate. An <arithmetic> rule becomes easier to remember when these interpretations fit together.
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