= Addition
{title2=$+$}
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Addition combines two <numbers> into their <sum>. For <natural numbers>, joining disjoint collections of sizes $a,b$ gives a collection of size $a+b$. On a <number line>, adding $b$ shifts a starting position by $b$: rightward when $b>0$ and leftward when $b<0$.
Starting at $2$ and adding $3$ reaches $5$. Starting there and adding $-3$ returns to $2$, because $3+(-3)=0$. <Additive inverses> turn this return move into an algebraic rule. With <real numbers>, <addition> is <commutative> and <associative>: exchanging two shifts or grouping several shifts differently leaves the final position unchanged.
\Image[/arithmetic-number-line.png]
{title=<Addition> shifts positions on a <number line>, while a negative multiplication factor changes both scale and direction}
The bottom panel applies $x\mapsto-3x/2$. Input $2$ becomes $-3$, and input $-2$ becomes $3$. <Multiplication> by a negative <real number> rescales distances from zero and reverses direction, giving a <reflection in mathematics> as well as a scale change. This is a different operation from shifting every input by one fixed amount.
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