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  "body": "\n## Definition\n\nLet $A$ and $B$ be [[abelian groups|082GHNPJ53]], written additively.\nA [[function|0EPAT51DXQ]] $f \\colon A \\to B$ is *additive* if\n$$f(x + y) = f(x) + f(y) \\quad \\text{for all } x, y \\in A.$$\nThe most common case is $A = B = \\R$.\n\n## Remarks\n\nExample: $f \\colon \\R \\to \\R$, $f(x) = -5x$, is additive, since $-5(x + y) = -5x + (-5y)$.\n\nNon-example: $f(x) = x^2$ is not additive, since $f(1 + 1) = 4$ while $f(1) + f(1) = 2$.\n\nEvery additive function satisfies $f(0) = 0$, $f(-x) = -f(x)$ and $f(nx) = n f(x)$ for $n \\in \\Z$.\nBetween real vector spaces, an additive function satisfies $f(qx) = q f(x)$ for every rational $q$, so it is $\\Q$-linear.\nFor arbitrary groups, the same condition defines a group homomorphism.\n",
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