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 integer [2013/09/03 00:40]nikolaj integer [2014/03/21 11:11] Line 1: Line 1: - ===== Integer ===== - ==== Definition ==== - | @#FFBB00: $\mathbb Z \equiv \mathbb N\times\mathbb N\ /\ \{\langle \langle a,​b\rangle,​\langle m,​n\rangle\rangle\ |\ a+n = b+m )\}$ | - with $a,b,n,m\in \mathbb N$. - - ==== Discussion ==== - For $a \ge b$, we denote $\langle a,b\rangle$ by $a-b$. The structure of the non-negative integers is then that of the natural numbers. - - For $a < b$, we have $(b-a)>​0$ and we denote $\langle a,b\rangle$ by $-(b-a)$. - - So if $[\langle a,​b\rangle]$ is the equivalence class of $\langle a,b\rangle$ with respect to the equivalence relation $\{\langle \langle a,​b\rangle,​\langle m,​n\rangle\rangle\ |\ a+n = b+m )\}$, we have - - * $0 \equiv [\langle0,​0\rangle] = [\langle1,​1\rangle] = \dots = [\langlek,​k\rangle]$ - - * $1 \equiv [\langle1,​0\rangle] = [\langle2,​1\rangle] = \dots = [\langlek+1,​k\rangle]$ - - * $-1 \equiv [\langle0,​1\rangle] = [\langle1,​2\rangle] = \dots = [\langlek,​k+1\rangle]$ - - * $2 \equiv [\langle2,​0\rangle] = [\langle3,​1\rangle] = \dots = [\langlek+2,​k\rangle]$ - - * $-2 \equiv [\langle0,​2\rangle] = [\langle1,​3\rangle] = \dots = [\langlek,​k+2\rangle]$ - - * $3 \equiv [\langle0,​3\rangle] = \dots$ - - where $k$ is any natural number. - - === Theorems === - The integer $-[\langle a,​b\rangle]$ is the additive inverse of $[\langle a,​b\rangle]$ and can be computed as - - $-[\langle a,​b\rangle]=[\langle b,​a\rangle]$ - - === Reference === - Wikipedia: [[http://​en.wikipedia.org/​wiki/​Integer|Integer]] - ==== Context ==== - === Subset of === - [[Quotient set]] - === Refinement of === - [[Rational number]] - === Requirements === - [[Arithmetic structure of natural numbers]]
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