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integer [2013/09/03 00:39] nikolaj |
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| - | ===== 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 $-(b-a)$. | ||
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| - | 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 | ||
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| - | * $ 0 \equiv [(0,0)] = [(1,1)] = \dots = [(k,k)] $ | ||
| - | |||
| - | * $ 1 \equiv [(1,0)] = [(2,1)] = \dots = [(k+1,k)] $ | ||
| - | |||
| - | * $ -1 \equiv [(0,1)] = [(1,2)] = \dots = [(k,k+1)] $ | ||
| - | |||
| - | * $ 2 \equiv [(2,0)] = [(3,1)] = \dots = [(k+2,k)] $ | ||
| - | |||
| - | * $ -2 \equiv [(0,2)] = [(1,3)] = \dots = [(k,k+2)] $ | ||
| - | |||
| - | * $ 3 \equiv [(0,3)] = \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 | ||
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| - | $-[\langle a,b\rangle]=[\langle b,a\rangle]$ | ||
| - | |||
| - | === Reference === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Integer|Integer]] | ||
| - | ==== Context ==== | ||
| - | === Subset of === | ||
| - | [[Quotient set]] | ||
| - | === Refinement of === | ||
| - | [[Rational number]] | ||
| - | === Requirements === | ||
| - | [[Arithmetic structure of natural numbers]] | ||
