Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
integer [2013/09/03 00:39] nikolaj |
integer [2014/03/21 11:11] (current) |
||
---|---|---|---|
Line 1: | Line 1: | ||
===== Integer ===== | ===== Integer ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | @#FFBB00: $ \mathbb Z \equiv \mathbb N\times\mathbb N\ /\ \{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a+n = b+m )\} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ \mathbb Z \equiv \mathbb N\times\mathbb N\ /\ \{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a+n = b+m )\} $ | |
with $a,b,n,m\in \mathbb N$. | with $a,b,n,m\in \mathbb N$. | ||
Line 12: | Line 12: | ||
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 | 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 [(0,0)] = [(1,1)] = \dots = [(k,k)] $ | + | * $ 0 \equiv [\langle0,0\rangle] = [\langle1,1\rangle] = \dots = [\langle k,k\rangle] $ |
- | * $ 1 \equiv [(1,0)] = [(2,1)] = \dots = [(k+1,k)] $ | + | * $ 1 \equiv [\langle1,0\rangle] = [\langle2,1\rangle] = \dots = [\langle k+1,k\rangle] $ |
- | * $ -1 \equiv [(0,1)] = [(1,2)] = \dots = [(k,k+1)] $ | + | * $ -1 \equiv [\langle0,1\rangle] = [\langle1,2\rangle] = \dots = [\langle k,k+1\rangle] $ |
- | * $ 2 \equiv [(2,0)] = [(3,1)] = \dots = [(k+2,k)] $ | + | * $ 2 \equiv [\langle2,0\rangle] = [\langle3,1\rangle] = \dots = [\langle k+2,k\rangle] $ |
- | * $ -2 \equiv [(0,2)] = [(1,3)] = \dots = [(k,k+2)] $ | + | * $ -2 \equiv [\langle0,2\rangle] = [\langle1,3\rangle] = \dots = [\langle k,k+2\rangle] $ |
- | * $ 3 \equiv [(0,3)] = \dots $ | + | * $ 3 \equiv [\langle0,3\rangle] = \dots $ |
where $k$ is any natural number. | where $k$ is any natural number. | ||
=== Theorems === | === Theorems === | ||
- | $-[\langle a,b\rangle]$ is the additive inverse of $[\langle a,b\rangle]$ and can be computed as | + | The integer $-[\langle a,b\rangle]$ is the additive inverse of $[\langle a,b\rangle]$ and can be computed as |
$-[\langle a,b\rangle]=[\langle b,a\rangle]$ | $-[\langle a,b\rangle]=[\langle b,a\rangle]$ | ||
Line 33: | Line 33: | ||
=== Reference === | === Reference === | ||
Wikipedia: [[http://en.wikipedia.org/wiki/Integer|Integer]] | Wikipedia: [[http://en.wikipedia.org/wiki/Integer|Integer]] | ||
- | ==== Context ==== | + | ==== Parents ==== |
=== Subset of === | === Subset of === | ||
[[Quotient set]] | [[Quotient set]] | ||
=== Refinement of === | === Refinement of === | ||
[[Rational number]] | [[Rational number]] | ||
- | === Requirements === | + | === Context === |
[[Arithmetic structure of natural numbers]] | [[Arithmetic structure of natural numbers]] |