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The set of integers, denoted by the symbol $\mathbb{Z}$, is the set $\{\dots3,2,1,0,1,2,3,\dots\}$ consisting of the natural numbers and their negatives.
Mathematically, $\mathbb{Z}$ is defined to be the set of equivalence classes of pairs of natural numbers $\mathbb{N}\times\mathbb{N}$ under the equivalence relation $(a,b)\sim(c,d)$ if $a+d=b+c$.
Addition and multiplication of integers are defined as follows:

$(a,b)+(c,d):=(a+c,b+d)$

$(a,b)\cdot(c,d):=(ac+bd,ad+bc)$
Typically, the class of $(a,b)$ is denoted by symbol $n$ if $b\leq a$ (resp. $n$ if $a\leq b$), where $n$ is the unique natural number such that $a=b+n$ (resp. $a+n=b$). Under this notation, we recover the familiar representation of the integers as $\{\dots,3,2,1,0,1,2,3,\dots\}$. Here are some examples:

$0=$ equivalence class of $(0,0)=$ equivalence class of $(1,1)=\dots$

$1=$ equivalence class of $(1,0)=$ equivalence class of $(2,1)=\dots$

$1=$ equivalence class of $(0,1)=$ equivalence class of $(1,2)=\dots$
The set of integers $\mathbb{Z}$ under the addition and multiplication operations defined above form an integral domain. The integers admit the following ordering relation making $\mathbb{Z}$ into an ordered ring: $(a,b)\leq(c,d)$ in $\mathbb{Z}$ if $a+d\leq b+c$ in $\mathbb{N}$.
The ring of integers is also a Euclidean domain, with valuation given by the absolute value function.
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integers by matte ✓
positive, strictly positive, etc by matte ✘
valuation by pahio ✘
defines, synonym by pahio ✓
Comments
also
I'd add a note about how (a,b) maps to ab in the isomorphism.
Zahlen
I would mention why Z is used to denote the integers. It was first used by Germans, since they call the set "Zahlen", which means "numbers".
Re: Zahlen
Actually, the notation Z (along with similar use of Q,R and C) were introduced, or at least popularized, by Bourbaki.
Re: Zahlen (and S(n))
The symmetric group on n symbols is written S(n). I vaguely remember
hearing once that the S does not stand for symmetry but actually
comes from spiegel (mirror in German). Is this right?