
# 1.2: Algebraic Operations with Integers


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On $$\ZZ$$, the set of integers, there are two basic binary operations, namely addition (denoted by $$+$$) and multiplication (denoted by $$\cdot$$), which satisfy the following well known properties:

1. Commutativity of addition and multiplication \begin{aligned} \forall a,b\in\ZZ:\quad a+b&=b+a\\ a\cdot b&=b\cdot a\end{aligned}
2. Associativity of addition and multiplication \begin{aligned} \forall a,b,c\in\ZZ:\quad (a+b)+c&=a+(b+c)\\ (a\cdot b)\cdot c&= a\cdot (b\cdot c)\end{aligned}
3. Distributivity of multiplication over addition \begin{aligned} \forall a,b,c\in\ZZ:\quad a\cdot (b+c)=a\cdot b+a\cdot c.\end{aligned}

In the set $$\ZZ$$ there are identity elements for the two operations $$+$$ and $$\cdot$$, and these are the elements $$0$$ and $$1$$ respectively, that satisfy the basic properties \begin{aligned} \forall a\in\ZZ:\quad a + 0 &= 0+a = a\\ a\cdot 1 &= 1\cdot a = a\ .\end{aligned}

The set $$\ZZ$$ allows additive inverses for its elements, in the sense that for every $$a\in\ZZ$$ there exists another integer in $$\ZZ$$, denoted by $$-a$$, such that $a+(-a)=0.$ While for multiplication, only the integer $$1$$ has a multiplicative inverse in the sense that 1 is the only integer $$a$$ such that there exists another integer, denoted by $$a^{-1}$$ or by $$\dfrac{1}{a}$$, (namely $$1$$ itself in this case) such that $a\cdot a^{-1}=1.$

From the operations of addition and multiplication one can define two other operations on $$\ZZ$$, namely subtraction (denoted by $$-$$) and division (denoted by $$/$$). Subtraction is a binary operation on $$\ZZ$$, i.e., defined for any two integers in $$\ZZ$$, while division is not a binary operation and thus is defined only for some specific pairs of integers in $$\ZZ$$. Subtraction and division are defined as follows:

1. $$\forall a,b\in\ZZ$$, $$a-b$$ is defined to be $$a+(-b)$$
2. Given $$a,b\in\ZZ$$, where $$b\neq 0$$, if $$\exists c\in\ZZ$$ such that $$a=b\cdot c$$ then $$\dfrac{a}{b}$$ is defined to be $$c$$.