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Mathematics LibreTexts

1.2: Algebraic Operations with Integers

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    28626
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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 \(c\).