Algebraic Operations With Integers

The set Z of all integers, which this paper is all about, consists of all positive and negative integers as well as 0. Thus is the set given by

Z = { -4, -3, -2, -1, 0, 1, 2, 3, 4,... }  

While the set of all positive integers, denoted by N, is defined by

N = {1, 2, 3, 4, ...}. 

On Z, there are two basic binary operations, namely addition (denoted by +) and multiplication (denoted by ·), that satisfy some basic properties from which every other property for Z emerges.

1. The Commutativity property for addition and multiplication

a + b = b + a 

a · b = b · a

Example:

• 2+3=3+2=5 
• 2×3=3×2=6

2. Associativity property for addition and  multiplication

(a + b) + c = a + (b + c) 

(a · b) · c = a · (b · c)

Example:

• (2+3)+4 = 2+(3+4) = 9
• (2×3)×4 = 2×(3×4) = 24

3. The distributivity property of multiplication over addition

a · (b + c) = a · b + a · c

Example: 2(3+4) = 2×3 + 2×4 = 14

In the set Z there are "identity elements" for the two operations + and ·, and these are the elements 0 and 1 respectively, that satisfy the basic properties

a +0 = 0+ a = a 

a · 1=1 · a = a

for every a in Z.

The set Z allows additive inverses for its elements, in the sense that for every a in Z there exists another integer in Z, 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 1/a, (namely 1 itself in this case) such that

a · 1/a = 1.

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

1. $a - b$ is defined by $a + (-b)$, i.e. $a - b = a + (-b)$ for every $a$, $b$ in $Z$.
2. $ \frac{a}{b}$ is defined by the integer $c$ if and only if $a = b · c$.

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