Wednesday, July 11, 2018

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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