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## Definition: Order Relation for Integers - Positive and Negative Integers

According to the definition of integers, we can represent any integer $$x$$ by two natural numbers $$a,b$$, formally $$x=[a,b]$$. In order to avoid too many different cases, we can assume without loss of generality at least one of these natural numbers to be equal zero. There are three different possibilities to have two natural numbers, at least one being equal to zero: $$[a,0],[0,0]$$ or $$[0,a]$$ for some natural number $$a > 0$$.

Therefore, based on the order relation for natural numbers, we call an integer $x$

• positive integer $x > 0,$ if and only if $x=[a,0]$ and $a > 0,$1
• zero $x=0,$ if and only if $x=[0,0],$
• negative integer $x < 0,$ if and only if $x=[0,a]$ and $a > 0.$

for some natural number $$a > 0$$.

Based on the definition of subtraction of integers, we can define the order relation for integers as follows:

• $x > y$ if and only if $x – y > 0,$
• $x = y$ if and only if $x – y = 0,$
• $x < y$ if and only if $x – y < 0.$

1 In all three cases, the first $0$ is the integer zero and the $0$ in the brackets means the natural zero.

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