# Definition: Order Relation for Rational Numbers - Positive and Negative Rational Numbers

According to the definition of rational numbers, we can represent any rational number $$x$$ by two integers $$a,b$$, with $$b\neq 0$$, formally $$x=\frac ab$$.

Based on the order relation for integers, we have three cases of representing a rational number, called:

• positive rational number $x > 0,$ if and only if $x=\frac ab$ and $(a > 0\wedge b > 0)\vee(a < 0\wedge b < 0),$1
• zero $x=0,$ if and only if $x=\frac 0b,~b \neq 0,$
• negative rational number $x < 0,$ if and only if $x=\frac ab$ and $(a < 0\wedge b > 0)\vee(a > 0\wedge b < 0).$

Based on the definition of subtraction of rational numbers, we can define the order relation for rational numbers 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 $x-y < 0.$

Corollaries: 1
Definitions: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Proofs: 17 18 19
Propositions: 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

Github: #### Footnotes

1. The first $0$ in all three cases means the rational zero, the second $0$ means the integer zero