Page 19 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Notes
if x = ( , )a b , y = ( , )c d , we have x > y.
)
if x – y = ( , ) (a b c , ) (ad bc ,bd > 0,
d
whence we find
(ad – bc) bd > 0, i.e., ad > bc, b > 0, d > 0.
Similarly, x < y if ad < bc, b > 0, d > 0.

and x = y if ad = bc.
The Trichotomy Law holds for rational numbers, as usual, i.e., given two rational numbers x, y
either x > y or x = y, or x < y.

Also the order relation is compatible with addition and multiplication. For,
x > y x + z > y + z
and x > y, z > 0 xz > yz, x, y, z Q.

Representation of Rational Numbers

A rational number of the form ( ,1)a can be identified with the integer a Z, and written simply
as a.
Further, since

b
a
( ,1) ( ,1) ( ,1) (1, ) ( ,1,1 ) ( , )
b
a
a
b
b
a
we obtain a method of representing the rational number (a, b) by means of two integers.
We have ( , ) = ( ,1) ( ,1)
a
b
b
a
= a b or a|b, b 0.
With this notation, the sum and product of two rational numbers assume the usual meaning
attached to them, viz.,
a c ad bc
b d bd
a c ac
and , b 0, d 0
b d bd
a c
Also ad bc , b 0, d 0 .
b d
The system of rational numbers Q provides an extension of the system of integral Z, such that
(i) Q Z, (ii) addition and multiplication of two integers in Q have the same meanings as they
have in Z and (iii) the subtraction and division operations are defined for any two numbers in Q,
except for division by zero.

In addition to the properties described above the system of rational numbers possesses certain
distinctive characteristics which distinguish it from the system of integers or natural numbers.
One of these is the property of denseness (the density property), which is described by saying
that between any two distinct rational numbers there lies another rational number.

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