We define the absolute value x of a rational number x to be

We define the absolute value |x| of a rational number x to be :

|x| := x if x is positive.

|x| := x if x is negative.

|x| := 0 if x = 0.

The quantity |x y| is called the distance between x and y and is sometimes denoted

d(x, y), thus d(x, y) := |x y| where x y := x + (y).

Assuming that all the basic properties of order are true on Q the set of all rationals, show that :

i) |x + y| |x| + |y| (triangle inequality).

ii) y x y iff y |x|.

iii) d(x, z) d(x, y) + d(y, z).

i) |x| = max{x,-x}

|y| = max{y,-y}

So x+y <= |x| + |y|

And also -x + (-y) < = |x| + |y|

So max(x+y, -x+(-y)) <= |x| + |y|

hence |x+y| < = |x| + |y|

ii) y>= |x|

if and only if

y>= x and y >= -x

which is true if and only if

x<= y and x >= -y

iii) |x-z| = |x-y + y - z|

and now use i)

|x-y + y - z| < = |x-y| + |y-z|