Real Numbers

Abstract

We have defined the ordering x ⩽ y on the set Q of rational numbers; we have seen that this ordering makes Q a linearly ordered set, and that it is compatible with the additive group structure of Q, i.e. for each z ∈ Q the relation x ⩽ y is equivalent to x + z ⩽ y + z (that is, the ordering is invariant under translations). We recall the notation (which is used in any linearly ordered group)

$$ {x^{ + }} = \sup \left( {x,0} \right) $$

,

$$ {x^{ - }} = \sup \left( { - x,0} \right) = {\left( { - x} \right)^{ + }} $$

,

$$ \left| x \right| = \sup \left( {x, - x} \right) $$

|x| is called the absolute value of x, and ew have

$$ x = {x^{ + }} - {x^{ - }},\quad \left| x \right| = {x^{ + }} + {x^{ - }} $$

and the triangle inequality

$$ \left| {x + y} \right| \leqslant \left| x \right| + \left| y \right| $$

(1)

together with the inequality

$$ \left| {\left| x \right| - \left| y \right|} \right| \leqslant \left| {x - y} \right| $$

(2)

which is an immediate consequence of (1); moreover

$$ \left| {{x^{ + }} - {y^{ + }} \leqslant \left| {x - y} \right|} \right| $$

(3)

.