Negative of Absolute Value
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Contents |
[edit] Theorem
Let
be a real number.
Let
be the absolute value of x.
Then
.
[edit] Corollary
-
;
-
.
[edit] Proof
Either
or x < 0.
- If
, then
.
- If x < 0, then
.
[edit] Proof of Corollary
- First we show that
.
Suppose
.
Then from the above,
and
.
So x < y and − x < y, and so x > − y from Ordering of Inverses.
It follows that − y < x < y.
Now suppose that
.
If
then − y < x < y and so
.
Otherwise, if
then either x = y or − x = y and hence the result.
- Next we show that
.
Suppose − y < x < y.
Then x < y and − x < y.
For all x,
or
.
Thus it follows that
.
Now suppose that
.
If − y < x < y then
and hence
.
Else, if either − y = x or x = y then
and hence the result.

