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6.1
Definition (Absolute values.)
Recall that
if
![$x$](img35.gif)
is a real number, then the
absolute value of ![$x$](img35.gif)
, denoted by
![$\vert x\vert$](img699.gif)
, is
defined by
We will assume the following properties of absolute value, that follow easily
from
the definition:
For all real numbers
with
For all real numbers
, and all
and
![\begin{displaymath}
(\vert x\vert\leq a)\hspace{1ex}\Longleftrightarrow\hspace{1ex}(-a\leq x\leq a).
\end{displaymath}](img1433.gif) |
(6.2) |
We also have
and
6.3
Theorem.
Let
and let
. Then for all
we have
and
Equivalently, we can say that
and
Proof: I will prove only the first
statement. I have
6.4
Definition (Distance.)
The
distance between two
real numbers
![$x$](img35.gif)
and
![$y$](img1044.gif)
is defined by
Theorem 6.3 says that the set of numbers whose distance
from
is smaller than
is the interval
. Geometrically
this is clear from the picture.
I remember the theorem by keeping the picture
in mind.
Proof
For all
and
in R we have
and
so
Hence (Cf. (6.2))
6.7
Exercise.
Can you prove that for all
![$(x,y)\in\mbox{{\bf R}}^2\Big( \vert x-y\vert\leq
\vert x\vert-\vert y\vert\Big)$](img1451.gif)
?
Can you prove that for all
?
Remark:
Let
be real numbers with
and
.
Then
This result should be clear from the picture. We can give an analytic
proof as follows.
6.8
Examples.
Let
Then a number
![$x$](img35.gif)
is in
![$A$](img6.gif)
if and only if the distance from
![$x$](img35.gif)
to
![$2$](img19.gif)
is smaller than
![$5$](img1123.gif)
, and
![$x$](img35.gif)
is in
![$B$](img145.gif)
if and only if the distance
from
![$x$](img35.gif)
to
![$2$](img19.gif)
is greater than
![$5$](img1123.gif)
. I can see by inspection that
and
Let
If
![$x \in \mbox{{\bf R}}\setminus \{-1\}$](img1463.gif)
, then
![$x$](img35.gif)
is in
![$C$](img37.gif)
if and only if
![$\vert x-1\vert < \vert x+1\vert$](img1464.gif)
, i.e. if and only if
![$x$](img35.gif)
is closer to
![$1$](img481.gif)
than to
![$-1$](img1465.gif)
.
I can see by inspection that the point
equidistant from
![$-1$](img1465.gif)
and
![$1$](img481.gif)
is
![$0$](img48.gif)
, and that the numbers that are
closer to
![$1$](img481.gif)
than to
![$-1$](img1465.gif)
are the positive numbers, so
![$C = (0,\infty)$](img1467.gif)
.
I can also do this analytically, (but in practice I wouldn't)
as follows. Since the alsolute values are all non-negative
6.9
Exercise.
Express
each of the four sets below as an interval or a union
of intervals. (You can do this problem by inspection.)
6.10
Exercise.
Sketch the graphs of the functions from
![$\mbox{{\bf R}}$](img153.gif)
to
![$\mbox{{\bf R}}$](img153.gif)
defined by the
following equations:
(No explanations are expected for this problem.)
6.11
Exercise.
Let
![$f_1,\cdots ,f_7$](img1471.gif)
be the functions described in the
previous exercise. By looking at the graphs, express each of the
following six sets in terms of intervals.
Let
![$S_7=\{x\in\mbox{{\bf R}}\colon f_7(x)<{1\over 2}\}$](img1473.gif)
. Represent
![$S_7$](img1474.gif)
graphically on a number line.
Remark: The notation
for absolute value of
was introduced by
Weierstrass in 1841 [15][Vol 2,page 123]. It was first introduced
in
connection with complex numbers. It is surprising that analysis advanced so
far
without introducing a special notation for this very important function.
Next: 6.2 Approximation
Up: 6. Limits of Sequences
Previous: 6. Limits of Sequences
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Ray Mayer
2007-09-07