Next: 5.5 Brouncker's Formula For
Up: 5. Area
Previous: 5.3 Monotonic Functions
  Index
5.57
Notation (, .)
Let
be a bounded function from the interval
to
. We will denote the area of
by
. Thus
We will sometimes write
instead of
. Thus, for example
We will also write
and
for
and
respectively.
5.58
Lemma.
5.2
Let
and
be real numbers
such that
and
. Then
Proof:
Let
be a partition of , and let
be the partition of obtained by
multiplying the points of by .
Then
and
We know that
Hence by (5.59) and (5.60) we have
for every partition of . It follows from this and the last statement
of theorem 5.40 that
5.61
Exercise.
A
From lemma
5.58 we see that
whenever
, and
. Use this result to show that for
and
|
(5.62) |
5.63
Definition (.)
We will define a function
by
By exercise 5.61A we
have
|
(5.64) |
In this section we will extend the domain of to all of
in such a way
that (5.64) holds for all
.
5.65
Theorem.
Let
be real numbers such that
, and let
be a bounded function from
to
. Then
|
(5.66) |
Proof: We want to show
Since
and the sets and are almost
disjoint, this conclusion follows from our assumption about additivity
of area for almost disjoint sets.
I now want to extend the definition of to cases where may be less
than . I want equation (5.66) to continue to hold in all cases.
If in (5.66), we get
i.e.,
Thus we make the following definition:
5.67
Definition.
Let
be real numbers with
and let
be a bounded
function
from
to
. Then we define
5.68
Theorem.
Let
be real numbers and let
be a bounded non-negative real valued function whose
domain contains an interval containing
, and
. Then
Proof: We need to consider the six possible orderings for and . If
we already know the result. Suppose . Then
and hence
, i.e.,
. The remaining four cases are left as an exercise.
5.69
Exercise.
Prove the remaining four cases of theorem
5.68.
5.70
Definition (Logarithm.)
If
is any positive number, we define the
logarithm of by
5.71
Theorem (Properties of Logarithms.) For all
and all
we have
Proof: Let
. From lemma 5.58 we know that if
then
|
(5.74) |
If we get
so equation (5.74) holds in all cases. Let be arbitrary elements
in
. Then
Also
so
and it follows from this that
Hence
5.75
Lemma.
For all
,
.
Proof: The proof is by induction on . For the lemma is clear.
Suppose
now that the lemma holds for some
, i.e., suppose that
.
Then
The lemma now follows by induction.
If
then
and
Thus equation (5.72) holds whenever
. If
and
, then
so
Thus (5.72) holds for all .
5.76
Theorem.
Let
and
be numbers such that
. Then
Proof: By lemma 5.58
Logarithms were first introduced by John
Napier (1550-1632) in 1614. Napier made up the word logarithm from
Greek roots meaning ratio number, and he spent about twenty
years making tables of them.
As far as I have been able to find out, the earliest use of for
logarithms was by Irving Stringham in
1893[15, vol 2, page 107]. The notation
is probably more
common among mathematicians than , but since calculators almost
always calculate our function with a key called ``ln'', and calculate
something else with a key called ``log'', I have adopted the ``ln''
notation. (Napier did not use any abbreviation for logarithm.)
Logarithms were seen as an important
computational device for reducing multiplications to additions. The first
explicit notice of the fact that logarithms are the same as areas of hyperbolic
segments was made in 1649 by Alfons Anton de Sarasa (1618-1667), and this
observation increased interest in the problem of calculating areas of
hyperbolic segments.
5.77
Entertainment (Calculate .)
Using any computer or calculator, compute
accurate
to 10 decimal places. You should not make use of any special
functions,
e.g., it is not fair to use the ``ln" key on your calculator.
There are better polygonal
approximations to
than the ones we have
discussed.
The graph of the logarithm function is shown below.
We know that
and
it is clear that is strictly increasing.
If , then
From the fact that
for all
, it is clear that
takes on arbitrarily large positive and negative values, but the
function increases
very slowly.
Let
be the regular partition of into three subintervals.
Then
Now
and
i.e.,
|
(5.78) |
There is a unique number such that . The uniqueness
is clear because is strictly increasing.
The existence of such a number was taken as obvious before the nineteenth
century.
Later we will introduce the intermediate value property
which will allow us to prove that such a number exists.
For the time being, we will
behave like eighteenth century mathematicians,
and just assert that such a number exists.
5.79
Definition (.)
We
denote the unique number in
whose logarithm is
by
.
5.80
Exercise.
A
Prove that
. (We already know
.)
5.81
Entertainment (Calculate .)
Using any computing power you have, calculate
as accurately as you
can,
e.g., as a start, find the first digit after the decimal point.
Next: 5.5 Brouncker's Formula For
Up: 5. Area
Previous: 5.3 Monotonic Functions
  Index
Ray Mayer
2007-09-07