e (Euler’s number) Swiss mathematician LEONHARD
EULER (1707–83) introduced a number, today denoted e,
that plays a fundamental role in studies of compound
INTEREST, TRIGONOMETRY, LOGARITHMS, and CALCULUS,
and that unites these disparate fields. (EULER’S FORMULA,
for instance, illustrates this.) The number e has approximate value 2.718281828459045 … and can be defined
in any of the following different ways:
1. The number e is the limit value of the expression
raised to the nth power, as n increases
indefinitely:
2. If L(a) denotes the area under the curve y = 1/x
above the interval [1,a], then e is the location on the
x-axis for which L(e) = 1.
3. If f(x) is a function that equals its own DERIVATIVE,
that is,
f(x) = f(x), then f(x) is an EXPONENTIAL
FUNCTION with base value e: f(x) = e
x
.
4. e is the value of the infinite sum 1 +
+
+
+ ….
Definition 1 is linked to the problem of computing
compound interest. As we show below, definition 2
defines the natural logarithm, and definition 3 arises
from studies of natural growth and decay, and consequently the consideration of EXPONENTIAL FUNCTIONs.
The fourth definition arises from the study of TAYLOR
SERIES. One proves that all four definitions are equivalent as follows:
First consider the curve y = 1/x. It has the remarkable
property that rectangles touching the curve and just under
it have the same area if the endpoints of the rectangles are
in the same ratio r. For example, in the first diagram on
the opposite page, the rectangles above the intervals [a,ra]
and [b,rb] each have area
. By taking narrower and
narrower rectangles, all in the same ratio r, it then follows that the area under the curve above any two intervals of the form [a,ra] and [b,rb] are equal.
Following definition 2, let L(x) denote the area
under this from position 1 to position x. (If x is less
than 1, deem the area negative.) Notice that L(1) = 0.
Also, set e to be the location on the x-axis where the
area under the curve is 1: L(e) = 1.
Notice that the area under the curve from 1 to
position ab, L(ab), is the sum of the areas under the
curve above the intervals [1,a] and [a,ab]. The first area
is L(a) and the second, by the property above, equals
L(b). We thus have:
L(ab) = L(a) + L(b)
This shows that L is a function that converts multiplication into addition, which is enough to prove that it is
the LOGARITHMIC FUNCTION base e. We have L(x) =
log e (x). This function is called the natural logarithm
function and is usually written ln(x).
r – 1
––– r
1
–
3!
1
–
2!
1
–
1!
d
––
dx
e
n
n
n
=
+
→∞
lim
1
1
1
1
+
n
152
E
EULER (1707–83) introduced a number, today denoted e,
that plays a fundamental role in studies of compound
INTEREST, TRIGONOMETRY, LOGARITHMS, and CALCULUS,
and that unites these disparate fields. (EULER’S FORMULA,
for instance, illustrates this.) The number e has approximate value 2.718281828459045 … and can be defined
in any of the following different ways:
1. The number e is the limit value of the expression
raised to the nth power, as n increases
indefinitely:
2. If L(a) denotes the area under the curve y = 1/x
above the interval [1,a], then e is the location on the
x-axis for which L(e) = 1.
3. If f(x) is a function that equals its own DERIVATIVE,
that is,
f(x) = f(x), then f(x) is an EXPONENTIAL
FUNCTION with base value e: f(x) = e
x
.
4. e is the value of the infinite sum 1 +
+
+
+ ….
Definition 1 is linked to the problem of computing
compound interest. As we show below, definition 2
defines the natural logarithm, and definition 3 arises
from studies of natural growth and decay, and consequently the consideration of EXPONENTIAL FUNCTIONs.
The fourth definition arises from the study of TAYLOR
SERIES. One proves that all four definitions are equivalent as follows:
First consider the curve y = 1/x. It has the remarkable
property that rectangles touching the curve and just under
it have the same area if the endpoints of the rectangles are
in the same ratio r. For example, in the first diagram on
the opposite page, the rectangles above the intervals [a,ra]
and [b,rb] each have area
. By taking narrower and
narrower rectangles, all in the same ratio r, it then follows that the area under the curve above any two intervals of the form [a,ra] and [b,rb] are equal.
Following definition 2, let L(x) denote the area
under this from position 1 to position x. (If x is less
than 1, deem the area negative.) Notice that L(1) = 0.
Also, set e to be the location on the x-axis where the
area under the curve is 1: L(e) = 1.
Notice that the area under the curve from 1 to
position ab, L(ab), is the sum of the areas under the
curve above the intervals [1,a] and [a,ab]. The first area
is L(a) and the second, by the property above, equals
L(b). We thus have:
L(ab) = L(a) + L(b)
This shows that L is a function that converts multiplication into addition, which is enough to prove that it is
the LOGARITHMIC FUNCTION base e. We have L(x) =
log e (x). This function is called the natural logarithm
function and is usually written ln(x).
r – 1
––– r
1
–
3!
1
–
2!
1
–
1!
d
––
dx
e
n
n
n
=
+
→∞
lim
1
1
1
1
+
n
152
E