1. Bethany Bouchard
An Analysis on Euler’s constant e and its
functions in Number Theory, Complex Analysis,
and Transcendental Number Theory
“The difference… between reading a mathematical demonstration, and originating one
wholly or partly, is very great. It may be compared to the difference between the pleasure
experienced, and interest aroused, when in the one case a traveller is passively conducted
through the roads of a novel and unexplored country, and in the other case he discovers
the roads for himself with the assistance of a map.”
George Shoobridge Carr “A Synopsis of Elementary Results in Pure and
Applied Mathematics”
Word Count: 2,271
2. ABSTRACT
In Calculus, we learn about the special cases of the base of the natural logarithm, e.
However, many characteristics of e that are not frequently explored provide insight to
various areas of mathematics. In this essay, I will investigate the question: What is the
definition of e and to what extent has it furthered areas of Mathematics such as Number
Theory, Complex Analysis, and Transcendental Number Theory?
To answer this question, I first approached the several properties of e. These
attributes of Euler’s constant included the transcendence of e, the belonging of e to
irrational numbers, and the Taylor Series of e. Then, I examined these characteristics to
determine what place they had in other areas of mathematics, more specifically, number
theory, complex analysis, and transcendental number theory. There were two major
mathematical discoveries that were a result of the utilization of this constant, Euler’s
Identity and the transcendence of pi which disproved circle squaring.
After the relationships were discussed and proven, it became clearer to see the
significance of e in different areas of mathematics. Without its existence, it would have
been difficult, if not impossible, to ascertain knowledge in seemingly unconnected areas
of mathematics. The understanding of e allowed for a completion to the problem of circle
squaring and allowed for the uncovering of a formula that unites complex analysis and
trigonometry and some of the most significant numbers in mathematics history.
Word Count: 234
5. Introduction
In the education of calculus, we come across the unique function which is its own
derivative and anti derivative, x
e . There are many unexplored properties of the base of
the natural logarithm that are not taught or explored at a high school that display the
individuality of e. Many of these properties can be applied to various areas of
mathematics such as basic number theory, complex analysis, and transcendental number
theory. Through the discovery of various aspects of e, sometimes denoted as Euler’s
constant, great moments in mathematics have also been discovered such as the
universally known Euler’s identity, which is magnificent in its relationships between
commonly known numbers.
Hence, I shall investigate these characteristics of e by answering the question:
“What is the definition of e and to what extent has it furthered areas of Mathematics such
as Number Theory, Complex Analysis, and Transcendental Number Theory?”
The Origins of e
The mathematical constant e was first utilized indirectly by John Napier in 1618;
however Jacob Bernoulli was one of the first to apply it to compound interest in 1968.
While examining compound interest in money, Bernoulli determined that if he deposited
a dollar at 100% APR: annually becomes $2.00, biannually becomes $2.25, quarterly
becomes $2.44, monthly becomes $2.61, weekly becomes $2.69, daily becomes $2.71,
1
6. and continuously becomes $2.718 (Bentley 2008, p. 118). He then used this knowledge to
find the limit of
n
n
)
1
1( + as n approaches infinity.
The constant was accredited the occasionally used name “Euler’s constant” based
upon Leonard Euler’s application of the letter “e” to represent the value 2.71828…. In
Euler’s Introductio in Analysin Infinitorum, he defined e as having the value
...
!4
1
!3
1
!2
1
!1
1
1 ++++ . Given that a > 1
j
x
w
jwx
w
x
j
kwaw
=
=
=
+=1
Where w is an infinitely small number but not 0, k is a constant dependent on a, and x is a
real number.
Euler defined w as being an infinitely small number, but not 0, k as being a
constant dependant on a, and x as any real number. Note, the concept of the limit of an
equation can be observed through Euler’s definition of w. This means that as w nears 0,
then j nears infinity.
jjjwx
j
kx
kwaa )1()1( +=+==∴
Then, by using Newton’s binomial theorem where
k
n
k
n
x
k
n
x )
!
()1(
0=
∑=+
...)(
!3
)2)(1(
)(
!2
)1(
)(
!1
1 32
j
kxjjj
j
kxjj
j
kxj
a x −−
+
−
++=
2
7. ∴When w→0 then j→∞
!
)(
...
!3
)(
!2
)(
!1
1
...
321
1
32
n
kxkxkxkx
a
j
nj
j
j
j
j
j
j
n
x
+++++=
−
==
−
=
−
=
−
=
When x = 1, then
!
...
!4!3!2!1
1
432
n
kkkkk
a
n
++++++=
When k = 1 then e has the value of a
!
1
...
!4
1
!3
1
!2
1
!1
1
1
n
e ++++++=
(Burton 1999, p. 486-487)
This process ends with the definition of e according to Euler, which is similar to
Bernoulli’s explanation of e, where the value is equivalent to
n
n
)
1
1lim( + , as both
Bernoulli and Euler describe. The interesting part of Euler’s analysis is figuring out how
he discovered e, rather than defining it. While searching through Euler’s works,
specifically Introductio in Analysin Infinitorum, and other’s analyses on Euler’s work, it
became clear that there was no document of Euler’s discovery of e, only his individual
characterization and application of the base of the natural logarithm (Burton 1999, p.
486-487).
The Properties of e
3
8. The base of the natural logarithm, e, has several non trivial properties that set it apart
from many common constants used in mathematics. Some of these properties include e,
as a function, being its own derivative and therefore its own anti derivative shown
through the proof here.
h
e
ee
dx
d
h
ee
e
dx
d
h
eee
e
dx
d
h
ee
e
dx
d
h
h
xx
hx
h
x
xhx
h
x
hhx
h
x
1
lim
)1(
lim
)(
lim
lim
0
0
0
0
−
=
−
=
−×
=
−
=
→
→
→
+
→
Then, by knowing the definition of e, where
n
n n
e )
1
1(lim +=
∞→
we can replace
n
1
with h
h
h
he
1
0
)1(lim +=
→
Therefore, for all small values of h
e ≐ h
h
1
)1( +
he
he
h
hhh
+=
+=
1
])1[(][
1
We can then input this in the application of the fundamental theorem of calculus
4
10. This graph depicts the limit of
x
x
xf )
1
1()( += , which is the base of the natural
logarithm. As seen on the graph, the line continues infinitely along the horizontal
asymptote that is the exact value of e. In the base of the natural logarithm graph, the
vertical asymptote is the y-axis, displaying how the function slowly approaches infinity
as x nears 0. Unique properties that can be observed from this graph are the fact that the
natural logarithm of 1 is equal to 0. The constant was proven to be irrational be Leonard
Euler through his proof that e is equivalent to a series that continues infinitely and in his
proof of Euler’s formula which will be further discussed later.
!
1
...
!4
1
!3
1
!2
1
!1
1
1
n
e ++++++=
In another function, e
e
1
is the absolute maximum of the function x
xy
1
= at x = e.
x
xy
1
=
6
11. Since we can not apply the power rule, I will take the natural log of each side.
)ln1(
)ln1(
)ln(
)
1
ln(
)ln(
ln
11
ln
1
ln
2
1
2
1
22
1
12
1
xx
dx
dy
xxx
dx
dy
xxxx
dx
dy
x
xxxy
dx
dy
xx
dx
d
y
dx
dy
x
xdx
d
ydx
dy
x
x
y
x
x
x
−=
−=
−=
+−=
=
=
=
−
−
−−
−−
−
Now that we have the first derivative, we can find the root of the equation in order to find
the extrema.
ex
x
x
xx x
=
=
=−
−=
−
ln1
0ln1
)ln1(0
2
1
By using the first derivative test, we see that e is the global maximum of this
function, one of the most unique global extrema. Note that e is irrational, meaning that it
7
12. can not be represented by a fraction and continues after the decimal place infinitely. By
using this feature of the function x
xxf
1
)( = , we can calculate an accurate representation
of the value of e.
Euler’s Identity
In Euler’s publication of Introductio in Analysin Infinitorum in 1748, a proof of the
formula that relates cosine, sine, e, and i, is explained. However, it is debatable whether
Euler was the original attributer to the formula as there is proof that mathematicians
Roger Cotes and Johann Bernoulli were familiar with this equation. In A Concise History
of Mathematics by Dirk J. Smirk, it is stated that Johann Bernoulli initially discovered
this formula. Euler’s formula has been applied in the discovery of De Moivre’s formula,
nxinxxix n
sincos)sin(cos +=+ (Carr 2013, p. 174). This formula is only applicable
when x is a complex number and n is an integer, and is used in the understanding of
hyperbolic functions. Euler’s identity is an application of Euler’s formula in which x is
replaced with pi.
There are several ways that this formula has been proven. Euler had proven the
formula by defining sine and cosine with constant e and the imaginary number i, which,
at the time, was written by Euler as 2
1
)1(− . Another proof comes from the Taylor series
of ex
, sin x, cos x when ix is substituted for x. By using the power series expansion of
8
13. 1−±= vz
these functions, the convergence of the series provides evidence of Euler’s formula, also
known as ix
excis =)( . Note that cis(x) translate into “cosine plus isine” (Weisstein).
1. Euler’s Proof
In Euler’s Introductio, the introduction of the formula xixeix
sincos += begins with
Euler’s definition of cos v, sin v, and ez
.
iz
ii
ii
i
z
e
i
v
i
v
v
i
v
i
v
v
)1(
12
)
1
1()
1
1(
sin
2
)
1
1()
1
1(
cos
+=
−
−
−−
−
+
=
−
−+
−
+
=
We can see the similarities in the right hand sides of each equation, as defined by
Euler. Also it is important to note Euler’s interchangeable notation of 1− for i, the
imaginary number.
12
sin
2
cos
11
11
−
−
=
+
=
−−−+
−−−+
vv
vv
ee
v
ee
v
Here, Euler substitutes z for the right sides using 1−v , according to whether it
is a positive or negative difference and makes sin v and cos v in terms of ez
.
vve
vve
v
v
sin1cos
sin1cos
1
1
−−=
−+=
−−
−+
(Struik 1987, pp. 122-123)
9
14. We can see the end result of the formula, in both positive and negative
exponential form.
This diagram demonstrates the understanding of Euler’s formula using complex
numbers. Complex equations with complex numbers involve the combination of ℝ and i.
Hence, in the diagram, there is the horizontal axis labelled Real Numbers, and the vertical
axis labelled Imaginary Numbers. This diagram is expressed in exponential form which
means that the complex number is represented by ℝ
ix
e where the angle x must be in
radians (Storr 2014).
In the diagram, it is stated that xixeix
sincos += , where the blue line indicates the
complex number that results from the vertical and horizontal axis. As stated by Euler, as
previously discussed,
2
1sin
2
cos
11
11
−−−+
−−−+
−
=−
+
=
vv
vv
ee
v
ee
v
10
15. (Struik 1987, pp. 122-123)
where the sum of these is v
e 1−
. One thing that comes from the application of the
diagram is the various results that come from adding or subtracting the vertical and
horizontal component. We can see that a rearrangement of Euler’s formula can produce
four different results.
2. Taylor Series Proof
Another proof of Euler’s formula involves examining the Taylor series of each
component in the formula. A Taylor series is an infinitely long “series expansion of a
function” (Weisstein). The elements of Euler’s formula each have their own Taylor
series, which, I have found, end up revealing the relationship between the trigonometric
functions and the base of the natural logarithm.
...
!6!4!2
1cos
...
!7!5!3
sin
...
!4!3!2
1
642
753
432
xxx
x
xxx
xx
xxx
xex
−+−=
−+−=
++++=
Already, there is evidence of similarities between cos x and sin x. The
trigonometric function cos x is an even function meaning that the series only involves
even powers, and the function sin x is an odd function meaning that the series only
involves odd powers.
11
16. )
!7!5!3
(...)
!6!4!2
1(
)
!7!5!3
(...)
!6!4!2
1(
...
!5!4!3!2
1
...
!4
)(
!3
)(
!2
)(
1
753642
753642
5432
432
xxx
xi
xxx
e
ixixix
ix
xxx
e
ixxixx
ixe
ixixix
ixe
ix
ix
ix
ix
−+−+−+−=
−+−+−+−=
++−−+=
++++=
when i is applied into the series of ex
, two separate series are formed where i is not
present in one, and is multiplied to other. Through this, it becomes evident how ix
e is
composed of both cos x and sin x.
Euler’s identity, with the substitution of pi for x, connects some of most important
numbers in all of mathematics, arguably. It also shows the relationship between complex
analysis and trigonometry which becomes applicable in fields such as electrical
engineering where waves become a combination of sine and cosine curves to produce a
3-D graph like that of Euler’s formula.
Transcendence Theory
Transcendental numbers are defined as being not algebraic, not a root of any non-zero
polynomial (ax + b) with coefficients that can be represented as a quotient, or fraction
(Baker). This means that number can not be a zero of any equation to any exponential
degree. The most common transcendental numbers that are known to us and proven are e,
π , π
e , and 2
2 . The proof of the transcendence of pi involves the knowledge that e
is transcendental. From the proof of the transcendence of pi in 1882 by Lindemann comes
12
17. the end to a long unproven problem of squaring the circle which attempts to have
identical areas between a square and a circle (Weisstein).
Proof of the Transcendence of e
Charles Hermite was able to prove the transcendence of e in 1873 through working
through the assumption that e is algebraic and attempting to prove that. He began with the
integral ∫
−
x
t
dttfe
0
)( and multiplied it by ex
.
∫∫ ′+−= −−
x
txx
x
tx
dttfeexffedttfee
00
)()()0()(
In this instance, Hermite replaces f(t) with f’(t) and if continued with f’’(t), f’’’(t)
and so on, then parts of the equation on the right hand side will be able to cancel out.
Now, since proving the transcendence involves the usage of a polynomial, then we know
that f must be a polynomial. When it is a polynomial, )(xF can be used to replace the
functions ∑
∞
=0
)(
)(
j
j
xf in the previous equation after cancellations to get a new equation.
)()0(
0
)( xFFxe
x
dttftexe −=∫
−
With the assumption that e is algebraic, there exists a polynomial equation p(t)
with the coefficient )0( 0 ≠cc j and exponential degree n that is greater than or equal to 1
13
18. when p(e) = 0. This produces the series j
n
j
c
0=
∑ . Then, Hermite multiplied both sides of
the previous equation by the series and replaced x with j.
∫
∫
−
==
==
−
=
∑−=∑
∑−∑=∑
j
tj
j
n
j
j
n
j
j
n
j
j
j
n
j
j
tj
j
n
j
dttfeecjFc
jFcecFdttfeec
0
00
00
0
0
)()(
)()0()(
Hermite then chose a polynomial to replace f(t) where he chose
))...(2)(1()(
)(
)!1(
1
)( 1
nttttgwhere
tgt
p
tf pp
−−−=
−
= −
This particular choice was chosen to create an imbalance in the equation because
the left side is an integer that is not 0, and the right side is small. Also, there is the
freedom to make p any large prime number, and A is the absolute maximum value of
tg(t).
pj
j
n
j
j
tj
j
n
j
Aec
p
dttfeec )(
)!1(
1
)(
0
0
0 =
−
=
∑
−
≤∑ ∫
We check that )(
0
jFcj
n
j=
∑ is a non-zero integer as stated before and examine f(t) at t
= 0,1,2 and so on and get the same result that
pppjj
tgtinnttofcoefficien
p
j
nf )()(
)!1(
!
)( 1)( −−
−×
−
=
14
19. Therefore, )(
0
jFcj
n
j=
∑ is an integer but not a multiple of this expression, and since p
is prime, the expression is not valid and therefore e is transcendental. (A.F..
Transcendence of e.)
Application of the Transcendence of e
The transcendence of e may seem trivial in the grand scheme of mathematics, however,
the knowledge of this property of e has proven to be useful in proving the transcendence
of pi. By using Euler’s formula, in which pi is a degree of e when multiplied by i, the
imaginary number, we can use Ferdinand von Lindemann’s approach from 1882 (Mayer,
2006).
This discovery that pi is transcendental allows proof to circle squaring. This
problem consisted of the idea that in Euclidean geometry, a square could contain the
exact same area as a circle. Although there were mathematicians who recognized close
approximations, for circle squaring to be proven true, there needs to be a segment equal
to r×π . However, knowing that pi is transcendental means that this can not be the case
and thus, in Euclidean space, a circle can not be squared (Drexel, 1998).
15
20. Conclusion
In conclusion, I believe the discovery of e and many of its properties have proven
beneficial to the understanding and development of other areas of mathematics such as
number theory, analysis, and transcendence theory. Specifically, e has been applied to
areas such as electrical engineering where the usage of Euler’s formula comes into play,
and has also helped solve one of the three greatest problems in Classical geometry
(Drexel, 1998). The properties of e allows various areas of mathematics to come together
and form what has been deemed one of the most beautiful equations of mathematics
which unites universally known numbers and functions: e, i, pi, sine, and cosine.
However, there were some aspects of e that were unresolved in this
examination of the base of the natural logarithm such as its purpose in De Moivre’s
theorem and how that application influences the insight on hyperbolic functions. Also,
there remains the problem of how exactly e was discovered and how it came to be by
Euler, which could debatably be the most interesting part of e.
16
21. WORKS CITED
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the world. New York: Firefly Books Ltd.
Burton, D.M. (1999). The history of mathematics: An introduction. The McGraw-Hill
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Carr, G.S. (2013). A synopsis of elementary results in Pure and Applied Mathematics.
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Desmos Inc. (2011). The Desmos graphing calculator[Graph]. Caroline Hall. Retrieved
from https://www.desmos.com/calculator
Drexel University. (1998). Math forum: Squaring the circle. Retrieved from
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F., A. (n.d.). Transcendence of e. Retrieved from
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Mayer, S. (2006). The transcendence of pi. Retrieved from
sixthform.info/maths/files/pitrans.pdf
Storr, W. (2014). Complex numbers and phasors. Basic Electronics Tutorials Site.
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numbers.html
Struik, D.J. A concise history of mathematics. Toronto: General Publishing Company,
Ltd.
17
22. Weisstein, E.W. “Cis” From Math World – A Wolfram Web Resource.
http://mathworld.wolfram.com/Cis.html
Weisstein, E.W. “Taylor Series” From Math World – A Wolfram Web Resource.
http://mathworld.wolfram.com/TaylorSeries.html
Wikimedia Commons. File:Sine Cosine Exponential qtl1.svg. Retrieved from
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iaviewer/File:Sine_Cosine_Exponential_qtl1.svg
18
