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

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

TABLE OF CONTENTS

Introduction 1

The Origins of e 1

The Properties of e 3

Euler’s Identity 7

1. Euler’s Proof 8

2. Taylor Series Proof 10

Transcendence Theory 11

Proof of the Transcendence of e 12

Application of the Transcendence of e 14

Conclusion 15

Introduction

In the education of calculus, we come across the unique function which is its own

derivative and anti derivative, . 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,

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and continuously becomes $2.718 (Bentley 2008, p. 118). He then used this knowledge to

find the limit of 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

. Given that a > 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.

Then, by using Newton’s binomial theorem where

∴When w→0 then j→∞

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When x = 1, then

When k = 1 then e has the value of a

(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 , 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

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,

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as a function, being its own derivative and therefore its own anti derivative shown

through the proof here.

Then, by knowing the definition of e, where we can replace with h

Therefore, for all small values of h

e ≐

We can then input this in the application of the fundamental theorem of calculus

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Note that knowledge of the definition of e is necessary to find the derivative of ex

as is replaced with h in the fundamental theorem of calculus.

The fundamental theorem of calculus is used instead of calculating the derivative

using the commonly understood rules and patterns in calculus since no rules can be

applied to finding the derivative of this function, ex.

This graph depicts the limit of , 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

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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.

In another function, is the absolute maximum of the function at x = e.

Since we can not apply the power rule, I will take the natural log of each side.

Now that we have the first derivative, we can find the root of the equation in order to find

the extrema.

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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

can not be represented by a fraction and continues after the decimal place infinitely. By

using this feature of the function , 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,

(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,

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at the time, was written by Euler as . 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 these

functions, the convergence of the series provides evidence of Euler’s formula, also

known as . Note that cis(x) translate into “cosine plus isine” (Weisstein).

1. Euler’s Proof

In Euler’s Introductio, the introduction of the formula begins with

Euler’s definition of cos v, sin v, and ez.

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 for i, the

imaginary number.

Here, Euler substitutes z for the right sides using , according to whether it is

a positive or negative difference and makes sin v and cos v in terms of ez.

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(Struik 1987, pp. 122-123)

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 where the angle x must be in

radians (Storr 2014).

In the diagram, it is stated that , where the blue line indicates the

complex number that results from the vertical and horizontal axis. As stated by Euler, as

previously discussed,

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(Struik 1987, pp. 122-123)

where the sum of these is . 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.

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.

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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 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,

, , and . 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

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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 and multiplied it by ex.

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, can be used to replace the

functions in the previous equation after cancellations to get a new equation.

With the assumption that e is algebraic, there exists a polynomial equation p(t)

with the coefficient and exponential degree n that is greater than or equal to 1

when p(e) = 0. This produces the series . Then, Hermite multiplied both sides of

the previous equation by the series and replaced x with j.

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Hermite then chose a polynomial to replace f(t) where he chose

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).

We check that 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

Therefore, 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

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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 . 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).

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.

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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.

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