Eulers Equations

Greatest Equation, page 1

The Greatest Equation Ever
The History of Euler’s Identity
Jessica Mack

History of Mathematics
Instructors Adam Harbaugh and Jennifer McCarthy
North Carolina Summer Ventures in Science and Mathematics 2009
Friday, July 17, 2009

Greatest Equation, page 2

Abstract:
This report is a brief history of Leonhard P. Euler and his many contributions to theHistory of
Mathematics. This paper focuses on Euler’s Identity and how Euler was able to solve this concept.
Also called Euler’s Equation, Euler’s Identity is defined as

and is used as a

mathematical procedure to find and perform trigonometric functions without the use of
geometry. This article examines the mathematics of Euler’s Identity and the brief history behind
how it came intoexistence. This report will analyze what it means to be a ‘beautiful equation’ and
why Euler’s Identity can be classified as such. Finally, this report will briefly examine the different
uses of the Euler Identity in Complex Analysis and Trigonometry.

Greatest Equation, page 3
Leonhard Paul Euler (1707- 1783) was a Swiss mathematician who made tremendous
contributions in the field of mathematicsin topics such as Calculus, Geometry, Trigonometry,
Algebra, Graph Theory and Physics. He is considered to be one of the greatest mathematicians of
the 18th century and of all time. He was a student of Johann Bernoulli, one of Europe’s foremost
mathematicians of the early 1700’s. He was also a pioneer in the area of mathematical notation
and terminology from his book The Elements of Algebra,some of which is still used today. Euler
was the first to introduce the concept of a function, as well as , also known as Euler’s number or
2.718281828. However, out of all Euler’s many accomplishments to mathematics, the one that is
considered most remarkable is his equation, Euler’s Identity,

as written from his book,

Introductio in analysin infinitorum in 1748.
Euler’s Identity, alsoreferred to as Euler’s Equation or Euler’s Formula, is sometimes
written as

or

because

when theta is substituted for pi. We can conclude this
. When pi and theta, and

and

are substituted, the result is

. This Identity can be used to define trigonometric functions without the use
of geometry, which to many is amazing. It can define functions of a triangle, without ever havingto draw a triangle. It is a ‘beautiful equation’ because it encompasses five important
mathematical constants,

, The numbers

and

are the most concrete of numbers

and are the basis of counting. Pi is the basis of geometry in that the measure of the circle, the
most perfectly symmetrical shape, is dependent upon it. Euler’s number, or , is the heart of
Calculus and Complex Analysis,and the most accurate attempt to understand infinite numbers.
The square root of -1, the most mysterious irrational number, represents just how intricate
mathematics can become. These numbers all in one equation, all built on different concepts of
mathematics, create the most beautiful mathematical composition (Devlin 2009). Moreover, the

Greatest Equation, page 4
Identity is beautifulbecause it is a simple, brief, important, and surprising mathematical
statement (Nahin, 2006 ). It uses three basic arithmetic operations: addition, multiplication and
exponentiation, each occurring only once. It is generally considered the most important formula
of mathematics. According to Paul Nahin in his book, Dr. Euler’s Fabulous Formula, the Euler
Identity “sets the gold standard for
imathematical beauty”. The Identity is
considered “the greatest equation ever”
(Physics World, 2004). It is remarkable.
α
Euler’s Identity is defined as

-1 +i0

, where

is the ratio of any

circle’s circumference to its diameter while
at the same time being an irrational
-i
Figure 1: Complex Number Plane

number;

is a unique number such that the

value of

is equal to the...