Leonhard+Euler

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=Timeline=

1707: Leonhard Euler is born in Basel, Switzerland (Boyer, Carl B. __A History of Mathematics)__

1720: Enters the University of Basel at the age of 13 (Boyer, Carl B. __A History of Mathematics__)

1723: Receives his Master's Degree ("The Euler Archive - the works of Leonhard Euler online." Mathematics at Dartmouth)

1727: Able to get a seat in the department of medicine at the St. Petersburg Academy (Boyer, Carl B. __A History of Mathematics__)

1730: Switched departments form medicine to natural philosophy (Boyer, Carl B. __A History of Mathematics__)



1733: Became head of mathematics of the Academy at age 26 (Boyer, Carl B. __A History of Mathematics__)

1733: Eyesight began to deteriorate (Boyer, Carl B. __A History of Mathematics__)



1736: Calculates the infinite sum of reciprocal squares to be equal to

1737: Adopted use of the symbol π. (Boyer, Carl B. __A History of Mathematics__)

1741: By invitation of Frederick the Great, left Russia to join the Berlin Academy (Boyer, Carl B. __A History of Mathematics__)

1746: Showed to have infinite real values

1748: In his //Introductio in analysin infinitorum, shows that



//

1766: Being unhappy in Germany, Euler went back to his post in Russia (Boyer, Carl B. __A History of Mathematics__)

1766: Started to become fully blind. However, this would become his most prolific period. (Boyer, Carl B. __A History of Mathematics__)

1783: Died in St. Petersburg, Russia at the age of 76 (Boyer, Carl B. __A History of Mathematics__)



=Achievements=

“ Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind,” stated François Arago. (Bell 139) Without a doubt, this can be regarded as correct. Leonhard Paul Euler can be considered to be among the greatest of mathematicians. At the very least he is the most prolific. If one takes all of the pages he wrote throughout his career, they will find that they can fill a good eighty volumes, or about 25000 pages, and more are continually unearthed (Washington Post). He touched near all areas of mathematics and physics, and thus it would be impossible to summarize all of his achievements in such a brief space. Thus, I will only attempt to describe those that he is most famous for.

Complex Analysis
Euler was the first to have penned what is commonly regarded as the most beautiful equation in all of mathematics: e^i pi +1=0. Notice how it so elegantly combines five fundamental constants with the arithmetic operations of addition, multiplication, and exponentiation. This is based off of his formula, eix= cos(x)+isin(x), which I shall leave without proof. However, it is through this equation that imaginary exponents can be calculated, not to mention complex trigonometric functions. In addition, this representation allows complex functions to be graphed through polar coordinates, a fundamental portion of complex analysis. Furthermore, it was through this equation that i^I can be shown to have infinitely many //real// values, which was also shown by Euler, which I will show here:

E^i pi = cos( pi )+isin( p )= -1+i0= -1 I^I = (-1)^1/2^i =e^i pi ^1/2^i =e^(i*i*1/2* pi ) =e^(- pi /2) =1/(e^( pi /2) However, note that the trigonometric functions are periodic, such that f(x+2k p ), where f(x) is one of the six basic trigonometric functions and k is an integer. Therefore,

e^3i pi = cos(3 pi )+isin(3 pi )= -1= e^5i pi = e^7i pi …

Hence, we reach a general solution:

I^i= 1/(e^((2k+1) pi /2)), where k is an integer.

As we can see, despite its counter-intuitiveness, Euler discovered an equation that was not only mathematically beautiful but also extensively applicable in modern scientific fields.

Number Theory
Through a bit of manipulation of Maclaurin series, Euler was able to calculate the correct answer to the Basel problem (the infinite sum of the reciprocals of perfect squares) to be pi ^2/6. Furthermore, he also showed that this was related to the product of the reciprocal prime numbers. Bernhard Riemann based his zeta function off of this and was thus able to propose his infamous hypothesis on the distribution of prime numbers (Riemann 1859). Should this be solved, it would provide deep insight to number theorists everywhere, but we mustn’t forget Euler’s part in all of this.

Graph Theory
In Euler’s day, there was a problem concerning the seven bridges of the town of Konigsberg. They were constructed in such a way on various islands that it was unknown as to whether there could be a path including all bridges only once (Grilly 116). Euler was able to show this false. However, in doing so, he basically opened the field of graph theory, where points and the paths connecting them are studied. His work here would ultimately cumulate in another formula that bears his name:

V-E+F= 2, where V, E, and F are the vertices, edges and faces in a convex graph. For example, a cube has 8 vertices, 12 edges, and 6 faces; 8-12+6=2.

Notations
Many of the symbols used in mathematics were invented, or at the very least popularized by Euler. For example, it was he who first used f(x) to denote a function. In addition, he used, e, and i for their current definitions. Finally, he used to represent summations (Boyer 441).

=A Discussion Concerning Euler and Gauss=

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=An Explanation of the Given Proof= If we follow a definition of i without rigor, we can say that it is equal to sqrt(-1) Thus, i^2=-1 i^3=-i i^4=1

From algebra, we know that the slope of a line is equal to ∆y/∆x. As y can be rewritten as f(x), slope can also be written as f(x+∆x)-f(x)/∆x. This equation can be used to approximate the slope of tangent lines to curves, though in reality, these lines are secants. However, if we want to be exact, that is have a tangent, we will need to take a limit. lim (∆x ->0) f(x+∆x)-f(x)/∆x.

This is known as the derivative of f(x), and is denoted as f'(x). There are several rules that differentiation, as it is called, follows:

(f+g)'(x)= f'(x)+g'(x) (Addition Rule) (f-g)'(x)=f'(x)-g('x) (Subtraction Rule) (f*g)'(x)= f'(x)g(x)+g'(x)f(x) (Product Rule)

There are also some less theoretical rules:

If f(x)=c, where c is a constant, f'(x)=0 f(x)= sinx, f'(x)=cosx f(x)=cosx, f'(x)= -sinx f(x)=e^(cx), where c is a constant, f'(x)=c*(e^(cx))

All of the above rules can be proven, but at the present moment, one would have to inquire somewhere off of this page to find said proofs.

Now, let f(x) = (cosx+isin(x))*e^(-ix) By the product rule, f'(x)= (-sinx+icosx)*e^(-ix)+(cosx+isinx)*-i*e^(-ix) = (-sinx+icosx)*e^(-ix)+(-icosx+sinx)*e^(-ix) =e^(-ix)*(sinx-sinx+icosx-icosx) =e^(-ix)*0 =0

As f'(x) = 0, we know that f(x) must be a constant function, and thus f(x) = f(c), where c is an arbitrary constant Therefore, f(x) = f(0) =(cos0+isin0)*(e^(i0) =(1+i0)*e^0 =1*1 =1

Thus, f(x)=1.

From this, we see that (cosx+isinx)*e^(-ix)=1, or

(cosx+isinx)*e^(-ix)*e^(ix)=1*e^(ix)

Thus, e^(ix)= cosx+isinx

If we let x= pi

e^(i*pi)= cos(pi)+isin(pi) =-1+i0 =-1

Finally, if 1 is added, we get the final identity,

e^(i*pi)+1=0

Q.E.D.

=Political, Social, and Economic Background= "History of RUSSIA." Gascoigne ) || Poland was partitioned among the great central European empires (Gascoigne 2001) || Great Northern War results in Russia becoming the major Baltic power (Gascoigne 2001) || New city St. Petersburg was constructed near the start of his lifetime (Gascoigne 2001) ||
 * Political Background || Catherine the Great wanted to westernize Russia (<span style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; border-collapse: separate; font-family: Arial,helvetica,sans-serif; font-size: 12px; line-height: 18px;">"History of RUSSIA." Gascoigne ) || Russia and Turkey were warring at the end of the 18th century
 * Economic Background || Primary school was impoverished; learned little mathematics (O'Connor, Robinson 1998) || Most of Russia was living in serfdom (Gascoigne 2001) || Recent creation of Peter the Great's Beard Tax (Gascoigne 2001) || Period of increased Russian industrialization (Gascoigne 2001) || Lived through a series of expensive wars ||
 * Social Background || Father was a protestant minister (O'Connor, Robinison 1998) || Studying to be a theologian || Lived in the period of the Enlightenment || Extremely religious throughout his life (Economic Expert) || Design of the Bridges of Konigsberg would lead to a major field of study for Euler (Grilly 116) ||

=Quiz=

(Answers are at the very bottom of the page)


 * 1) In what year was Leonhard Euler born?
 * 2) 1707
 * 3) 1783
 * 4) 1784
 * 5) 1708
 * 6) Where did Euler spend most of his career?
 * 7) Minsk
 * 8) Berlin
 * 9) St. Petersburg
 * 10) Basel
 * 11) What is e^(i*pi) equal to?
 * 12) -2
 * 13) -1
 * 0
 * 1
 * 1) What city inspired Euler's graph theory?
 * 2) Kiev
 * 3) Konigsberg
 * 4) Moscow
 * 5) Paris
 * 6) What is the Euler characteristic for convex graphs?
 * 1
 * 2
 * 3
 * 4
 * 1) How many values does i^i have?
 * 1
 * 2
 * 3
 * 1) Infinite
 * 2) Which of the following notations did Euler not develop?
 * 3) Pi
 * 4) Sigma
 * 5) f(x)
 * 6) tan(x)
 * 7) Approximately how many pages did Euler write throughout his lifetime?
 * 8) 25000
 * 9) 50000
 * 10) 75000
 * 11) 250000
 * 12) Which famous problem in number theory is related to Euler's solution to the Basel problem?
 * 13) Poincaré Conjecture
 * 14) Goldbach Conjecture
 * 15) Gödel's Second Incompleteness Theorem
 * 16) Riemann Hypothesis
 * 17) In what year did Euler receive his Masters' degree?
 * 18) 1733
 * 19) 1737
 * 20) 1723
 * 21) 1726

=Works Cited= Bell, Eric T. Men of Mathematics. New York: Touchstone, 1937. Print. Boyer, Carl B. A History of Mathematics. New York: John Wiley & Sons, 1991. Print. "The Countless Achievements of a Math Master - washingtonpost.com." Washingtonpost.com - nation, world, technology and Washington area news and headlines. Web. 26 Feb. 2010. <http://www.washingtonpost.com/wp-dyn/content/article/2007/04/08/AR2007040800745.html>. "The Euler Archive - the works of Leonhard Euler online." Mathematics at Dartmouth -- Welcome. Web. 25 Feb. 2010. <http://www.math.dartmouth.edu/~euler/>. "Euler Biography." GAP System for Computational Discrete Algebra. Web. 3 Mar. 2010. <http://www.gap-system.org/~history/Biographies/Euler.html>. Gascoigne, Bamber. "History of RUSSIA." HistoryWorld - History and Timelines. 2001. Web. 03 Mar. 2010. G rilly, Tony. 50 Mathematical Ideas You Really Need to Know. London: Quercus, 2007. Print. Riemann, Berhard. Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Diss. Trans. David R. Wilkins. Monatsberichte der Berliner Akademie, 1859. Trinity College Dublin School of Mathematics. Web. <http://www.economicexpert.com/a/Leonhard:Euler.htm>."LEONHARD EULER." United States Naval Academy - Home Page. Web. 03 Mar. 2010. <http://www.usna.edu/Users/math/meh/euler.html>. <http://www.historyworld.net/wrldhis/PlainTextHistories.asp?gtrack=pthc&ParagraphID=ixd>."Leonhard Euler Euler's Number Formula Function Mathematics." Business, Economy, Market Research, Finance, Income Tax Informations. Web. 03 Mar. 2010. <http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf>.

Graphics Cited: Euler oil by Handmann. Digital image. Wikimedia Commons. Web. <http://commons.wikimedia.org/wiki/File:Leonhard_Euler.jpeg>. Euler on Swiss Currency. Digital image. Wikimedia Commons. Web. <http://commons.wikimedia.org/wiki/File:Euler-10_Swiss_Franc_banknote_(front_and_back).jpg>. Leonhard Euler Portrait. Digital image. Wikimedia Commons. Web. <http://commons.wikimedia.org/wiki/File:LeonhardEuler.jpg>. Portrait of Euler by Handmann. Digital image. Wikimedia Commons. Web. <http://commons.wikimedia.org/wiki/File:Leonhard_Euler_by_Handmann_.png>. Portrait of Euler. Digital image. Wikimedia Commons. Web. <http://commons.wikimedia.org/wiki/File:Leonhard_Euler_2.jpg>.

Answers to Quiz: (1,3,2,2,2,4,4,1,4,3)