a mathematical ode to euler: proving euler’s identity by james d. nickel copyright 2010 e i + 1...

Eulers Identity There is a famous formulaperhaps the most compact and famous of all formulas developed by Euler from a discovery of De Moivre: e i + 1 = 0. It appeals equally to the mystic, the scientist, the philosopher, the mathematician Edward Kasner & James Newman, Mathematics and the Imagination (1940). Copyright 2010 www.biblicalchristianworldview.net

Groundwork: The Binomial Formula He first recognized that he could apply the binomial formula to this definition. This formula is based upon Pascals triangle (1623-1662) and derived by Isaac Newton (1643-1727). Copyright 2010 www.biblicalchristianworldview.net 1707-1783

Amazing Connection: Combination Formula He began with the combination formula (studied in statistics and finite mathematics courses) for deriving the terms of a binomial expansion: Copyright 2010 www.biblicalchristianworldview.net 6! = 6 5 4 3 2 1 = 720 5! =5 4 3 2 1 = 120 4! = 4 3 2 1 = 24 3! = 3 2 1 = 6 2! = 2 1 = 2 1! = 1 0! = 1 Note the Pattern!

Combination Formula Note that all the factors from 1 to (n j) in the numerator cancel with those in the denominator, leaving only these factors in the numerator: n(n 1)(n 2) (n j + 1). In our example, we have 6(6 1)(6 3 + 1). Hence: Copyright 2010 www.biblicalchristianworldview.net

Some Pascalinian Words of Wisdom The eternal silence of these infinite spaces frightens me (p. 61). Two extremes: to exclude reason, to admit reason only (p. 74). The last proceeding of reason is to recognize that there is an infinity of things which are beyond it (p. 77). Copyright 2010 www.biblicalchristianworldview.net

The Pattern The sum of each row is the binary sequence (powers of 2). Row 0: 1 = 2 0 Row 1: 1 + 1 = 2 = 2 1 Row 2: 1 + 2 + 1 = 4 = 2 2 Row 3: 1 + 3 + 3 + 1 = 8 = 2 3 Row 4: 1 + 4 + 6 + 4 + 1 = 16 = 2 4 Copyright 2010 www.biblicalchristianworldview.net

A Fine Piece of Thinking When a baseball player works the count, gets his pitch, and drives a single into the outfield, seasoned baseball experts usually remark, That was a fine piece of hitting. From equation 2 to equation 8 we have followed Eulers fine piece of thinking. But, there are more of the amazing thoughts of Euler to come. Copyright 2010 www.biblicalchristianworldview.net

Limits Since we are looking for the limit of (1 + 1/n) n as n , we must let n increase without bound. Our expansion will have more and more terms. At the same time, the expression within each pair of parentheses will tend to 1 since the limits of 1/n, 2/n, 0 as n . Copyright 2010 www.biblicalchristianworldview.net

Convergence It can be shown that this series converges (i.e., reaches a limit) for all real values of x. In fact, the rapidly increasing denominators cause the series to converge very quickly. It is from this series that numerical values of e x are calculated. This series is programmed into modern scientific calculators when the e x key is punched. Copyright 2010 www.biblicalchristianworldview.net

Euler Nerve Euler continued to experiment with this power series for e x. He decided to see what happened to this series if he replaced x with the imaginary expression ix where i = -1. This substitution took some nerve on Eulers part since e x had always represented a real number. Now Euler was going to see what would happen to the expression e ix. Copyright 2010 www.biblicalchristianworldview.net

Eulers Leap Now Euler did what todays mathematicians consider unthinkable. He changed the order of the terms collecting all the real terms separately from the imaginary terms. With an infinite series, this could create trouble. Instead of converging to a limit, it may diverge. He got: Copyright 2010 www.biblicalchristianworldview.net

A Remarkable Connection In Eulers time, thanks to the work of Brook Taylor (1685- 1731), the power series of sin x and cos x (x in radians) was well known. Here it is: Copyright 2010 www.biblicalchristianworldview.net

Startling This equation expresses a startling, indeed, incredible link between the exponential function (albeit raised to an imaginary power) and basic trigonometry ratios. Euler now replaced x by -x and using the trigonometric identities cos (-x) = cos x and sin (-x) = -sin x, he rewrote the above equation as follows: Copyright 2010 www.biblicalchristianworldview.net

Interlude Every step that Euler has taken so far (and some steps were giant intuitive leaps) has been confirmed by the rigors of analysis in the 19 th and 20 th centuries. Euler, like many of his time, was a pioneer. He blazed many trials and left it for others to confirm his steps. Copyright 2010 www.biblicalchristianworldview.net

Hats off! Mathematicians have described this equation using words like remarkable, beautiful, a revelation, absolutely paradoxical, certainly true, incomprehensible, blew a few circuits in my head, and cannot be explained in words. Hats off and thank you, Professor Euler! Copyright 2010 www.biblicalchristianworldview.net You are welcome! Euler, It sometimes seems to me that my pencil is smarter than I.

e i + 1 = 0 Here we have an equation that connects the five most important constants of mathematics (0, 1, i, , e) and three of the most important mathematical operations (addition, multiplication, and exponentiation). The five constants connect four major branches of mathematics: arithmetic (0 and 1), algebra (i), geometry ( ), and analysis (e). Copyright 2010 www.biblicalchristianworldview.net

Unity in Diversity For the Biblical Christian, this formula serves as another resounding echo that can be traced back to the ultimate One and the Many, the Author and Sustainer of this striking connection this astonishing and wondrous unity and diversity in mathematics. Copyright 2010 www.biblicalchristianworldview.net

a mathematical ode to euler: proving euler’s identity by james d. nickel copyright 2010 e i + 1...

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