Leonhard Euler

Leonhard Euler was born in Basel, Switzerland in 1707, but his career as a mathematician was not limited to one country. Euler had already received his master's degree by 1723 from the University of Basel, and was unable to find a scholarly position because of his age. Journals written on his plight seem undecided on whether Euler moved to St. Petersburg, Russia because he was called there by Catherine the First or because of the urging of his friend-mentor Jean Bernoulli. Either way, Euler became an accomplished professor of physics and of mathematics within six years of the move. It was during his stay in Russia that he lost the use of his right eye, said to be caused by his observations of the sun.

Euler left Russia for Berlin in 1741 to become the Director of Mathematics at the Academy of Science. During his stay, he developed theories of logarthmic and trigonometric functions and of complex numbers, the most well known of which (the special number e) described below. He returned to St. Petersburg in 1766, and by that point he had lost the use of his second eye. What is truly incredible is that he was still able to function as a mathematician, concentrating his research on positions of the moon and the gravitational relationships between it, the sun and the Earth.

Although John Napier created the first table of logarithms and it was the English translation of his Descriptio that featured the computation of the logarithm of 10 using the base e (his original used a number close to 1/e as what we recognize now as a base), the introduction of the actual letter e to represent the base of the natural logarithm is due to Leonard Euler. This notation was first used in a work published 80 years after his death, although the manuscript had been written as early as the 1720s.

The special number e, like pi and i, is a mathematical constant, unique in that the slope of the graph of ex at any point is equal to ex itself. Note that when attempting to find the derivative of f(x) = log x, the following limit occurs:

lim (1+h)1/h
h->0

Using a table to evaluate the limit:

h (1+h)1/h
1 2
0.1 2.954
0.01 2.705
0.001 2.717
0.0001 2.718145…
0 Undefined
-0.9999 2.718418…
-0.999 2.720
-0.99 2.732

If you are given y = ex , you will find the inverse to be x = ey , which can be rearranged into y = logex. This log has its own notation, which is y = ln x, the natural log.

Summarily, formulas and their derivatives using the special number e and the natural log:

y = ex
y' = ex

y = ln x
y' = 1/x

y = ax
y' = ax ln a

y = af(x)
y' = af(x) ln a * f'(x)

The number e can easily be calculated to with accuracy to 25 decimal places using factorials, although the total number of places found to date is in the hundreds of millions.

 1/0! = 1.0000000000000000000000000 1/1! = 1.0000000000000000000000000 1/2! = 0.5000000000000000000000000 1/3! = 0.1666666666666666666666667 1/4! = 0.0416666666666666666666667 1/5! = 0.0083333333333333333333333 1/6! = 0.0013888888888888888888889 1/7! = 0.0001984126984126984126984 1/8! = 0.0000248015873015873015873 1/9! = 0.0000027557319223985890653 1/10! = 0.0000002755731922398589065 1/11! = 0.0000000250521083854417188 1/12! = 0.0000000020876756987868099 1/13! = 0.0000000001605904383682161 1/14! = 0.0000000000114707455977297 1/15! = 0.0000000000007647163731820 1/16! = 0.0000000000000477947733239 1/17! = 0.0000000000000028114572543 1/18! = 0.0000000000000001561920697 1/19! = 0.0000000000000000082206352 1/20! = 0.0000000000000000004110318 1/21! = 0.0000000000000000000195729 1/22! = 0.0000000000000000000008897 1/23! = 0.0000000000000000000000387 1/24! = 0.0000000000000000000000016 1/25! = +0.0000000000000000000000001 e = 2.7182818284590452353602875

Please note that this is only the tip of Euler's work and outreaching influence on the maths and sciences, in that he published over eight hundred significant papers during the course of his life.

### Citations

• "e (number)." Gale Encyclopedia of Science. Eds. K. Lee Lerner and Brenda Lerner. Vol. 2. 3rd ed. Detroit: Gale, 2004. 1 pp. 6 vols. Gale Virtual Reference Library. Gale. Thames Valley DSB. 25 Feb. 2008. <http://find.galegroup.com/ips/start.do?prodId=IPS>.
• "e, the number." World of Scientific Discovery. Kimberley A. McGrath and Bridget Travers. Online. ed. Detroit: Thomson Gale, 2007. Discovering Collection. Gale. Thames Valley DSB. 25 Feb. 2008. <http://find.galegroup.com/ips/start.do?prodId=IPS>.
• Notes from MCB4U, Spring 2007.
• Picture from American Mathematical Society, from website: <www.ams.org/ams/euler.html>.

# Leonhard Euler

Add a New Comment
page revision: 28, last edited: 05 Mar 2008 17:16