Leonhard Euler: Mathematical Biography

Leonhard Euler was born In Basel, Switzerland in 1707, son of a pastor and pastor’s daughter (different pastor!) Not only a mathematician, Euler was also a physicist and made contributions in the fields of mechanics, astronomy and optics. He grew up in the town of Reihen where his family was good friends with the Bernoullis.

Euler enrolled in the University of Basel at age 13 and had achieved his bachelor’s degree at the age of 15 then his masters’ degree only one year later. At this time, while under the private tutelage of Johann Bernoulli, the latter recognized his mathematical talent and successfully persuaded his father to allow him to give up studying to be a pastor.

In 1727, Euler became a professor at the ImperialRussianAcademy of Sciences in St. Petersburg and four years later became head of the Mathematics department after his good friend Daniel Bernoulli left. A severe fever in 1735 damaged his eyesight leaving him partially blind in one eye, eventually becoming almost totally blind.

In 1741, Euler took a post at the BerlinAcademy where he wrote around 380 articles and published two of his major works in the span of 25 years. Then Euler returned to the ImperialRussianAcademy of Sciences in St. Petersburg and died in 1783 of a brain hemorrhage. He was buried in St. Petersburg’s Alexander Nevsky Monastery.

Euler found a function that would generate prime numbers given values of x up to 39. Values up to 1000 give prime numbers at a rate of 58.1%.

Euler’s 1737 discovery relates the sequence of natural numbers to the primes:

It is easier to understand what is going here if we expand a few terms:

taking s=1, the left hand side is merely the harmonic series while the right comes from the series of prime numbers…

for values of s>1 the series converges while we know that the harmonic series diverges.

Euler was the first person to use the letter e for the base of the natural logarithm, published for the first time in his 1736 work, Mechanica. Although the symbol π had been used before Euler, it was he who popularized its use. Another notation first used by Euler is the letter i for the imaginary

In the field of geometry, he popularized the use of letters a, b, and c for the sides of the triangle and the capitalized versions to label the angles opposite each side. Also, using the letters r, and R for the radius of the inscribed and circumscribed circles respectively was his idea. Probably most notable notation introduced by Euler is the function notation, f(x).

Introductio in analysin infinitorum was published in 1748 and was one of Euler’s most important works. One element of this work is the founding of the general mathematical branch of analysis by combining differential calculus and the method of fluxions.

Developed by Euler, this is the general form of a type of linear equation used in the field of differential equations.

Euler studied probability quite a lot and published an account in 1751 called Memoirs in which many problems were published. One of the simplest examples concerned the drawing of tickets: n tickets are numbered from 1 to n . If 3 tickets are drawn at random, the probability that a sequence of 3 consecutive numbers is drawn is,

the probability that exactly 2 consecutive numbers are drawn is,

and the probability that none of the numbers drawn are consecutive is,

Wikipedia

Mathematical Mysteries by Calvin C. Clawson

A History of Mathematics by Carl B. Boyer