leonhard euler contributions to mathematics pdf

In fact, Euler proved that the problem has no solution, but in doing so he made the important conceptual leap of pointing out that the choice of route within each landmass is irrelevant and the only important feature is the sequence of bridges crossed.

The problem was to find a route through the city that would cross each bridge once and only once. The city of Königsberg in Prussia (modern-day Kaliningrad in Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The Seven Bridges of Königsberg ProblemĪlso in 1735, Euler solved an intransigent mathematical and logical problem, known as the Seven Bridges of Königsberg Problem, which had perplexed scholars for many years, and in doing so laid the foundations of graph theory and presaged the important mathematical idea of topology. He also showed that the infinite series was equivalent to an infinite product of prime numbers, an identity which would later inspire Riemann’s investigation of complex zeta functions. Euler’s friend Daniel Bernoulli had estimated the sum to be about 1 3⁄ 5, but Euler’s superior method yielded the exact but rather unexpected result of π 2⁄ 6. 1⁄ 1 2 + 1⁄ 2 2 + 1⁄ 3 2 + 1⁄ 4 2 … (a zeta function using a zeta constant of 2). It was called the Basel problem after the Bernoulli’s had tried and failed to solve it, and asked what was the precise sum of the of the reciprocals of the squares of all the natural numbers to infinity i.e. The discovery that initially sealed Euler’s reputation was announced in 1735 and concerned the calculation of infinite sums.

He seemed to have an instinctive ability to demonstrate the deep relationships between trigonometry, exponentials and complex numbers. In fact, in a recent poll of mathematicians, three of the top five most beautiful formulae of all time were Euler’s. Another such discovery, often known simply as Euler’s Formula, is e ix = cos x + isin x. This equation combines arithmetic, calculus, trigonometry and complex analysis into what has been called “the most remarkable formula in mathematics”, “uncanny and sublime” and “filled with cosmic beauty”, among other descriptions. He even managed to combine several of these together in an amazing feat of mathematical alchemy to produce one of the most beautiful of all mathematical equations, e i π = -1, sometimes known as Euler’s Identity.

His efforts to standardize these and other symbols (including π and the trigonometric functions) helped to internationalize mathematics and to encourage collaboration on problems. Much of the notation used by mathematicians today – including e, i, f( x), ∑, and the use of a, b and c as constants and x, y and z as unknowns – was either created, popularized or standardized by Euler. Mathematical notation created or popularized by Euler His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights and measures and even the theory of music. Today, Euler is considered one of the greatest mathematicians of all time. Petersburg of Peter the Great and Catherine the Great.ĭespite a long life and thirteen children, Euler had more than his fair share of tragedies and deaths, and even his blindness later in life did not slow his prodigious output – his collected works comprise nearly 900 books and, in the year 1775, he is said to have produced on average one mathematical paper every week – as he compensated for it with his mental calculation skills and photographic memory (for example, he could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last). But, partly due to the overwhelming dominance of the Bernoulli family in Swiss mathematics, and the difficulty of finding a good position and recognition in his hometown, he spent most of his academic life in Russia and Germany, especially in the burgeoning St. Like the Bernoulli’s, he was born in Basel, Switzerland, and he studied for a while under Johann Bernoulli at Basel University. Leonhard Euler was one of the giants of 18th Century mathematics.

Leonhard euler contributions to mathematics pdf