Euler’s number, e, emerged during the renaissance period, which was characterized by emergence of new knowledge in Europe. During the 16th century, an important discovery was made; Michael Stifel, a German mathematician, expanded the knowledge on geometric progression and the respective exponents (Maor, 2009). Stifel, nonetheless, only examine integral values relating to exponents. In the 17th century, John Napier developed the idea of logarithms. His basic idea was that any positive number could be written in exponential form, and it would be possible to find the multiplication or division of the number by simply adding or subtracting the powers. In writing numbers to exponential form, a certain fixed number or the base is used. This was critical in making numerical computations easy. Napier’s logarithms decrease with each increase in numbers. This is due to the fact that in his system, Log 107= 0. In 1624 Henry Briggs, a professor of Geometry, published a revised system of logarithms (Maor, 2009). In this system, the logarithm of 1 was taken as 0, and the logarithm of 10 taken as powers of 10.
Napier seems to be the first person to have made note of the letter e in mathematics. In his work, Descriptio, he noted that the logarithm of 10 to the base e was 2.302585 (Maor, 2009). However, the critical question remains as to how the number e came into existence. The number e is derived from the mathematics of finance. In the calculation of compound interest, mathematicians were interested in knowing how long a sum of money would take to grow to a certain amount. In order to find the number e, the principal is compounded for 1 year, at the rate of 100%. As “n” (the number of times the principal is compounded) increases, the figure obtained increase but in less significant digits (Maor, 2009). In 1737, Euler proved that e is an irrational number. In his work Introductio, Euler emphasized on the role of the numbers e and ex in numerical analysis. In 1781, Euler estimated the number e to 16 places. Another important mathematician is Jacob Bernoulli, who attempted to establish the value of the limit of the equation n as n approaches infinity.
There are a number of interesting facts about e. The first interesting thing is that no matter how large the value of n (as it tends to infinity), the figure obtained will always be approximately 2.718281828. The number e is the limit of the equation n as n approaches infinity (Maor, 2009). Although it appears to have a repeating block of decimals, 1828, the number is irrational. It is non-repeating and non-terminating. The number e is not or cannot be a solution to any polynomial equation having integer coefficients.
The number e is applied in probability theory. Bernoulli successfully applied the number e in probability theory. This is known as the “misplaced envelope” problem. The number e is applied in measuring the decay rate of an exponential function (Pickover, 2009). This is critical in biological growth models that help in estimating the population of organisms that show a proportional rate of change to their number. The number e has numerous applications in calculus. For instance, it can help in finding the derivatives and in evaluation of limits. Its applications in differential equations are immense. Euler’s number is applicable to complex numbers (Pickover, 2009). The number helps in extending the use of ex even when the values of x fall under complex numbers. The number e is useful in giving different representations. For instance, it can be represented as an infinite product, a limit of a sequence, an infinite series, or as a stochastic representation.
Maor, E. (2009). E: The story of a number. Princeton, N.J: Princeton University Press.
Pickover, C. A. (2009). The math book: from Pythagoras to the 57th dimension, 250 milestones in the history of mathematics. Sterling Publishing Company.