A BRIEF INTRODUCTION TO THE NUMBER e

Blindness was not an obstacle for him to deliver magnificent works. Along with his perseverance in studying mathematics, he lost the sight of his right eye and then his left eye as well 31 years later. He discovered a branch of mathematics called topology. He also contributed to the study of geometric shapes, where he derived Euler’s formula. Who is he? His name is Leonhard Euler, a Swiss, who lived between 1707 and 1783. One of his other works, which will be discussed in this post, is an irrational number denoted by e. The number proves to be widely used in mathematics and natural sciences.

Consider the sequence (un), where for every n ∈ . The values of u_n ​​for some values ​​of n are shown in the following table.

Note that as the values of n increase, so do the values of u_n. The sequence is monotone increasing. It can also be proved that for every n ∈ , u_n < 3. This means that the sequence is bounded above.  One of the theorems in mathematics says that “a monotonically increasing sequence that is bounded above converges”. By applying the theorem just-mentioned, we can conclude that (u_n) converges. As the values of n increase, the values of u_n approach a certain number denoted by e. Stated another way, . Rounded to 10 decimal places, e ≈ 2.7182818285.

In integral calculus, the area bounded by the curve y = 1/x, the x-axis, the line x = 1, and the line x = e is 1. (In the figure below, the area of ​​the green region is L = 1.)

The number e is always related to a real-valued function called the natural logarithm function. The domain of the function is D = {x ∈ ℝ| x > 0} and for every x ∈ D the function maps x into . By defining the natural logarithm function that way, it can be proved that ln e = 1.

The number e has been applied in various fields, some of which are as follows.

1. In statistics, the number e is often found in probability distributions or probability density function, such as normal distribution, Poisson distribution, and exponential distribution.
2. In the field of psychology, the formulation of the learning curve uses e.
3. In finance, the number e is used in continuous compounding.
4. In physics, the number e is used in the equation of the exponential radioactive decay.

Whether or not in his lifetime Euler predicted that the number e would be of significant importance in the development of science, what is certain is that despite his handicaps, he contributed something so valuable to our life today. Blindness did not prevent him from being such a productive mathematician.