Elective Math- Trigonometry
Venus Abigail D. Gutierrez

              The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithms. It is the limit of (1 + 1/n)n as n becomes large, an expression that arises in the study of compound interest, and can also be calculated as the sum of the series e = 2 + 1/2 + 1/(2 × 3) + 1/(2 × 3 × 4) + 1/(2 × 3 × 4 × 5) + …

             The constant can be defined in many ways; for example, e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the pointx = 0 is equal to 1. The function ex so defined is called the exponential function, and itsinverse is the natural logarithm, or logarithm to base e. The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1and x = k, in which case, e is the number whose natural logarithm is 1. There are also morealternative characterizations.

            Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ—the Euler–Mascheroni constant, sometimes called simply Euler's constant. It is also known as Napier's constant, but Euler's choice of the symbol e is said to have been retained in his honor. The number e is of eminent importance in mathematics, alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers; and it istranscendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is 2.71828182845904523536028747135266249775724709369995... (sequence A001113 in OEIS). The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact e):



          The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 November 1731. Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and the first appearance of e in a publication was Euler's Mechanica (1736). While in the subsequent years some researchers used the letter c, ewas more common and eventually became the standard.