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The Number e: 2.7182818284590452...

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If you asked someone to name his or her favorite mathematical constant, you would probably get some quizzical looks. After a while someone may volunteer that the best constant is pi. Pi, typically denoted by the Greek letter pi Π is the mathematical constant defined as the ratio of a circle’s circumference to its diameter. Pi is quite well known, due to its appearances throughout geometry and trigonometry, and has an infinite nonrepeating decimal representation that begins 3.1415926.

But this is not the only important mathematical constant. A close second, if not contender for the crown of most ubiquitous constant is e. This number shows up in calculus, number theory, probability and statistics.

Value of e

Like pi, e is an irrational real number. This means that it cannot be written as a fraction, and that its decimal expansion goes on forever with no repeating block of numbers that continually repeats. The number e is also transcendental, which means that it is not the root of a nonzero polynomial with rational coefficients. The first fifty decimal places of e = 2.71828182845904523536028747135266249775724709369995.

Definition of e

The number e was discovered by people who were curious about compound interest. In this form of interest, the principal earns interest and then the interest generated earns interest on itself. It was observed that the greater the frequency of compounding periods per year, the higher the amount of interest generated. For instance, we could look at interest being compounded:

  • Annually, or once a year
  • Semiannually, or twice a year
  • Monthly, or 12 times a year
  • Daily, or 365 times a year
The total amount of interest increases for each of these cases.

A question arose as to how much money could possibly be earned in interest. To attempt to make even more money we could in theory increase the number of compounding periods to as high a number as we wanted. The end result of this increase is that we would consider the interest being compounded continuously.

While the interest generated increases, it does so very slowly. The total amount of money in the account actually stabilizes, and the value that this stabilizes to is e. To express this using a mathematical formula we say that the limit as n increases of (1+1/n)n = e.

Uses of e

The number e shows up throughout mathematics. Here are a few of the places where it makes an appearance:

  • It is the base of the natural logarithm. Since Napier invented logarithms, e is sometimes referred to as Napier's constant.
  • In calculus the function ex has the unique propery of being its own derivative.
  • Expressions involving ex and e-xcombine to form the hyperbolic sine and hyperbolic cosine functions.
  • Thanks to the work of Euler, we know that the fundamental constants of mathematics are interrelated by the formula e+1=0, where i is the imaginary number which is the square root of negative one.
  • The number e shows up in various formulas throughout mathmatics, especially the area of number theory.

The Value e in Statistics

The importance of the number e is not limited to just a few areas of mathematics. There are also several uses of the number e in statistics and probability:

  • The number e makes an appearance in the formula for the gamma function.
  • The formulas for the standard normal distribution involves e to a negative power. This formula also includes pi.
  • Many other distributions involve the use of the number e. For example, the formulas for the t-distribution and chi-square distribution both contain the number e.
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