Euler’s Number, commonly referred to as the mathematical constant e, is an irrational number of immanent importance in mathematics alongside four other numbers: 0, 1, pi, and i. These five numbers cross all borders in mathematics and help describe natural phenomena in our world. But, lets not get hung up on all five, were here to talk about one of these numbers; Euler’s Number.
History of Euler’s Number
John Napier, the man who made common use of the decimal point, was a Scottish mathematician, physicist, astronomer and astrologer, and also the 8th Laird of Merchistoun. In 1618, Napier published a work on logarithms which contained the first reference to the constant e (at the time, it was not called “e”).
It was not until Jacob Bernoulli, whom the Bernoulli Principle is named after, attempted to find the value of a compound-interest expression that e’s numerical value was discovered.
*Note: The equation from which Bernoulli derived e can be found here under “History”.
Euler’s number was actually expressed as b until Euler himself published his work Mechanica, which used e instead of b as the variable. Eventually the letter ‘e’ made its way as the standard notation of Euler’s number.
How e is used
Euler’s number can be used in a variety of ways, the most common being compound-interest problems.
Equation: A=Pert
The equation describes a bank account that gives continuously compounded interest on the dollar amount.
Euler’s number is particularly useful in calculus when dealing with exponential and logarithmic equations. It simplifies many equations because ex is its own derivative, a unique quality which other numbers do not possess.
Euler’s number can also be described as a Taylor series and because it is stable even when x in ex is complex, it allows us to extend the definition of e into complex numbers dealing with sine and cosine. Here is a list of identities involving e that are particularly useful.
- eix = cos(x) + isin(x)
- ei(pi) = -1
- loge(-1) = i(pi)
- (cos(x) + isin(x))n = einx = cos(nx) + isin(nx)
Aside from the “textbook” applications, Euler’s number can be used to calculate anything from radioactive decay to population increase. The applications are endless!
Conclusion
Well, there you have it, an explanation of Euler’s number. If you want to dig deeper, as this was only a general description, see the web page I give as my source below.
Remember, Euler is watching you!
Source:
e (mathematical constant) – Wikipedia
