The number *e*, also called the natural number and Euler’s number, starts with the six numbers 2.71828 and extends into infinite more decimal places. Mathematicians indicate the various appearances of *e *in natural patterns, but its more practical usage appears in calculations of exponential growth, including the continuous compound interest formula *A* = *Pe ^{rt}. *

We may credit Scottish scientist John Napier (1550-1617) for introducing the idea of *e* in his efforts to create a model to simplify multiplication by transforming multiplication into addition. This idea would eventually promulgate in the logarithmic functions of today’s mathematics, and their addition property, whereby *log _{b}(xy)* =

*log*

_{b}(x) + log_{b}(y).Back to *e*, Napier himself did not define or explicitly acknowledge the number. This task passed to Gottfried Leibniz (1646-1716), one of the founders of calculus, who called it *b.* Its present designation as *e* comes from Leonhard Euler (1707-1783), who also played with the imaginary number i. ^{[1]}

Today computers can calculate *e* to millions if not billions of decimal places, saving mathematicians much time and agony. Nevertheless in the beginning the value of *e *was hand-computed.

A friend of Leibniz, Jacob Bernoulli (1655-1705) drew close to its discovery when he tried to solve during his investigation of compound interest, and determined that the answer lay between 2 and 3. ^{[2]} Euler, in his *Introductio in Analysin infinitorum, *found the real value of *e* using this limiting process. He also used summation to prove its value, in the equation .

Later calculations by Euler and French mathematician Charles Hermite (1822-1901) confirmed *e* as an irrational and transcendental number, meaning that one cannot state *e* as the quotient of non-zero integers *p* and *q* nor as the root of a single variable, non-zero polynomial equation with integer coefficients.

*e* ranks as one of the five most fundamental numbers of mathematics, falling in beside *i, *π, 1, and 0. In a previous post, I discussed the history and meaning of the number *i. *A future one will deal with π. In a final post, we will bring these five numbers together in the remarkable equation known as Euler’s identity.

##### [1] A history in verse of the number e

##### [2] The number e

I am afraid my mathematics is not strong but from this I gather e extends to an infinite number of decimal places. I think I must go back at some point and read your earlier blogs before you post about pi.

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Yes, that is correct. e is an irrational number that cannot be expressed as a fraction of two non-zero integers. Expressed in decimal form, the number would have infinite non-repeating digits to the right of the decimal point. An example of a number that can be expressed as a fraction and that has an infinite number of repeating digits to the right would be the decimal for 1/3, which is 0.33333… 1/3 is a rational number because 1) we can write is as a fraction, and 2) while its decimal form extends to an infinite amount of decimals, the digits repeat.

I’m a bit of a math enthusiast, so I’m glad you’ve taken an interest in my math posts. I recommend the short imaginary number series. In my opinion, it’s a fascinating history.

Thank you for stopping by!

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