The mathematical constant e appears in many settings around mathematics. It holds importance in mathematics along side 0, 1, pi and i. e is an irrational number, meaning it cannot be described as a ratio of two integers. As an irrational number, the decimal place value is infinite. Early mathematicians began calculating the digits of e to 18 decimal places. Leonhard Euler used a technique called a continued fraction to create a fractional representation of e. This representation is pictured below and helps establish a pattern of convergence. The convergence pattern helped to formulate a decimal approximation and develop an understanding of the properties of this constant.
Here is the continued fraction fully written out.
However, using the computing power of our modern day, X. Gourdon was able to calculate up to 1.250 billion digits! You can read his letter here.
The On- Line Encyclopedia of Integer Sequences provide a decimal expansion of e to 50 digits.
In addition to being an irrational number, e is in fact it is a transcendental number. A transcendental number is a number that is not a root of any non-zero polynomial with rational coefficients. If you would like to learn more about transcendental numbers, head over to Mrs. Durrant’s page!.
The YouTube channel Numberphile released a great video on the mathematics behind the number e. Click the screen shot below to watch thier video!
Unlike its fellow constant pi, e is unique in that it was not derived using geometric relations. Instead, it was first derived using compound interest.
Jacob Bernoulli discovered this constant by examining this question about compound interest:
An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?
What do you think happens? Use the compund interest calculator found here to draw some conclusions for yourself!
The expression Bernoulli found, and you learned about by watching the Numberphile video, for compound interst is
$1.00x(1 + 1/n)^n.
He continued by asking, what happens if you compound more and more until you eventually reach continuous compounding. In mathematics, we can interpret continuous compounding usign the limit function. Bernoulli did so and took the limit of our function. He found the limit to be between 2 and 3, but we now know that the limit is e.
In addition to being expressed as the limit of the above function, e can also be calculated as the sum of an infinite series.
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