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'''Defining ''e''''' <br />
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=== Defining <math>e</math> ===
  
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There are a number of constants in mathematics that are defined by geometry in some way. For example, Archimedes' constant π is defined as the ratio between the circumference and diameter of a circle. Pythagoras' constant sqrt(2) is defined as the length of the diagonal of a square with sides of 1. However, ''e'' is based on formulae rather than geometry.<br /><br />
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There are a number of constants in mathematics that are defined by geometry. For example, Archimedes's constant <math>\pi</math> is defined as the ratio between the circumference and diameter of a circle. Pythagoras's constant <math>\sqrt2</math> is defined as the length of the diagonal of a square with sides of 1. Unlike these constants, however, <math>e</math> is more easily expressed through limits and formulas.
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When Jacob Bernoulli was studying compound interest in 1683, his work led him to a formula:
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[[File:Bernoulli.png|125px|thumbnail]]
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Let's observe what happens to the value of the formula when different values of ''n'' are used:
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Let us consider the formula for compounding interest:
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{| class="wikitable"
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|-
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
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! n !! (1+1/n)^n
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<math>\begin{align}
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|-
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  P' = P\left(1+\frac rn\right)^t
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| 5 || 2.48832
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\end{align}</math>
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|-
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| 50 || 2.69159
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Where <math>P</math> is initial principle, <math>P</math> is final principle, <math>r</math> is the interest rate,  
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|-
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and <math>n</math> is the number of times the interest is compounded. To make analysis simpler, we'll set <math> P = r = 1 </math> and <math> t = n </math>.
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| 100 || 2.70481
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|-
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| 100,000 || 2.71827
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Let us observe what happens to the value of <math>P'</math> when different values of <math>n</math> are used:
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| 1,000,000 || 2.71828
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
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|}
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<math>\begin{array}{|c|c|}\hline
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As can be seen from the table, the value of the formula converges towards a constant number, which in this case is ''e''. Therefore, we can see that Bernoulli's formula is one way to calculate the value of ''e''. <br /><br />
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n & P'\\\hline
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Leonhard Euler was the one who defined this constant with the symbol ''e'' in one of his papers in 1731. He proved that the constant was irrational by finding a formula that was never ending, thereby proving it was irrational. He was also able to find the first 18 digits of ''e'' using a different formula he came up with to define the number. This particular formula is based on the Taylor Series of ''e<sup>x</sup>''. The Taylor Series of ''e<sup>x</sup>'' is:
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5 & 2.48832\\\hline
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[[File:Extaylor.jpg|350px|thumbnail]]
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50 & 2.69159\\\hline
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Since ''e'' is the same as taking ''x=1'' in the series, we get the following formula for ''e'':
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100 & 2.70481\\\hline
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[[File:Eseries.png|350px|thumbnail]]
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100,000 & 2.71827\\\hline
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Therefore, we can see another way of calculating the value of ''e''. Bernoulli's formula is more complicated since the numbers can get complicated quickly because of taking powers of irrational numbers. But Euler's formula is much simpler since it uses relatively common fractions and addition to find the value.
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1,000,000 & 2.71828\\\hline
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\end{array}</math>
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By careful analysis of this table, it can be seen that value of this formula seems to approach some number. The number that this sequence approaches is called <math>e</math>. From this, we can extract a limit that can be used to express <math> e </math> from the formula for P':
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
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<math>\begin{align}
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  e := \lim_{n\to\infty} \left(1+\frac1n\right)^n
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\end{align}</math>
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Leonhard Euler was the one who defined this constant using the symbol ''e'' in a paper he wrote in 1731. Surprisingly, the agreed upon theory is that he named the number ''e'' not after himself but because it was the next letter in the variables he was using.
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Upon further study of this constant, Euler found the following fractional expansion for <math> e </math>:
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
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<math>\begin{align}
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  e = 2 + \frac1{1 + \frac1{2 + \frac1{1+\frac1{1+\frac1{4 + \ddots}}}}}
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\end{align}</math>
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With coefficients <math> 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \cdots, 1, 2n, 1, \cdots </math>. He used this expansion to show that the number is irrational.
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He was later able to find the first 18 digits of <math> e </math> using a different formula he came up with to define the number. This particular formula was based on the Taylor Series of <math> e^x </math>:
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
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<math>\begin{align}
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  e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots
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\end{align}</math>
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Obviously, evaluating this series at <math>x = 1</math> yields approximations of <math> e </math>. This method of calculating <math> e </math> is much easier to work out by hand than the initial formula based on compounding interest as it does not require taking large powers of already long fractions.
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<big>References</big><br>
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Haran, B. [Numberphile]. (2016, December 19). ''e (Euler's Number) - Numberphile'' [Video File]. Retrieved from https://youtu.be/AuA2EAgAegE <br />
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Maor, E. (1994). ''E: The Story of a Number.'' Princeton, NJ: Princeton University Press. Retrieved from http://webwork.utleon.edu.mx/Paginas/Libros/E%20-%20The%20Story%20of%20a%20Number.pdf <br />
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O'Connor, J. J., & Robertson, E. F. (2001, September). The number e. Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/e.html <br />
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Sandifer, E. (2006, February). How Euler Did It. Retrieved from http://eulerarchive.maa.org/hedi/HEDI-2006-02.pdf
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Sýkora, S. (2008, March 31). Mathematical Constants and Sequences. In ''Mathematical Constants and Sequences''. Retrieved from http://www.ebyte.it/library/educards/constants/MathConstants.html <br />
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[[Walther_MA279_Fall2018_topic3|Back to Mysteries of the Number e]]
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[[Category:MA279Fall2018Walther]]

Latest revision as of 23:54, 2 December 2018

Defining $ e $

There are a number of constants in mathematics that are defined by geometry. For example, Archimedes's constant $ \pi $ is defined as the ratio between the circumference and diameter of a circle. Pythagoras's constant $ \sqrt2 $ is defined as the length of the diagonal of a square with sides of 1. Unlike these constants, however, $ e $ is more easily expressed through limits and formulas.


Let us consider the formula for compounding interest:

                $ \begin{align} P' = P\left(1+\frac rn\right)^t \end{align} $

Where $ P $ is initial principle, $ P $ is final principle, $ r $ is the interest rate, and $ n $ is the number of times the interest is compounded. To make analysis simpler, we'll set $ P = r = 1 $ and $ t = n $.


Let us observe what happens to the value of $ P' $ when different values of $ n $ are used:

                $ \begin{array}{|c|c|}\hline n & P'\\\hline 5 & 2.48832\\\hline 50 & 2.69159\\\hline 100 & 2.70481\\\hline 100,000 & 2.71827\\\hline 1,000,000 & 2.71828\\\hline \end{array} $


By careful analysis of this table, it can be seen that value of this formula seems to approach some number. The number that this sequence approaches is called $ e $. From this, we can extract a limit that can be used to express $ e $ from the formula for P':

                $ \begin{align} e := \lim_{n\to\infty} \left(1+\frac1n\right)^n \end{align} $

Leonhard Euler was the one who defined this constant using the symbol e in a paper he wrote in 1731. Surprisingly, the agreed upon theory is that he named the number e not after himself but because it was the next letter in the variables he was using.


Upon further study of this constant, Euler found the following fractional expansion for $ e $:

                $ \begin{align} e = 2 + \frac1{1 + \frac1{2 + \frac1{1+\frac1{1+\frac1{4 + \ddots}}}}} \end{align} $

With coefficients $ 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \cdots, 1, 2n, 1, \cdots $. He used this expansion to show that the number is irrational.


He was later able to find the first 18 digits of $ e $ using a different formula he came up with to define the number. This particular formula was based on the Taylor Series of $ e^x $:

                $ \begin{align} e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots \end{align} $

Obviously, evaluating this series at $ x = 1 $ yields approximations of $ e $. This method of calculating $ e $ is much easier to work out by hand than the initial formula based on compounding interest as it does not require taking large powers of already long fractions.


References

Haran, B. [Numberphile]. (2016, December 19). e (Euler's Number) - Numberphile [Video File]. Retrieved from https://youtu.be/AuA2EAgAegE
Maor, E. (1994). E: The Story of a Number. Princeton, NJ: Princeton University Press. Retrieved from http://webwork.utleon.edu.mx/Paginas/Libros/E%20-%20The%20Story%20of%20a%20Number.pdf
O'Connor, J. J., & Robertson, E. F. (2001, September). The number e. Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/e.html
Sandifer, E. (2006, February). How Euler Did It. Retrieved from http://eulerarchive.maa.org/hedi/HEDI-2006-02.pdf Sýkora, S. (2008, March 31). Mathematical Constants and Sequences. In Mathematical Constants and Sequences. Retrieved from http://www.ebyte.it/library/educards/constants/MathConstants.html



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