# e day is ’ere

#### Somerset Times Edition

Week 2, Term One, 2018

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Wednesday 7 February sees the digits in the date align exactly to a very important letter in Mathematics. I am referring to the number "e" which approximates to 2.718 (or 2/7/18).

e is often referred to Euler’s number or Napier’s constant (he used it in his logarithms) but was, in fact, discovered by Swiss Mathematician Jacob Bernoulli in the 17th Century and is another example of Stigler’s law of eponymy, where a discovery is not named after its original discoverer. Speculation abounds around whether or not the letter e was used in honour of Euler but there is no evidence for this. The letter was probably chosen simply because a, b, c and d were already taken. Bernoulli encountered the number e while studying compound interest. The expression for calculating compound interest, where the interest on a principal also gains interest is the principal multiplied by (1+ 1/n)n, with n representing the number of compounding periods. For example, if your money was compounded monthly (n=12), the expression would equal 2.61. As n gets very large, its value approaches e = 2.718…
The nautilus shell takes the shape of a logarithmic spiral (which Bernoulli called spira mirabilis) whose growth is modelled by the function r = aebθ. This implies that as the size of the spiral increases, the shape of each successive curve is unaltered.

e crops up widely in mathematics and like its more famous cousin π, it is both irrational (cannot be expressed as a fraction) and transcendental (is not the solution to any equation).

Another interesting property of e is that it can also be expressed as a sum of an infinite series as follows; e = 1 + 1/1 + 1/1x2 + 1/1x2x3 + 1/1x2x3x4 ……

e is found in both pure and applied mathematics. Population growth, cooling objects and radioactive decay are all linked to this magical number which is also connected to the distribution of prime numbers. Even the normal distribution curve (bell curve) is a function involving e (and π). That seems so random!

Our students first meet e when studying the Calculus which involves both rates of change and accumulation. The function y = e^x has a unique property as shown in the following graph: When x = 1, the y value is 2.718 and the gradient (rate of change) of the line at that point is also 2.718. Furthermore the area under the curve to this point (accumulation) is also 2.718.

e also appears in Euler’s identity which is often called the most beautiful equation in mathematics as it links the five important constants, e, i, π, 1 and of course 0 as follows: e^iπ + 1 = 0

In 2004, Google founders Sergey Brin and Larry Page paid e a major compliment in their IPO prospectus when they valued their stock, to the nearest dollar, at \$e billion or \$2,718,281,828 thus showing that any stock valuation is inherently irrational. Google later used e in a recruitment drive, inviting applicants to search for “{first 10-digit prime found in consecutive digits of e}.com". This clue led to a website with further challenges to overcome for potential candidates. The first 10-digit prime in the digits of e is 7427466391, which starts at the 99th digit.

So please take it easy on 7 February as this date alignment will not occur for another 100 years, unless you prefer not to use the American notation – in which case it will occur all over again this year on 2 July.

And while I’m thinking irrationally…. we should not forget Phi day this year on 1/6/18 which happened in January in the US but will also surface again elsewhere in June. Make that a golden day.

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