The Evasive Irrationals

[Conan Edogawa 837]

 

A quick note: The Standard has been put on hiatus indefinitely following the lack of growth seen in DCC. This would have been the article I would have published in May, but a decision was reached to put The Standard on hiatus then.

 

The Evasive Irrationals, part I

Pi is nearly omnipresent in mathematics - a mere flip in your textbook will confirm that fact, with it being used most obviously in trigonometry and consequently calculus, but also making the rounds in statistics and even number theory.

The area under this bell curve, or a Gaussian distribution, is sqrt(pi).

And it isn’t confined to mathematics, either. It’s popularly celebrated on Pi Day, 3/14, with citizen and corporation alike celebrating the wonderful history of Pi. Even the Bible makes mention of pi, in First Kings:

"And he [Hiram] made a molten sea, ten cubits from the one rim to the other it was round all about, and...a line of thirty cubits did compass it round about....And it was an hand breadth thick...." — First Kings, chapter 7, verses 23 and 26

For anyone wondering, the value measured appears to be approximately 3.14, the first two digits of Pi. There are also preposterous attempts to remember the digits of Pi, with Rajveer Meena holding the Guiness World Record at a lofty 70,000 digits. With all the popularity of Pi, though, comes a certain opposition towards it - namely, the Tau movement, which hails Tau as being superior to its brother, mainly on the accounts of it simplifying equations, the unit circle, and trigonometry. In my opinion, Tau is certainly interesting, but its detriments nearly outweigh the benefits, and so the benefits of a change are minimal are best. Nevertheless, it’s gotten some interest from some titans of the mathematical YouTube community, namely Vi Hart, 3Blue1Brown, and Numberphile. 

With all the love pi receives, however, most other constants languish in neglect. The Golden Ratio commands considerable respect from the math community and the public, with it being presented most prominently in geometry and architecture. However, E, Euler’s number, still remains(mostly) in the dominion of geeks, and other captivating constants, such as γ(the Euler-Mascheroni constant), the Feigenbaum constants(α and δ), and Apery’s constant(ζ(3)) being scarce noticed outside of higher mathematics - that’s all good and well, but it’s time to spot a light on and appreciate the usefulness of these hidden but intriguing constants. For 

 

Euler’s number

Leonhard Euler

We first head on to perhaps the most popular of these constants - e. To go back to its origins, we start with a simplistic finance scenario - consider investing $1.00 into your account with 8% interest, compounded annually. Say you let it sit there for 4 years. Just a little number-crunching arrives  you at (1 + .08)4, namely $1.36. You get angry at your low interest(yes, I know it is quite high in real life, but we will abandon reality for just a second.), and you demand it to be raised to be 100%. They comply, and your further demand it to be compounded monthly. Then grumble and comply, and so you’d have (1 + 1/12)12 ($2.61) dollars by the end of the year. How about daily? Then it’d be (1 + 1/365)365 dollars, or 2.714 dollars. As one may see, the limit as n approaches infinity approaches e, or 2.71828.. Jacob Bernoulli attempted this same question, and thus made the first discovery of e in 1683. However, Leonhard Euler first introduced it as the base for natural logarithms, and later on in calculations concerning cannons. The constant c was used occasionally, but only a short while after, e has firmly supplanted it as its representation.

The limit, expressed formally.

So why is it so important? Well, when attempting to find the integral of 1/x, you find a few rather striking properties: this function, at L(1), is 0, L(x) > 0 when x> 1, L(x) < 0 when x < 1, and L(x) is undefined when x <= 0. More importantly, it seems as though L(a) + L(b) = L(ab), confirming it to be a logarithmic function. All we have to do to find its base is to locate when the area under 1/x equals 1, and that point is 2.718, otherwise known as e.

E is the unique number greater than 1 for which the area under 1/x equals 1.

Furthermore, e is crucial in calculus, as e^x is the only nontrivial function where it equals its own derivative, thus giving rise to its importance in differential equations, most notably the most simplest one, f’ = f.)

E also shares a few sectors of math with pi, namely.. 

Statistics. The function above is the PDF(prob. density function) of the normal distribution with a mean of 0 and a standard deviation of 1.

Complex analysis. For the mathematically inclined, e^x can be represented as a Taylor series(a representation of a function using an infinite sum of functions)

E’s Taylor series

Using the Taylor series for cos(x) and sin(x), one can arrive at..

And plug in pi to receive the elegant equation

Combining what are, perhaps, the 4 most important constants in math - 0, 1, pi, and e.

Number theory. E is transcendental(i.e, it is not the root of an algebraic equation, like y = x2. Other numbers in this captivating group of numbers are pi and ln(2). ) More impressively, it was the first number that was proved to be transcendental that was not created to be transcendental.

And that’s all for part I, folks! In our next issue, I’ll cover the other constants I mentioned above, but in the meantime, celebrate E day(2/7, or February 7th) and admire its enlightening and elegant uses.

Here’s an interesting fun fact before I go - Donald Knuth, a prestigious and perspicacious computer scientist, founder of TeX and Metafont(a font rendering system that uses geometric equations to render glyphs(i.e a linguistic symbol), uses a versioning system based on e., where version numbers will progress in line with the digits of e.