Nerdology 101: Mathematical Constants

I'm going to start making some (not all) my posts related to the epicness of nerdiness. So, let's see how this goes.

This post will discuss some common (and not so common) mathematical constants. Personally, I think that it is important that a well educated nerd knows the basic mathematical constants. If you are a nerd of any kind, you should read this (ignore the calculus parts if you don't understand it). I will divide this into an basic and advanced section (for thoose of you not so mathematically inclined, skip the advanced section). So, let's get started!!

Basic:

Archimedes' Constant: When you ask someone to name a mathematical constant, the one that they will most likely answer is pi. I mean, who doesn't like pi(not to be confused with 'pie')? Most people who know what pi is (if you don't, continue to read this post, otherwise you are a n00b) can name the first few digits: 3.14 or 3.14159 if they have above average nerdiness. I, myself know 3.14159265358979, which is the first 15 digits, which is pretty good. However, when you think of people who know pi, you think of the people that can go 3.14159265358979323846264338327950288... or the pi song. Now, for those of you don't know what pi is used for, it is used for calculations relating to circles or any shape containing circles (sphere, cylinder, etc...). Pi represents the ratio of a circle's diameter to its circumference. In terms of calculus, it is the limit as the number of sides of a polygon, of the perimeter of the number of sides circumscribed around a circle over the diameter of the circle.

Napier's Constant: Ah, the natural number. It is pretty well known, as it is used to describe exponential growth, and has its own special logarithm name: The Natural Logarithm. While this number is pretty amazing, it fails to compare to pi's fan base, even though it is more useful for normal people in day to day life than pi. That, and e is incredibly useful in upper levels of math (i.e. Calculus). However, it is not completely ignored, for in 2004, Google announced that it was going to raise $2,718,218,828, which is e multiplied by a billion rounded to the nearest dollar. The decimal representation of this number is 2.71828182845904523536028747135266249. In terms of calculus, it is the limit as n approaches infinity, of 1 plus 1 over n raised to the power of n.

Phi: You probably didn't recognize this constant did you? Does the Golden Ratio ring a bell? It's the same thing! It is the ratio that is most aesthetically pleasing to the eye, it is greatly encouraged in the world of art. In actuality, it is a specific ratio of a smaller side to a larger side, and it commonly found throughout nature (in humans, plants. etc...). So, as you can imagine, this is quite an important number, although similar to e, it is commonly forgotten and ignored. It's decimal representation is 1.6180339887... In terms of Calculus, it is the limit as n approaches infinity of a Fibonacci sequence of n +1 over a Fibonacci sequence of n.

i: I know what you're thinking: "Pssshhh, i doesn't exist". Technically, you're right. i is the mathematical constant of the square root of negative 1, making it the building block of imaginary numbers. Most people either hate or love this constant. This number allows the exploration of a whole different area of mathematics, so its natural that some people hate it. You might be wondering, why the heck do we care about an imaginary number? Well, for 1 thing, there are mathematical applications, plus, if i is raised to an even power, it becomes a real number, making it so that i can be used to help equations that cannot otherwise be solved. Still think it's useless? Well, the quanity e raised to the power of pi times i, plus 1 equals 0. PUT THAT IN YOUR PIPE AND SMOKE IT!!! (Euler FTW)

Pythagoras's Constant: You most likely don't know what this constant is, even though you are actually already familiar with it: the square root of 2. Now, you may be wondering, why the heck is that considered a constant? Well, Pythagoras discovered it first, so naturally he named it after himself. Using the Pythagorean Theorem on a right triangle with legs of 1, the hypontanuse is the square root of 2. Now, you may be wondering what this constant is actually used for. Well, it is common in trigonometry because square root of 2 over 2 equals the cosine and sine of 45 degrees (or pi over 4 radians). It also used in the aspect ratio of paper sizes so that when you cut a straight line parallel to its short side, that the 2 sheets will have the same ratio (Bet that you didn't think that paper was so complicated!)

Advanced:

Brun's constant: This one is pretty uncommon, and in fact, I can pretty much guarantee that unless your friends are mathematicians, they will not know this constant. This constant notates the sum of the reciprocals of twin pairs of prime numbers that are different by 2. Specifically, this number is approximately 1.902160583104. This proves that there is an infinite number of pairs of prime numbers (two primes that differ by 2). Outside of mathematics, this number has little practical use.

Mills' Constant: Another prime number constant, this number raised to the power of 3 time n generates a prime number. This constant is approximately 1.3063778838630806904686144926... Any prime number generated by this number raised to 3 times n, where n is a positive integer, is called a Mills Prime (i.e. 2, 11, 1361, 2521008887, etc...). This number is useful for finding prime numbers.

Plastic Number (Also called the Silver Number): Is the only real solution to x cubed equal to x plus 1. It is approximately 1.324717957244746025960908854. It is also the largest Pisot-Vijayaraghavan number (meaning that it is a real cubic solution greater than 1, whose other solutions are less than 1 in magnitude).

There are a lot more mathematical constants, but I only touched on a few of them. Now, go and impress/buffalo your friends with your new-found knowledge!!

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P.S. Blogger decided to be annoying, and screwed up the format of my post (extending the page indefinitely to the left) and it took me a while to remedy it.