Math trivia by Richard Bleil
Today, I had an interesting revelation. Your phone number is a part of the value of pi.
There are 10,000,000,000! different possible phone numbers in the current US phone system (10 digits from 0 to 9), where the “!” notation means “factorial”. Mathematically this means some number, times the same number minus 1, times the same number minus 2 and so on. So, 4! would be 4*3*2*1, or 24. 10,000,000,000! is SUCH a large number that, frankly, I can’t even calculate it. But, in the end, it will be a FINITE number. Huge, yes, and it may be SO large that to our eyes, yours and mine, it LOOKS infinite, but it’s not.
Pi is an irrational number. That means that the digits of pi go on forever. Never ending. Most people know pi to 3.14. I happen to know pi to 3.14159. On my calculator, the value of pi reads 3.141592654. Computer scientists have figured out the value of pi to over 31 trillion digits, and counting. And this, by the way, is NOWHERE NEAR the true absolute value of pi, because there are an INFINITE number of digits.
Pi is an “irrational number”. The formal definition means that it cannot be expressed as a simple fraction (22/7 is actually a decent approximation to pi, but even that gives 3.143, which is already wrong in the third decimal point). What this means is really two-fold. First of all, the number of digits after the decimal point go on forever. The second part of this, however, is that THESE DIGITS NEVER REPEAT! Take the simple fraction 50/27, for example. This is a rational number, because it can be written as a simple fraction. In decimal form, however, this comes to 1.851851851851… This is not rational because the digits end, but rather because the digits “851” begin to repeat over and over again. I cannot imagine an application where we will need a trillion digits of pi (although in astronomical calculations the distances are large enough that it is possible we would need so many digits for navigational purposes), but computer scientists are still trying to expand the known digits because they are looking to see if they can, indeed, find a pattern.
So, here is the logic. Your phone number is one of ten billion factorial possible combinations of digits, but this number is so small compared to the number of different digits in pi that, somewhere in those digits, whether we currently know enough digits or not (and we probably do not), your unique phone number, including the area code, does show up somewhere.
Is your mind blown? Let’s take this further. Not only does it show up once or twice, but it shows up …are you ready for this?…an INFINITE number of times.
Infinity is kind of freaky like that. The implications are just mind boggling. And, yes, my mind is boggled. Many (or most) players of fantasy games play with this kind of thing frequently. I knew a character who had a box. He could open it, take the box inside of it out, place the box that was on the outside into the box that was in the inside and close the lid. This can’t happen in the real world. In any real world box, the box itself, no matter how thin it is, has some finite thickness. We cannot make a box where that thickness is so small that the volume inside of the box equals the volume outside of the box to allow this to occur (without some kind of “expanding box” trick that defeats the purpose of this discussion), so it is impossible to make a box that can both hold and fit into another box.
But, we could if the thickness were “infinitesimally small”. We’ve been discussing infinitely large number, but there are also infinitely small numbers. These are numbers that are so small that they are actually zero, without actually being zero, much like my love-life.
Notice that the fraction I gave as an example of a rational number had three digits that repeated. A fraction like 1/3 has just one, 0.33333333 where the 3 repeats forever. We can have one digit repeat forever, or two, or three, or fifty-eight trillion, or any number that we wish, and if that repeats, the number is rational. So, if we have a fraction of two numbers that have digits that go on forever, because it’s a fraction, it’s a rational number. Take this same fraction and add, say, 1 to the numerator, and the slice of rational space between the two numbers would be infinitesimally small. In this infinitesimally small slice, there are an infinite number of irrational numbers between the two rational ones.
If anybody is smoking dope while reading this entry…their heads just exploded.