So this is a highly math-y, highly conceptual post about an idea I came up with today for giving an actual value to the expression 1/0. In this process I create an entirely new number set with a whole bunch of cool new properties. If you’re a math-y person, or you want to be bored, or you like cool new things, then read on and enjoy! 🙂
I was recently thinking that it was kind of silly that an expression such as , so simplistic in nature, returns a value– actually, sorry, not a value, but rather the lack of one, namely: undefined. In fact, I considered it quite disconcerting that something as elegant as also returns that ugly word, undefined; it’s not even a number for crying out loud. Because of this, I decided to propose a simple solution. Similar to the case of imaginary numbers, in which, I believe it would be quite appropriate to create a new number set called the extra-universal number set which allows to have a value, z. I term this set extra-universal because it technically resides outside of the universe, since conceptually the idea of multiplying 0 by a number to reach a finite value sounds impossible, but is not that conceptual discrepancy also present with i? Now, what would be the applications of this new number set? Let us see.
We’ll take a look at some simple arithmetic:
We see from that simple example that the value of z resides somewhere between 1 and -1. We also see – from examining the positive side of the pattern and z, as well as looking at the negative side of the pattern and z – that z must simultaneously be greater than 1 and less than -1. In beautiful, symbolic form:
What else can we do with this fun number? Well, now we can define. It is as follows:
You will notice I did not include the middle term. That is because it is so wonderful I feel that it is fitting of its own place in this document. First we will define the general term of . It is simply equivalent to xz. And now the solution to , an expression ordinarily accompanied by three answers (undefined, 1, and 0), has a value: 0.
We can go even further than this. Division of any number y by any value of z is 0:
The inverse of z, z-1 results quite elegantly in 0, thus the series of equations above. z2 is merely z itself, while the square root of z is also z. This means that any “z-number” n such that n = xz multiplied by z remains n. In this way, z acts as an extra-universal 1. There is now a rival to 1.
Now z has a very interesting property when it comes to multiplication between it and fractions. Consider the following case:
And now the queer thing happens, because suddenly we’re left with:
Is that not most remarkable? When multiplying z by a fraction of any sort, one can completely forget about the denominator. This means something like this is perfectly possible:
That, ladies and gents, is simply wonderful. This means that any number between 0 and 1 when multiplied by z is simply equivalent to z. The rational numbers do not exist in the extra-universal number set. We can expand the concept to improper fractions:
Something interesting happens though, when the division of the fraction (note it has to result in a whole number for this to apply) is evaluated before multiplying it by z.
In other words, rational multiplication inside of the extra-universal number set is conditionally associative. This conditionality arises from the fact that inside of the extra-universal number set, . However, interestingly enough, does equal 1.5 inside of the extra-universal number set.
Something quite remarkable, and frankly quite beautiful happens when one considers the case of z divided by itself:
Which, based on what we proved above, is equivalent to 0. This means that .
And with that, not as a closing thought, but hopefully as an opening one, I leave you with the concept of z and the extra-universal number set. I’ve opened up what most likely is a pandora’s box in the world of math, and I sincerely doubt this concept will turn into a reality, but it is fun to theorize and postulate, and maybe, if you have the spare time, you might consider expanding it further.