*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. *z*^{2} 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:

Expanding *z*:

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.

*Tours yruly*

I had fun sifting through this. Unfortunately, my nature disallowed me from reading it without stopping all the time to make sure you didn’t make a mistake XD sorry, not trying to find something to call you out on….

Anyway, great job!

Anna B.

P.S. You’ll drive yourself mad if you venture into Algebra….

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lol. I’m glad you enjoyed it. =P I don’t mind that you checked for mistakes. It’s good to look over my work. Several times when I was editing, I did a double-take, just because of how radically different from other math extra-universal numbers are =P

~Michael Hollingworth

Disce Ferenda Pati – Learn to endure what must be borne

P.s. Nah. We’ll save going mad for when I try to blend the concept with imaginary and transcendental numbers XP

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*jaw drops* Oh. My. Storms. Michael your brain is amazing and I am amazed and very very fascinated and if you come up with anything else I will read it and be amazed even more.

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lol. It wasn’t that hard to come up with. =P I had fun X) I’ll probably explore it with Algebra next. But that’ll take a

lotlonger. XDLikeLike