Extra-universal numbers

Math is fun X)

Math is fun X)

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 aspastedGraphic.png , 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 pastedGraphic_1.png 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 whichpastedGraphic_2.png, I believe it would be quite appropriate to create a new number set called the extra-universal number set which allows pastedGraphic.png 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:

pastedGraphic_3.png

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:

pastedGraphic_4.png

What else can we do with this fun number? Well, now we can definepastedGraphic_5.png. 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 pastedGraphic_5.png. It is simply equivalent to xz. And now the solution to pastedGraphic_7.png, 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:

pastedGraphic_8.png

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:

pastedGraphic_9.png

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:

pastedGraphic_12.png

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:

pastedGraphic_13.png

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.

pastedGraphic_14.png

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, pastedGraphic_15.png. However, interestingly enough,pastedGraphic_16.png 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:

pastedGraphic_17.png

Which, based on what we proved above, is equivalent to 0. This means that pastedGraphic_18.png .

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

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4 thoughts on “Extra-universal numbers

  1. 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….

    Like

    • 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

      Like

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