Hello,

This is my first post here on the community, so I apologise for any errors I make and if this is frankly just an unworthy discussion topic. I discovered something neat, shared it with my teacher and a few others who found it interesting also, and now I'm hoping to find some help in finding the rest of its applications. Nevertheless, let's begin.

Last year (junior year), I was dosing off in math class and went home that night having absolutely no clue what we learned. I later found out we worked on piecewise functions. That night, as I attempted to do the homework, I was stumped. Setting up all of these equations into one scenario was ridiculous! I persevered. I tried using certain exponents that would decrease or increase or do anything at all to help, depending on where x was, with no avail. Nothing was quite working, until I got this hunch about using absolute values and some weird fractions. I went to class the next day, showed my teacher what I found and she said I was on to something. I still didn't do the homework right, though,, so I lost a few points there, but whatever.

What I created, I initially called binary mathematics but I was nearly certain that must be a title already taken, and fearing plagiarism, I gave my little bits I use the name existence values. The way they work is this:

Start with an x and a p. The p is here is a point somewhere along the x-axis. You could think of it as a vertical line if you wish. Let's call that p the point of transition, or the point of discontinuity (p is actually a hole you will later find out). When we have a piecewise function, we have function 1 bound to the left of p, and function 2 bound to right of p. Either one can claim point p as there own, it does not matter. Simple, enough, right?

Now erase these piecewise pieces and leave that p in your head. What I'm going to throw at you now is one of my existence values. It looks kind of goofy at first, but I'll explain (I use v for existence values because e was already taken by that weird number we use for stuff like compound interest and natural logs, neither of which I enjoy dealing with, considering I do not bank often and logarithms are the antimatter of fun):

v = (|x - p|/(x - p) + 1)/2

If that looks too weird at first, here it is in words: The absolute value of x minus p, divided by x - p, plus 1, and all of that divided by 2. How this function work is just like a yes/no statement. If x is greater than p, then |x - p| is positive, its denominator is the same and positive so that fraction creates a 1, that 1 is added to the other one, and then that newly created 2 is divided by 2 to create a 1. If x is less than p, then |x - p| is positive because it's an absolute value, its denominator is negative but off the same "value" so that fraction creates a -1, that -1 is added to the 1 to create a 0, and as we all know 0/2 = 0. Child's play, right? It's simple arithmetic.

Now what you're probably asking is, "Matthew, why are you wasting my time with this foolish piece of rubbish?" Well here's why. What these existence values are are little bits of greater than/less than statements we can actually incorporate into a function. What I just showed you was a greater than bit. If exceeded p, we had a 1, a yes, and if it was less than p, we received a 0, a no. If we wished to have it the other way, a less than bit, we just swap the locations of x and p in the existence value (try it out if you're skeptical). Finally, we can get back to the piecewise dilemma.

To make use of these existence values, all you have to do is multiply them by a function. Let's start with a simple one: f(x) = 3x. Ugh, you know what? I feel like that 3x just isn't working for me today. He needs to take a break. He works too hard, going all the way to negative infinity, and as we all know, negativity just isn't good to have in our lives. I'll go ahead and help him out and cut out all of that negativity and a little extra. If we set p as 2 and use our greater than existence value and multiply 3x by our v so that f(x) = 3xv, what we create is a slightly altered f(x), in which 3x works just as he did before, but he doesn't start until 2 units to the right, at which point he resumes where he was. (I would add in a picture, but it's late and I have an athletic competition tomorrow).

Now that 3x has cut out all of the negativity in his life, I think he needs a friend, someone that completes him. 3x couldn't handle the negativity, but I think his pal x^2 can. He has a way of turning a negative situation into a positive one. First off, let's make another existence value and rename the other one we had. For the original one, let's make v sub g, and this new one is v sub L. Why? Because the first one is a greater than bit and this one is a less than bit. For the new one, just swap where x and p were so they're 2 - x. If we take f(x) and add in our new pal and multiply him by the new existence value, he will exist to the left of point p and no more and each piece of this new f(x) can operate independent of the other. f(x) = (v sub L)x^2 + (v sub g)3x. What we just made was a very simple piecewise-sort-of function, merging our functions together into one, and not having to use statements off to the side to bind where our pieces fit (don't say that the existence values are off to the side, because we could very well substitute them in here unlike a piecewise function).

Why does any of this matter? Boy, I wish I had more time to write out everything that I've found so far, but the over-arching theme I have found for the applications is in creating a function that a computer or graphing calculator can process by simply plugging in your p-values and slopes that would typically require human logic to work out. The initial problem I created these to solve was one amount of taxes one pays according to the different tax rates at different tax brackets. What I have also recently done is use them to calculate the Joules of energy inserted/required in a substance to produce a certain change in temperature K, including as the substance goes through phase changes. All of this can be done by using multiple existence values, along with tricks on how to make the last value of a function stick all the way to the right or left, without the rest of its changes continuing on after p.

Like I said, I wish I had more time to explain well and add in pictures of values working in their prime, but I do not, so for now, tell me what you think. Am I on to something? Does this seem practical or at least interesting? Could this have uses in programming or massive computation (both of which I know little to nothing about, but hope this could be that extra piece they might be needing)? And most of all, did someone already do this?

I appreciate any feedback and hope to make more posts about this topic if it's worthy to display my current findings and what I later learn.

Thanks for your insight,

Matthew