The Functional Equivalency of Zero and Infinity

As I’ve done some more looking at the Factor Field, I’ve discovered an interesting phenomenon. Of course, when I say “I discovered” this I don’t think it means that I’m the first person to think this, but only that I’ve come up with it on my own. Of course, that doesn’t mean I’m not the first person to think of it, but I’ll be the first to admit that I’m not a professional mathematician and only dabble in it for fun, so I’m pretty much clueless as to where the vast majority of theorists are on this subject!

In any case, I’ve come to the conclusion that zero and infinity are functionally equivalent. This isn’t just due to the fact that you can’t divide numbers by either zero or infinity (e.g., 1/0 is “undefined”), but because when you look at the Factor Field you can SEE that these things are graphed IDENTICALLY.

That last part is important to emphasize. These are identical graphs. Which means that if we stipulate that if method A = Graph 1, and method B = Graph 1, then method A = method B, then we have to say that zero = infinity.

So let me show you what I mean. Here’s a cross section of the Factor Field. I’m getting this by using the following VBScript code, which you can run at home if you have Windows and Microsoft Excel:

set objExcel = CreateObject("Excel.Application")
Set objWorkbook = objExcel.Workbooks.Add()
Set objWorksheet = objWorkbook.Worksheets(1)
objExcel.Visible = True

' Use if you want to fill up entire spreadsheet for 2007 format
'MaxWidth = 16384
'MaxDepth = 1048576	

' Use if you want to fill up entire spreadsheet for 2003 format
'MaxWidth = 256
'MaxDepth = 65536   

' Use for a practical demonstration
MaxWidth = 201
MaxDepth = 201

a = 1

for x = (MaxDepth-1) to 1 step -1
	objExcel.Cells(a,1).Value = x
	a = a + 1
next

for x = 1 to (MaxWidth - 1)
	for i = (MaxDepth-x) to 1 step -x
		objExcel.Cells(i,x+1).Interior.ColorIndex = 1
	next
next

a = MaxDepth
for x = 2 to (MaxWidth)
	objExcel.Cells(a,x).Value = (x-1)
next

for x = MaxDepth to (MaxDepth * 2)
	for i = MaxDepth + x to 1 step x
		objExcel.Cells(i,x+1).Interior.ColorIndex = 1
	next
next

a = -1

for x = MaxDepth + 1 to ((2 * MaxDepth) - 1)
	objExcel.Cells(x,1).Value = a
	a = a - 1
next

for x = 1 to (MaxWidth - 1)
	for i = (MaxDepth + x) to ((MaxDepth * 2) - 1) step x
		objExcel.Cells(i,x+1).Interior.ColorIndex = 1
	next
next

wscript.Echo "Done."

Now I should note that this code is fairly “hacked together” so it’s not elegant, but it does the job. And what it does is make this graph (if you scroll to row 201 of the Excel spreadsheet):

Now you can see what I’ve done to modify the Factor Field is to extend the pattern into the negative numbers. But with one important thing missing. There are no cells filled in on the y = 0 line (which holds the number for the x-axis of the graph in the picture). But clearly the pattern continues, because if we look at it with the y = 0 line filled in, it fits the patterns established in every column:

Clearly, if we just look at the pattern itself, the graph wants the y = 0 line to be filled in its entirety. But this causes an immediate problem, because what the Factor Field represents is factors of numbers. That is, when you look at y = 1, you see only x = 1 as a valid answer. That’s because 1 has only 1 as a factor. When you look at y = 2, you see both x = 1 and x = 2, because 2 and 1 are both factors of 2, etc. That’s why when you see y = 9, for instance, you see x = 1, x = 3, and x = 9, because 1, 3, and 9 are the only factors of 9.

But when you look at y = 0, you get x = 1, 2, 3, 4…to infinity.

Because this graph functions to show us factors, that means that you can take the furthest out x value and it will be equivalent to the y value. Or, to state it the other way, y is equal to the highest x value. So if the highest x value is 9, then y = 9. Therefore, if the highest x value is infinity, then the graph holds for when y = infinity.

Infinity = zero on this graph.

But it’s not just that. I’ve been thinking of the mini “proof” I gave about an infinitely long prime number, and it turns out that the “proof” exists only because I stipulated that a number must have itself as a factor by virtue of the identity axiom. But let’s jettison that for a moment and ask ourselves what the proof really requires. Let me restate it here, modified slightly:

1. Let n = 1 x 2 x 3 x 4 … x infinity.

2. Let w be a factor of some number, c.

3. If w is a factor of c, then the first number greater than c that w can also be a factor of is c + w. (e.g., 7 is a factor of 14; the next number greater than 14 that 7 can also be a factor of is 14 + 7, or 21.)

4. Since n is the product of every positive whole number, every positive whole number is a factor of n.

5. n + 1 can have only the factor of 1 (via 3). (Reasoning: if 2 is a factor of n + 1, then 2 cannot be a factor of n; since 2 is a factor of n, the next possible number after n that 2 can be a factor of is n + 2. This holds true for all numbers, therefore the only possible number that can be a factor of n + 1 is 1.)

6. Since n + 1 has only the factor of 1, then n + 1 MUST BE the number 1.

7. If n + 1 = 1, then n = 0.

8. Conclusion: 1 x 2 x 3 x 4 … x infinity = 0.

Pretty amazing, huh?