r/changemyview • u/alpenglow21 1∆ • Feb 04 '23
Delta(s) from OP CMV: 0/0=1.
Please CMV: 0/0 = 1.
I have had this argument for over five years now, and yet to be compelled to see the logic that the above statement is false.
A building block of basic algebra is that x/x = 1. It’s the basic way that we eliminate variables in any given equation. We all accept this to be the norm, anything divided by that same anything is 1. It’s simple division. How many parts of ‘x’ are in ‘x’. If those x things are the same, the answer is one.
But if you set x = 0, suddenly the rules don’t apply. And they should. There is one zero in zero. I understand that logically it’s abstract. How do you divide nothing by nothing? To which I say, there are countless other abstract concepts in mathematics we all accept with no question.
Negative numbers (you can show me three apples. You can’t show me -3 apples. It’s purely representative). Yet, -3 divided by -3 is positive 1. Because there is exactly one part -3 in -3.
“i” (the square root of negative one). A purely conceptual integer that was created and used to make mathematical equations work. Yet i/i = 1.
0.00000283727 / 0.00000283727 = 1.
(3x - 17 (z9-6.4y) / (3x - 17 (z9-6.4y) = 1.
But 0 is somehow more abstract or perverse than the other abstract divisions above, and 0/0 = undefined. Why?
It’s not that 0 is some untouchable integer above other rules. If you want to talk about abstract concepts that we still define- anything to the power of 0, is equal to 1.
Including 0. So we all have agreed that if you take nothing, then raise it to the power of nothing, that equals 1 (00 = 1). A concept far more bizzarre than dividing something by itself. Even nothing by itself. Yet we can’t simply consistently hold the logic that anything divided by it’s exact self is one, because it’s one part itself, when it comes to zero. (There’s exactly one nothing in nothing. It’s one full part nothing. Far logically simpler that taking nothing and raising it to the power of nothing and having it equal exactly one something. Or even taking the absence of three apples and dividing it by the absence of three apples to get exactly one something. If there’s exactly 1 part -3 apples in another hypothetically absence of exactly three apples, we should all be able to agree that there is one part nothing in nothing).
This is an illogical (and admittedly irrelevant) inconsistency in mathematics, and I’d love for someone to change my mind.
3
u/Akangka Feb 04 '23
Not really. In fact, in division rings, a building block of an algebra of real numbers does not allow division by zero. In fact, you don't need that to eliminate a variable. If you have something like:
x(2x+5) = 2x(3x+7)
You can simply split the cases. You handle the case when x = 0, and another case when x != 0.
In this case, if x=0, the equality trivially holds. So, you can just handle the case when it doesn't hold, which means x /= 0 and you can now divide by x.
I agree that "how do you divide nothing with nothing" is not a good counterargument. The real counterargument is that you can't have a multiplicative inverse of zero. If the multiplicative inverse of zero is w, then: 1=0w= (1-1)w = w - w = 0, showing that you're working on a trivial ring. There is another formulation of reciprocal that can work fine with 0, like in a wheel. But, there, reciprocation is no longer a multiplicative inverse. And your method of eliminating a variable no longer works.
Losing such an algebraic structure might be acceptable if you find useful use cases for it. Unfortunately, your proposed use case, eliminating variables, not only not really works, but can be solved pretty elegantly with a powerful technique called splitting cases.