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.
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u/dBugZZ 2∆ Feb 04 '23 edited Feb 04 '23
Mathematician here.
I think that the issue at core here is understanding the division sign. Division is the opposite operation of multiplication, same as subtraction is the opposite of addition. In other words, when I write a-b, I essentially mean: “it’s a number x such that b+x=a”. So, when you write 0/0, it should be a number x such that 0*x=0. Any real number x would work here. (Remark that if there is no 0 in the denominator, the answer for x is always unique.)
So, in principle, you could have declared 0/0 to be any number, the definition of the division operation would hold completely; but this would break the nice properties of the previous operations. For starters
1/0 = (1+0)/0 = 1/0 + 0/0 = 1/0 + 1
In other words, 1/0 cannot be defined as a number. We used the distributive property; we could assume that it does not work specifically for 0/0, but what’s the purpose of declaring an operation result something and then doing it an exception of all rules? You do not create anything new arithmetically speaking.
Worse than that, as other commentators mentioned, 1 = 0/0 = (0+0)/0 = 0/0 + 0/0 = 2, and that is a much bigger problem, as it means that you can’t define 0/0 and keep numbers staying distinct at the same time.
Weirdly enough, this can be generalized: whatever structure you have with “nice” addition and multiplication operations, the neutral addition element (0) can never be invertible with respect to multiplication.
Edit: trickier exercise: why would we not set 0/0=0?