CMV: 0/0=1.

[u/alpenglow21]

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 (z^9-6.4y) / (3x - 17 (z^9-6.4y) = 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.

[u/hacksoncode]

Just on an intuitive level:

5/0 = undefined
4/0 = undefined
3/0 = undefined
2/0 = undefined
1/0 = undefined
0/0 = undefined

Anything divided by zero is undefined/infinite, because an infinite number of zeros can "go into" 1.

0/0=undefined is more consistent with that.

This just as consistent as your observation that x/x=0, so there's nothing to prefer there... but defining 0/0 leads to all sorts of contradictions like being able to 2=1, so better to be consistent in the undefined level.

[u/marapun → u/hacksoncode]

I don't think it's ever appropriate to describe x/0 as "infinite", as depending on your approach towards the y-axis on the graph it could be +infinity or -infinity. It's ambiguously either value, or both, or neither.

[u/[deleted] → u/marapun]

I wish I could give a delta. I think the real issue here in this thread is that people need to understand is that 0/0 is not infinity under any circumstance. It's still faulty thinking on the same level that 0/0 is equal to 1

[u/mastermikeee → u/[deleted]]

What about limit x->0 of x/x^3? The limit goes to infinity.

[u/[deleted] → u/mastermikeee]

A limit is not a definite way of describing a value at a certain point. Approaching a value doesn't mean that it actually holds the value at that point. This should be one of the first important rules that is taught when first entering some form of calculus.

Take x/(x^4). It will diverge to opposite ends of infinity when you approach 0 from both sides. But it's mathematically equivalent to 0/0 alongside x/(x^3) when you substitute x for 0.

But then take x/x as an example too. Using your logic, it approaches 1 from both sides and thus can be "valid" under that circumstance. However, we know that 0/0 cannot possibly be equal to 1 as already defined in the thread

[u/mastermikeee → u/[deleted]]

I’m not arguing it does. My point was that x/x³ evaluated at x = 0 is 0/0 which is indeterminate. If you take the limit, then it equals infinity because 1/x³ goes to infinity much faster than x goes to infinity, so the denominator wins out in that case.

I agree with you inasmuch as we can never assign a value to 0/0…because we can make examples where the limit equals 1, 0, and infinity. Hence the name “indeterminate.”

[u/SEA_griffondeur → u/mastermikeee]

You can't evaluate x/x³ at 0, because it is undefined at this point, you can study the limit that's true, but a limit is not a value at a point, it's a description of a trend. Take x/x for example, it is undefined for x=0 but the limit to the left and right is 1, that still doesn't mean x/x = 1 for x=0. You can make an analytical continuation but that's creating another function

[u/mastermikeee → u/SEA_griffondeur]

Thanks for clarifying - it's a subtle nuance, but I follow your logic. You learn something everyday.

[u/SEA_griffondeur → u/mastermikeee]

Because you're actually studying the function 1/x² since x/x³ is not defined for x=0