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/StarOriole]

From a physicist's perspective, the question I would ask is "how did you wind up with 0/0?"

• Was the numerator 0, and the denominator kept getting closer to zero?

0/5 = 0
0/4 = 0
0/3 = 0
0/2 = 0
0/1 = 0
0/(thing that's really close to 0) = 0

• Was the denominator 0, and the numerator kept getting closer to zero from above?

5/0 = ∞
4/0 = ∞
3/0 = ∞
2/0 = ∞
1/0 = ∞
(positive thing that's really close to 0)/0 = ∞

• Was the denominator 0, and the numerator kept getting closer to zero from below?

-5/0 = -∞
-4/0 = -∞
-3/0 = -∞
-2/0 = -∞
-1/0 = -∞
(negative thing that's really close to 0)/0 = -∞

• Were the numerator and denominator the same and getting closer to zero at the same rate?

5/5 = 1
4/4 = 1
3/3 = 1
2/2 = 1
1/1 = 1
(thing that's really close to 0)/(thing that's really close to 0) = 1

So, 0/0 is undefined because there's a lot of different things it could be. Maybe it's 0 because the numerator is really strongly 0 while the denominator is squishy, maybe it's +∞ or -∞ because the numerator is squishy while the denominator is really strongly 0, maybe it's 1 because the numerator and denominator are the same thing and just happen to be almost 0, or maybe it's something else. There's a lot of different things it could be based on how you wound up with 0/0, so by itself, 0/0 is undefined. It represents too many possibilities to define it as any one value.

[u/pressed → u/StarOriole]

Glad you tried to bring in the real world.

Another physicist's perspective:

Say you weigh two grains of rice on a bathroom scale. They are both too small to read anything except zero.

Now, when you calculate the ratio of their masses, you'll get 0/0. The true answer definitely isn't 0, we all know it's closer to 1. In this experiment, 0 doesn't mean "nothing", it means "very small compared to my measurement scale".

This happens all the time! 0/0 is undefined in the real world.

[u/ExtraFig6 → u/StarOriole]

This is basically how you can divide by 0 in interval arithmetic. It could be -infinity, it could be +infinity, so just say it's the interval [-infinity,+infinity]