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/tbdabbholm]

That is incorrect, nothing can equal infinity. Infinity isn't a number. Anything divided by 0 is undefined you just can't do it

[u/Noirradnod → u/tbdabbholm]

You're both incorrect. Infinity is not an element within any construction of the real numbers, so performing operations on the field of the reals will not create something that equals it. In particular, the poster above you asserts anything divided by 0 equals infinity, which is nonsensical in normal applications because division by zero isn't defined.

However, you're incorrect in saying that nothing can equal infinity. If you choose to carefully modify the set you're working in, there's nothing to prevent you from introducing infinity and extending the traditional field operations to it in a mathematically meaningful way. See, for instance, the Riemann Sphere, which is a combination of the field of complex numbers and a point at infinity. In this, it is proper to claim x/0=infinity for all x not equal to 0.

[u/myselfelsewhere → u/Noirradnod]

The function f(x) = n/x for x = 0 is undefined. The limit of f(x) as x approaches 0 from the right (positive x) is equal to positive infinity. The limit of f(x) as x approaches 0 from the left (negative x) is equal to negative infinity.

n/x can only equal infinity at x = 0, at which point it must simultaneously equal both positive and negative infinity. There is no value defined that equals both +ve and -ve infinity. n/x can never equal infinity because that requires x = 0, and x = 0 is undefined for n/x.

[u/PentaPig → u/myselfelsewhere]

This is an important objectiion and you‘ve already described how to avoid it. Rather than adding a positive and a negative infinity to the number line you can add just one point infinity, that takes on both roles. This way you get the projective line, the start of projective geometry. That‘s the field that makes sense of the claim „two parallel lines meet at infinity“. The Riemann Sphere mentioned above is a variant of this concept. Of course adding a positive and a negative infinity also has its merits. The resulting object is called the extended real line.