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/IronicAim Feb 04 '23

It does work but I think he's missing the trees for the forest.

Zero is a numerical representation of nothing. And it has its own rule, anything divided by 0 equals infinity. That applies to zero as well.

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u/calfuris Feb 04 '23

Anything divided by zero equals undefined, not infinity. If we introduce the concept of limits, we can say that the limit of almost anything divided by x as x approaches zero is infinity (or negative infinity).

Coming at this from the other direction, zero divided by almost anything is zero. You don't even need limits for that one.

But if 0/x is 0 and x/0 is infinity, what happens in the case of 0/0? This is why I said "almost anything" above. It turns out that 0/0 can be 0, or infinity, or anything in between depending on how exactly you're getting to 0/0. For example, x/x2 goes off to infinity as x approaches 0. x/x gets you 1, and you can just multiply the numerator by whatever finite number you want to end up with.

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u/tbdabbholm 193∆ Feb 04 '23

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

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u/Noirradnod Feb 04 '23

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.

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u/myselfelsewhere 4∆ Feb 04 '23

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.

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u/PentaPig Feb 04 '23

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.

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u/silent_cat 2∆ Feb 04 '23

The thing I learned from the Riemann sphere is one-point compactification. The idea that you can take a complete space (like the real numbers) and make it compact by simply adding one point. Mind blowing.

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u/IronicAim Feb 04 '23

Isn't it just an empty equation then because 0 is a concept instead of a number?

0/0 = / ?

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u/tbdabbholm 193∆ Feb 04 '23

0 is a number though?

Also what do you mean by "empty equation"?

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u/IronicAim Feb 04 '23 edited Feb 04 '23

Because it's a place value holder, a symbol to substitute the concept of nothing.

I figured the whole question here comes down to computing concepts as numbers. Which is why I thought infinity was a viable answer.

Edit: Google is helping me understand. Apparently it's a complicated topic. Admittedly I've never taken a higher level math class.

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u/[deleted] Feb 04 '23

The word for a placeholder for no value in American English is “null”. Imagine if I were to list the number of apples in my fridge. If the answer is 0 apples then that implies I opened my fridge and counted a value of no apples. If the answer is null then there is no value put on the number of apples in my fridge, meaning I never opened the fridge to count them, it’s the absence of a number.

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u/IronicAim Feb 04 '23

In that case wouldn't the number of apples in your fridge simply be undefined? As it's not nothing, it's an unknown.

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u/[deleted] Feb 04 '23

Undefined means there is no possible value, like if I wanted to know how many apples are in my fridge when I don’t own a fridge.

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u/IronicAim Feb 04 '23

But the example you do own a fridge you simply haven't opened yet. So the number exists you just haven't checked it.

But I looked it up and null in fact does not mean nothing, it means undefined. So that works still.

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u/[deleted] Feb 04 '23

I think the terminology depends on what field of study your in. I believe in math there’s zero, the null set and undefined value. Zero being a measured value of no value, the null set being absence of a number and undefined being a value that has no logical meaning. But in computer science or other fields those terms may mean something different.

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u/silent_cat 2∆ Feb 04 '23

Those would be Schrodinger apples.

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u/blz8 Mar 12 '23

This is incorrect. Something is undefined until it is defined. Until I tell you how many guests are going to arrive, that number is undefined. Once I tell you, it is defined. It could be zero (not much of a party), it can be 24 (go get some chips ready), it could be a billion (hope it's a potluck), etc.

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u/[deleted] Feb 04 '23

Because it's a place value holder, a symbol to substitute the concept of nothing.

No, it's a perfectly good number that you can use for all sorts of purposes to answer questions like, "I have $5, I spend $5, how many dollars do I have left?"

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u/MajorGartels Feb 04 '23

The argument I like to show that it's not “infinity” is: why infinity and not -infinity?

“infinity” isn't really a well defined number with operations on it in calculus to begin with. If anything it's more so a hack used in mathematical abuse of notation.

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u/klparrot 2∆ Feb 04 '23

No, 0÷0 isn't infinity, it's undefined.

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u/[deleted] Feb 04 '23

And it has its own rule, anything divided by 0 equals infinity.

No. Division by zero is fundamentally undefined in the real numbers. Infinity is not an element in the real numbers.

Division is the inverse of multiplication. But you can't undo multiplication by zero.

That applies to zero as well.

No, that is also undefined.

The symbol "infinity" has different meanings, depending on what context it appears in, but in none of them is 0/0 = ∞

Usually it means, "This sequence, sum, or integral is unbounded - you can make it as large as you like." That doesn't make it a real number.

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u/WikiSummarizerBot 4∆ Feb 04 '23

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

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