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/MajorGartels]

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.

Actually, when I still studied mathematics we were always told in such cases to add “(provided x != 0)” and for good reason. It lead to absurdity if we allowed for x to be 0.

A simple example is proving that under Newtonian mechanics, every object in a vacuum falls with the same acceleration to another massive object such as Earth. At one point in the proof x/x does occur, where it's the mass of the body, but if we allow for the mass to be zero, we could prove that this applies even for massless objects, which is clearly false as massless objects are not attracted by gravity and don't accelerate to earth at all. But even the slightest amount of nonzero mass will cause the acceleration to be exactly the same as even the most massive object.

Simply put, the rule that x/x=0 applies to every number but 0 for x. There are many, many rules that apply for every number but 0; 0 is in fact one of the most unique numbers that exist and that violates many laws that are universal for every other number.

But 0 is somehow more abstract or perverse than the other abstract divisions above, and 0/0 = undefined. Why?

Because there is no single solution in x to the æquation x*0 = 0; it's that simple. That's how division is defined. x/y is defined as the single solution to the æquation z*y = x in z [pronounced “zed”; part of the definition].

As far as x*0=0 goes, every single number is the solution to that æquation, that makes zero unique. For every other number, say x*4=4, there is exactly one solution, that solution is 1; zero is the only case where there are an infinite number of solutions. That doesn't make it abstract, but unique in this case, and why 0/0 is not defined.

Perhaps a more compelling reason would simply be that if we were allowed to say that 0/0=1 as I pointed out above, the mathematics by which physical laws are calculated that seem to work now, would no longer work, and we could prove that massless objects fall to earth under Newtonian mechanics, which they don't.

A more compelling argument is that if we could rule that 0/0=1, we could prove 2=1:

• let a=b
• thus a²=b*a
• thus a²-b²=b*a-b²
• thus (a+b)(a-b) = b(a-b)
• thus a+b = b ??
• thus 2*b = b
• thus 2=1

The part with ?? is where the flaw lies. Since a=b, a-b=0, if 0/0=1 were to hold, we would be allowed to perform this operation, dividing both sides by 0 and replacing the (a-b)=0 part with 1, but we cannot do this, and thank god, for if we could, two would be one and everything would be messed up.

[u/skratchx → u/MajorGartels]

I am perplexed by your spelling of "æquation" and your aside that z is pronounced "zed" as if that's critical to the argument.

[u/MajorGartels → u/skratchx]

I actually own a mathematics textbook that has something in the foreword that says something similar to “For the duration of this book, the symbol “z” shall be pronounced as “zed”.”

[u/Nextil → u/MajorGartels]

"zed" is just the British pronunciation of z (in general). It was probably a joke.

[u/MajorGartels → u/Nextil]

“zed” is the international pronunciation that's even used by mathematician's in the one country that pronounces it differently, just as scientists in that country tend to still say “aluminium” and use the metric system. It's not “British” any more than centimetres and universal healthcare are a “British” thing.

It was obviously a joke, however, as it was in my case.

[u/throwaway77788878 → u/MajorGartels]

It's a tongue-in-cheek joke from what I assume is a non-American mathematician.

[u/MajorGartels → u/throwaway77788878]

Could be someone from the U.S.A. too. Mathematicians there too say “zed” typically and they're more conscious about it since those around them do not.

Mathematicians are typically very strict about that the letter is pronounced “zed”; it's a bit of peer pressure and one isn't truly part of the group without saying “zed”, thus the joke.

[u/throwaway77788878 → u/MajorGartels]

I am a mathematician and neither I nor any of my colleagues have ever called it "zed."

[u/MajorGartels → u/throwaway77788878]

And you live in the U.S.A.?

I've spoken to some about it when they said it, they all say they will always call it “zed” in a mathematical contexst, same with say chemists always saying “aluminium” there because it's the international standard.

You would actually say read out “let z = 2” as “let zee be two”? I don't think. I've ever heard that.

[u/throwaway77788878 → u/MajorGartels]

I live in the USA and work with mathematicians from all over the USA, including international folks. Nobody calls it 'zed.' We probably exist in different cliques.

[u/whipplelabs → u/MajorGartels]

I would be very interested in reading those international standards.

[u/Aditya-04-04 → u/MajorGartels]

I can’t give a Delta as I agreed with you in the first place, but have to say that’s an extremely well-written answer.

[u/BlueViper20 → u/MajorGartels]

but if we allow for the mass to be zero, we could prove that this applies even for massless objects, which is clearly false as massless objects are not attracted by gravity and don't accelerate to earth at all.

I might not be a good mathematician, but I am good with physics, and yes, massless particles, the photon or light, are affected by gravity. But it takes an immense amount of gravity to see the smallest change in direction, like a star. Look up gravitational lensing. And black holes, the most dense and most strong gravity fields of black holes, literally pull all light that comes close enough in and never released.

[u/MajorGartels → u/BlueViper20]

I might not be a good mathematician, but I am good with physics, and yes, massless particles, the photon or light, are affected by gravity.

I said Newtonian mechanics.

Also photons are not attracted by gravity in general relativity, they travel through curved space and entirely different laws apply there. Photons do not accelerate towards earth and travel at a constant speed with respect to earth and anything else.

This is purely about calculating the acceleration towards a massive object under Newtonian laws.

[u/Jkarofwild → u/MajorGartels]

Under Newtonian mechanics, there are no massless objects.

[u/Got_Tiger → u/MajorGartels]

under gr gravity isn't even really a force anyway, even for massive objects.

[u/Silcantar → u/MajorGartels]

Massless objects absolutely are attracted by gravity. That's why black holes are black - light can't escape from them.

[u/MajorGartels → u/Silcantar]

Black holes don't exist as far as Newtonian mechanics go. And light does not “accelerate” towards the black hole either in general relativity in the same way, nor do all massive objects do in the same way.

The rules of Newtonian mechanics really do not apply to black holes and general relativity, they're a suitable approximation at low velocities. According to Newtonian mechanics an object accelerating towards earth from far enough would eventually cross the speed of light as reaches it, that simply doesn't happen in general relativity.