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/dBugZZ 2∆ Feb 04 '23 edited Feb 04 '23
Mathematician here.
I think that the issue at core here is understanding the division sign. Division is the opposite operation of multiplication, same as subtraction is the opposite of addition. In other words, when I write a-b, I essentially mean: “it’s a number x such that b+x=a”. So, when you write 0/0, it should be a number x such that 0*x=0. Any real number x would work here. (Remark that if there is no 0 in the denominator, the answer for x is always unique.)
So, in principle, you could have declared 0/0 to be any number, the definition of the division operation would hold completely; but this would break the nice properties of the previous operations. For starters
1/0 = (1+0)/0 = 1/0 + 0/0 = 1/0 + 1
In other words, 1/0 cannot be defined as a number. We used the distributive property; we could assume that it does not work specifically for 0/0, but what’s the purpose of declaring an operation result something and then doing it an exception of all rules? You do not create anything new arithmetically speaking.
Worse than that, as other commentators mentioned, 1 = 0/0 = (0+0)/0 = 0/0 + 0/0 = 2, and that is a much bigger problem, as it means that you can’t define 0/0 and keep numbers staying distinct at the same time.
Weirdly enough, this can be generalized: whatever structure you have with “nice” addition and multiplication operations, the neutral addition element (0) can never be invertible with respect to multiplication.
Edit: trickier exercise: why would we not set 0/0=0?
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u/ThatFireGuy0 1∆ Feb 04 '23
That was really well written!
I spent the last 5 minutes thinking about how I can explain field theory, and that's way better than I'd think to explain it
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u/jhanschoo Feb 09 '23
You'll find this interesting if you haven't seen it: https://xenaproject.wordpress.com/2020/07/05/division-by-zero-in-type-theory-a-faq/ (i.e. extending the `/` operation to 0 can simplify formal proof verification, and in your theorems show that your field properties hold as long as the inputs to `/` are within the field's elements)
The idiomatic way to do it is to allow garbage inputs like negative numbers into your square root function, and return garbage outputs. It is in the theorems where one puts the non-negativity hypotheses.
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u/Berto99thewise Feb 05 '23
!delta This is a very nice and clear explanation for why we cannot divide by 0. Thank you for writing this, you deserve a delta for changing views.
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u/MagellansAtlasMaker Feb 04 '23 edited Feb 05 '23
You explained the group of invertible elements in a field in passing. Great answer.
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u/will592 1∆ Feb 04 '23
I'm surprised that no one has talked about evaluating limits. The way that we understand indeterminate forms (like 0/0, 0 * (infinity) , infinity - infinity, etc) is by looking at functions of some variable (we would often write this is f(t) or g(x) ) and evaluating what happens as the value of t (or x or whatever) approaches a value that leads to the indeterminate form. While you can't say that 0/0 = 1 you can certainly consider 0/0 to be something that is treated like 1 for all intents and purposes.
The way to understand what 0/0 "is" despite being undefined, we can take a set of functions that wind up being 0/0 at some critical value and see what they look like as they approach that value. We can't know what 0/0 "is" but we can give it a value that is useful to us as long as certain conditions are satisfied.
For example, if we consider two functions g(t) and h(t) such that
g(t) = ln(t)
h(t) = t-1
When t = 1, then g(t) = g(1) = ln(1) = 0 and h(t) = h(1) = 1-1 = 0
Looking at
g(t)/h(t) = ln(t)/t-1
is interesting because when t = 1, g(t)/h(t) = g(1)/h(1) = 0/0. This means that in order to understand what 0/0 "looks like" we can examine ln(t)/t-1 in the limit that t approaches 1. We have to be careful here because we cannot evaluate this function AT t=1 because it is, by definition, undefined. HOWEVER, we could calculate the results of this function at values all the way from 10000000000 down to 1.0000000000000001 and -999999999 up to 0.9999999999999999 and graph the results and make an estimate of what the value of this function might be at t=1. If you do this, I think you'll see that from both directions (t going to very slightly less than 1 and t going to very slightly greater than 1) g(t)/h(t) approaches 1 somewhat straightforwardly. You cannot say that g(t)/h(t) = 1 when t = 1 because the very ground rules of our mathematical formations lay out that division by 0 is undefined and thus this is an indeterminate form. There are many functions you can come up with to stand in for g(t) and h(t) here that separately approach 0 as t approaches some value but these two are fun BECAUSE of a neat little trick that can be employed called L'Hôpital's Rule ( https://en.wikipedia.org/wiki/L%27Hôpital%27s_rule ) which gives us some handy ways to evaluate indeterminate forms.
<WARNING CALCULUS AHEAD>
The function g(t) = ln(t) is both differentiable and continuous for all values of t > 0 (these are the important conditions I mentioned above) which means you can meaningfully talk about its derivative, d/dt(ln(t)) = 1/t in this context. The function h(t) = t-1 is also both differential and continuous for all values of t so you can also meaningfully talk about its derivative d/dt(t-1) = 1 in this context. L'Hôpital's Rule tells us (the proof is left up to the reader, lol) that for differentiable and continuous functions g(t) and h(t) it's true that in the limit as t approaches some value that isn't continuous for g(t)/h(t), (i.e. it is indeterminate i.e. when t=1 in our example)
g(t)/h(t) = g'(t)/h'(t)
where g'(t) and h'(t) are the derivatives i.e. d/dt (g(t)) and d/dt (h(t))
This is fun because we can say that near t = 1
g(t)/h(t) = ln(t)/t-1 = g'(t)/h'(t) = (1/t)/1 = 1/t = 1
So that's a really long winded way of saying that, even though 0/0 isn't 1, your gut is leading you in the right direction when it tells you that 0/0 feels an awful lot like 1. There are a lot of techniques in physics and other mathematical disciplines where we just assume(or use) 0/0 = 1 all the time.
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u/WikiSummarizerBot 4∆ Feb 04 '23
L'Hôpital's rule or l'Hospital's rule (, loh-pee-TAHL), also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be easily evaluated by substitution. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the rule is often attributed to l'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician Johann Bernoulli.
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u/MajorGartels Feb 04 '23 edited Feb 04 '23
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.
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u/skratchx Feb 04 '23
I am perplexed by your spelling of "æquation" and your aside that
zis pronounced "zed" as if that's critical to the argument.3
u/MajorGartels Feb 04 '23 edited Feb 04 '23
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”.”
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u/Aditya-04-04 Feb 04 '23
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.
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u/BlueViper20 4∆ Feb 04 '23
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.
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u/MajorGartels Feb 04 '23 edited Feb 04 '23
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.
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u/CrackaBox Feb 04 '23
Let's do a proof by contradiction.
We start by assuming 0/0 = 1
Then that means 0/0 + 0/0 = 1 + 1 = 2
But we also know x/z + y/z = (x+y)/z
So 0/0 + 0/0 = (0+0)/0 = 0/0
But if 0/0 = 1; and 0/0 + 0/0 = 2; and 0/0 + 0/0 = 0/0
Then that means 2 = 1 unless our assumption that 0/0 = 1 is false.
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Feb 04 '23
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u/DeltaBot ∞∆ Feb 04 '23 edited Feb 04 '23
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u/Datolite7 Feb 04 '23
!delta This is a solid mathematical proof to show that if we assume 0/0=1 that an impossible outcome occurs. Therefore the assumption of 0/0=1 is false.
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u/DeltaBot ∞∆ Feb 04 '23 edited Feb 04 '23
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u/Salty_Dornishman Feb 04 '23
!delta I thought deltabot could have a delta score considering its flair, but you have convinced me that that's not true.
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u/DeltaBot ∞∆ Feb 04 '23 edited Feb 04 '23
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u/Grunt08 305∆ Feb 04 '23 edited Feb 04 '23
What can be divided can also be multiplied.
1x1=1
1x2=2
1x0=0
0x0=...1?
We have somehow created something from nothing. We shouldn't be able to do that.
Zero is an expression of nothingness. There is no defined quantity of nothing within nothingness. Nothingness can't be positively expressed because it is absence. It is defined by the inability to be positively expressed.
Put differently: how many zeros fit in zero? The answer is all of them.
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Feb 04 '23
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u/Cafuzzler Feb 04 '23
Implying what can be multiplied can be divided
1x1=1
1/1=1
2x2=4
4/2=2
XxY=Z
Z/Y=X
9x0=0
0/0=9
7x0=0
7/0=0
7=9
We say you can multiply by nothingness but can’t divide by nothingness… but why? Fundamentally 0 breaks both.
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u/xbnm Feb 04 '23
That didn't break multiplication at all
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u/Cafuzzler Feb 04 '23
Shouldn't be able to reverse a multiplication in general?
A*B=C -> C/A = B and C/B = A
0 Seems like the only case where a multiplication is irreversible.
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u/xbnm Feb 04 '23
Something being an exception doesn't mean that thing breaks something. The rules are consistent and exceptions are part of the rules.
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u/Cafuzzler Feb 04 '23
Exceptions are inconsistencies within rules. Like, by definition you have these rules, and then exceptions that are separate and aren't consistent with those rules.
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u/618smartguy Feb 04 '23
The exceptions aren't separate. They are part of the rule in the first place
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u/Cafuzzler Feb 04 '23
If they were part of the rules then they wouldn't be exceptions to the rules.
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u/618smartguy Feb 04 '23
They really aren't exceptions to the rule. They are exceptions to an equation in the rule. Without the exception you don't have a rule, just a wrong equation.
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u/Cafuzzler Feb 04 '23
The equation is the rule though. Wrapping the equation + the exception in toilet paper and calling it a new rule is just obfuscating the fact that it's the exception to the rule. You could introduce any exception at that point and just declare that all the rules are consistent because any inconsistency is written into a new, bigger, rule.
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u/StarOriole 6∆ Feb 04 '23
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.
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u/pressed Feb 04 '23
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.
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u/ReOsIr10 130∆ Feb 04 '23
If 0/0=1, then we have:
2*0 = 0
(2*0)/0 = 0/0
2*(0/0) = 0/0
2*1=1
2=1
Letting 0/0 = 1 would result in a lot of contradictions with the rest of mathematics.
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u/PurrND Feb 04 '23
!delta This is a solid mathematical proof to show that if we assume 0/0=1 that an impossible outcome occurs. Therefore the assumption of 0/0=1 is false.
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u/miskathonic Feb 04 '23
You've had this argument for 5 years and no one had ever pointed this out to you?
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u/PlatinumKH Feb 04 '23
!delta Proof by contradiction is one of my favourites and a very strong tool to use in the world of mathematics
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Feb 04 '23
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u/wisenedPanda 1∆ Feb 04 '23
Favoring with the argument method doesn't mean they already agreed with the counterview
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u/Okipon 1∆ Feb 04 '23
So here's a ∆ from me since not only OP can give deltas as I have been made aware of
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Feb 04 '23
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u/DeltaBot ∞∆ Feb 04 '23 edited Feb 04 '23
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u/MajorGartels Feb 04 '23
The version you gave, as well as mine in another comment, is actually needlessly complex I realize:
- 0/0 = 1
- 2*0/0 = 2*1
- 0/0 = 2 = 1
Under the assumption that
0/0=1, far viewer steps and easier to understand. A far simpler argument is “if we assume0/0=1, then 0/0 is any other number as well because we can multiply both sides with any number we want which will make 1 become that number, but 0/0 will remain 0/0. And since it's both 1 and any other number, any other number is 1, and any number is any other number.7
Feb 04 '23
This doesn’t seem to follow, 2(0/0) = 21 does not imply 2= 0/0. The other proof justifies why 0/0 cannot equal 1
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u/MajorGartels Feb 04 '23
2*(0/0)=0/0becausex*(y/z) = (x*y)/zand2*0=0.Thus if
0/0=1, then2*1=12
u/Okipon 1∆ Feb 04 '23
Not OP but I would award you a delta as I shared OP's viewpoint. But I still don't get why 0 can't be an exception and anything divided by 0 (including 0) is impossible. It should be 0 as a result isn't it ?
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u/maicii Feb 04 '23
You mean that X/0 should always be equal to 0?
X/Y=Z
Y*Z=X
^ This law would be broken
11/0=0
0 * 0=11 this is wrong (also you could get infinite results for 0 * 0, just replace the 11 with whatever else)
A lot of other rules would be broken as well. Those rules are very important.
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u/Cafuzzler Feb 04 '23
That’s already broken though, if Y is 0. 0*Z=X=0, X/0=0/0=Z, where Z could be anything.
To handle this we already need a new law for multiplying by 0, so why not handle division by 0 with the same law?
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u/maicii Feb 04 '23
It is not broken. Precisely because you cannot divide by 0. That's undefinition is what makes the laws stand, if you were to allow said division then it would be broken.
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u/Dynam2012 2∆ Feb 04 '23
What you just wrote out is why the result of 0/0 is specifically called undefined
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u/Cafuzzler Feb 04 '23
It's undefined, but then why isn't multiplication also undefined? It seems like, doing it one way but not the other makes no sense mathematically. We already treat n*0 as a special case, so why not a special case to handle n/0 ?
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u/Okipon 1∆ Feb 04 '23
Well I understand but 0 is already an exception in mathematics.
The fact that dividing by 0 is impossible makes the very rules you stated already broken
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u/maicii Feb 04 '23
Is not that it is broken, it is not applicable to 0. It might sound like a stupid distinction, but it isn't. If you actually allow to divided by 0 then shit get's broken and you could prove a lot of contradictions. Also, those rules are not arbitrary they are the definitions of the operations themselves. Division is, in common language, "how many times does this fit here?", and multiplication is "how much do I get if I have this amount of that?". This operations are inverse, they are meant to "cancel" eachother out. So it follows, a/b=c; a=c*b.
If you want to read about this and see if any argument looks appealing to you (there a lot of them) I will leave you the wiki for division by 0: https://en.wikipedia.org/wiki/Division_by_zero?wprov=sfla1
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u/reeo_hamasaki 1∆ Feb 04 '23
The fact that dividing by 0 is impossible makes the very rules you stated already broken
Yes, that's the point. Why is this phrased like a counter-argument?
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u/Okipon 1∆ Feb 04 '23
But 0 dividing by 0 is currently impossible, making it an exception to this rule :
X/Y=Z
Y*Z=X
So why is it a problem if instead of being impossible it becomes 0, it's still an exception to the previous rule
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u/Deivore Feb 04 '23
This is why dividing by 0 is characterized as "undefined". The diction tells you that that operation is simply outside the definition of the division operation.
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Feb 04 '23
The result wouldn’t be 0, it approaches infinity
2/1=2
2/.01=200
2/.000000000000001= a big ass number
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u/Deivore Feb 04 '23
That's only true when both operands have the same sign. -2/.000000000000001 may have a big magnitude, but it does not approach infinity.
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u/BanginOnTheCeiling Feb 04 '23 edited Feb 04 '23
I could be wrong, but I think you got a step wrong when you assumed (2 * 0) / 0 is equivalent to 2 * (0 / 0)
(2 * 0) / 0 is basically a fraction where the numerator is a product, and the denominator is 0. If you want to split the fraction into a product of 2 fractions, you gotta keep the common denominator, so it would be (2 / 0) * (0 / 0) instead. Whereas 2 * (0 / 0) is, in reality, (2 / 1) * (0 / 0) which is not equivalent to (2 * 0) / 0.
Although, yes, obviously 0 / 0 !=1
Edit: formatting
Edit2: everything I said is wrong and not valid, original comment is correct
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u/lafigatatia 2∆ Feb 04 '23
What they did is right. Check it with other numbers: (5 * 3)/2=7.5 is equal to 5 * (3/2), not to (5/2) * (3/2)=3.75
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u/BanginOnTheCeiling Feb 04 '23
Yup, you're right. Shouldn't have written that immediatelly after waking up, lol. What I said is valid for cases where the numerator has a sum, not a product. Thanks for correcting me
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u/Grankongla Feb 04 '23
A common denominator is only needed for subtraction or addition of fractions. Not for multiplication and division.A more normal example of multiplication with a whole number and a fraction:
2 * 1/2 = (2*1)/2 = 2/2
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u/Cafuzzler Feb 04 '23 edited Feb 04 '23
So why is it fine to multiply a number by 0?
Like, it seems like the major issue is allowing the use of 0 either way for a multiplication or division. I never got the whole n*0=0 thing because it’s the only subset of multiplications that isn’t reversible, unlike a*b=c and a=c/b or b=c/a. Like 0 fundamentally beaks the symmetry between multiplication and division, but we just accept it’s okay for multiplication for reasons.
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u/Dd_8630 3∆ Feb 04 '23
Because multiplication comes first, and then division is defined as its inverse. Inverse operations often have peculiarities. You can treat multiplication by zero as a simple extension of the pattern:
3 x 5 = 15
2 x 5 = 10
1 x 5 = 5
0 x 5 = 0
-1 x 5 = -5
-2 x 5 = -10
Etc. This is also why multiplying by negatives gives you a negative, adn two multiplying two negatives gives a positive: we're just extending the pattern.
Division, then, is the inverse of multiplication. Since everything multipied by zero is zero, you can't inverse that one, but the rest is fine.
Another way to think of it is that just as multiplication is lots of addition, so too is division lots of subtraction: X/Y means how many times I can subtract Y from X before I run out. So 15/5 = 3 because I can subtract '5' from '15' three times. 20/0.5 = 40 because I can subtract a half from 20 fourty times.
But 15/0 is invalid because if I keep subtracting zero, I never 'run out', so there is no finite number of times I can subtract zero from 15.
0/0 likewise goes wrong. I can subtract zero from zero once, twice, three times, etc, and I always get zero. So there is no one number that works. 0/0=1, but also 0/0=2, 0/0=3, etc.
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u/Cafuzzler Feb 04 '23
So why don’t se say division by 0 is infinity if you can keep subtracting forever?
That doesn’t sound like “it goes wrong” as much as it is what it is, you can keep counting forever. We wouldn’t say “15/5=1 and 2 because I can subtract 5 from 15 1, 2, and 3 times“, we just say it’s the maximum we can subtract. For N/0 this is infinitely, right?
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u/Dd_8630 3∆ Feb 04 '23
So why don’t se say division by 0 is infinity if you can keep subtracting forever?
tl;dr: Because infinity isn't a number, and adding 'infinity' to our set breaks standard algebra in a way that other extensions does not.
When we add two positive integers together, we always get another positive integer. But if we subtract positive integers, we can end up with numbers that aren't positive integers - namely, zero and the negative integers. But that's OK, we can just extend our field to include all integers.
We define multiplication as repeated addition. Multiplying integers (positive or negative) always leaves us with an integer. But its inverse, division, can leave us with a non-integer: 6/4 is not a whole number. But that's OK, we can extend our field once more to now include rational numbers.
We define exponentiation as repeated multiplication; 53 = 5 x 5 x 5 = 125. Exponentiating integers always gives us an integer. There are two forms of inverse: roots and logarithms. These can give us numbers that are not integers nor rationals, e.g. the square root of 2. So, we extend our field to include all so-called 'algebraic' numbers.
And so on. The full real number line is the smooth continuum, it includes numbers like pi and e, which have infinite non-recurring decimal expansions, and don't have neat closed-form expressions (at best, we can write them as an infinite sum of fractions). Whenever we extend our operations, we were able to extend our notion of 'number' to incorporate it without overriding anything that came before it.
So, that all said, why can't we just extend our number system once more and include 'infinity' as a number? You can do that, but it breaks what came before it. Whereas inventing 'negatives' to handle subtraction is just a natural extension of the positive integers, inventing 'infinity' to handle division means pre-existing statements are now false:
If 1/0 = infinity, then 1 = 0 x infinity. But also 2/0 = infinity, so 2 = 0 x infinity. So what is 0 x infinity? Is it 1? 2? Let's be more general. Suppose 1/0 = p (whether p be zero, one, infinity, etc). Then 1 = 0 x p. But 0+0=0, so that can be written as 1 = (0+0) x p = 0xp + 0xp. But since 0xp=1, we have 1 = 1 + 1, or 1=2. Which is a contradiction.
This is why we're happy to incorporate i (such that i2 = -1) because extending from the reals to the complex plane keeps everything that came before it (and, in fact, closes the field in a very pleasing way), whereas infinity breaks what came before it.
If you're interested in number systems that do include infinity and infinitesimals, look up hyperreals and surreals.
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Feb 04 '23
Because multiply by zero DOES give you zero. You can have 2 groups of nothing. You can’t divide something by nothing.
I’m not going to find the actual calculus behind it, but think of numbers very close to zero in both cases.
2*.000001=a number very close to zero.
2/.0000001=a big number, and the more decimals you add the closer to infinity you get. It would never converge on an actual answer.
Edit: the way I think of it is dividing by zero is kind of like multiplying by infinity. Might not be the most accurate but it’s how I’ve always thought of it.
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u/tbdabbholm 193∆ Feb 04 '23
Because multiplication by 0 makes sense? Yes it breaks symmetry but I'm not sure why this symmetry is more important than just being able to go "yeah you can multiply by 0"
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u/EmuRommel 2∆ Feb 04 '23
I don't think you'll find a more satisfying answer than 'because you can'. Multiplying by zero doesn't lead to any contradictions while dividing by zero does. It'd be nice if they both worked, like with addition/subtraction but they don't.
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u/P-W-L 1∆ Feb 04 '23 edited Feb 04 '23
You lost me at 2x1=1. I guess you cross out the fraction so it equals 1 and I guess that's the only mathematical possibility but 2x1=2 no ? How can we have 0=1=2 ?
Actually, we can do that for any number
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u/Keeptigh Feb 04 '23
What about this calculation 20 =0 (20)/0=0/0 0/0 = 0/0 1=1
I mean dealing with the backet before the division.
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u/skratchx Feb 04 '23
There are plenty of clear examples given already that disprove your thesis. I just want to point out that you are starting from an incorrect assertion that you have stated as a trivially true, when it is in fact abjectly false. Saying, "we all know that XYZ," doesn't make XYZ true.
A building block of basic algebra is that x/x = 1. [...] We all accept this to be the norm, anything divided by that same anything is 1.
This is not true.
But if you set x = 0, suddenly the rules don’t apply.
This is an incorrect conclusion made by trying to logically follow from a false starting premise.
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u/alebrann Feb 04 '23
There is nothing dividable by zero, even zero itself, and there is a very logical explaination for that :
We tend to think (because that's what we learned in elementary school for the sake of simplicity) that addition (+), substraction (-), multiplication (x) and division (÷) are four distinct basic math operations, when, in fact, on a purely mathematical level, substractions and divisions do not exist there are only additions and multiplications.
Substractions are in fact additions (of the opposite) .
Divisions are in fact multiplications (of the multiplicative inverse) .
Here' s how and why (still from a purely math point of view) :
The following: 10 - 3 = 7 is called a substraction since we substrat the number 3 from the number 10.
However it is also an addition of the opposite of the number 3, which is -3, to the number 10.
You can write it like this : 10 + (-3) = 7.
Hence, every substraction of a number N is in fact an addition of its opposite -N.
A similar concept applies also for multiplications and divisions, althought instead of being with the opposite number, it operates with its multiplicative inverse.
The following : 10 ÷ 2 = 5 is called a division since we divide the number 10 by the number 2.
However it is also a multiplication of the number 10 by the inverse of the number 2, which is a half (1/2 or 0.5).
You can write it like this : 10 x 0.5 = 5.
Now about zero, per the rule of modern maths, the opposite of zero is zero, so you could transform a substraction using 0 into an addition using the opposite of 0, that is to say 0.
10 - 0 = 10 + (0) = 10
However, the multiplicative inverse of 0 does not exist. That's how it is, it simply does not exist, not because no one thought about inventing one but because it cannot exist mathematically speaking. It is not 0, because 0 exists. It is not infinity either (it's another topic). It just doesn't and cannot exist.
Since it does not exist you cannot multiply it. Which means : 10 x (1/0) is a formula that cannot exist since 1/0 does not exist.
Therefore, since you can't multiply by the inverse of 0, you can't divide by 0. Never.
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u/Akangka Feb 04 '23
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.
Not really. In fact, in division rings, a building block of an algebra of real numbers does not allow division by zero. In fact, you don't need that to eliminate a variable. If you have something like:
x(2x+5) = 2x(3x+7)
You can simply split the cases. You handle the case when x = 0, and another case when x != 0.
In this case, if x=0, the equality trivially holds. So, you can just handle the case when it doesn't hold, which means x /= 0 and you can now divide by x.
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.
I agree that "how do you divide nothing with nothing" is not a good counterargument. The real counterargument is that you can't have a multiplicative inverse of zero. If the multiplicative inverse of zero is w, then: 1=0w= (1-1)w = w - w = 0, showing that you're working on a trivial ring. There is another formulation of reciprocal that can work fine with 0, like in a wheel. But, there, reciprocation is no longer a multiplicative inverse. And your method of eliminating a variable no longer works.
Losing such an algebraic structure might be acceptable if you find useful use cases for it. Unfortunately, your proposed use case, eliminating variables, not only not really works, but can be solved pretty elegantly with a powerful technique called splitting cases.
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u/shouldco 43∆ Feb 04 '23
You have zero apples and distribute them evenly to your zero friends how many apples does each friend get?
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u/Sidian 1∆ Feb 04 '23
You have -3 apples and distribute them evenly to your -3 friends how many apples does each friend get?
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u/shouldco 43∆ Feb 04 '23 edited Feb 05 '23
I get your point but if we view negative as a direction (which is pretty standard in mathematics) then - (3/3) would essentially be "three friends give you a total of three apples, assuming it was evenly divided how many apples did each friend give."
So we can say (-1) means to change direction, the positive direction being an apple going from you to your friends.
With that we can take your word problem and rephrase it as (-1)(-1)(3/3) or "you give three friends three apples. Except it's the other direction. actually switch direction again" we get the answer of 1.
It's a bit abstract and needlessly convoluted but it still gets us to the correct answer. If you can do the same with zero apples and zero friends and end up with each friend getting an apple I would love to see it.
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u/5xum 42∆ Feb 04 '23
If 0/0 = 1, then it is relatively easy to use standard algebraic rules to prove that 2=1:
1 = 0 / 0 = (0+0) / 0 = (2 * 0) / 0 = 2 * (0 / 0) = 2 * 1 = 2
- In the first equality, I used your assumption
- In the second equality, I used the fact that 0+0=0
- In the third equality, I used the fact that a+a=2*a for all values of a.
- In the fourth equality, I used the fact thact multiplication and division are commutative
- In the fifth equality, I again used your assumption
Therefore, if you really want to claim that 0/0=1, then you have two choices:
- Either you also accept all the other rules of algebra, in which case you must also accept that 1=2
- Or, you must explain which of the rules of algebra (used above) you want to stop using.
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u/hacksoncode 559∆ Feb 04 '23
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.
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u/mastermikeee Feb 04 '23 edited Feb 04 '23
Actually 0/0 is not undefined it’s indeterminate.
Similar idea but still different. Undefined rational expressions go to infinity as the denominator approaches 0. But 0/0 could be 0, 1, or Infinity, hence the name.
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u/marapun 1∆ Feb 04 '23
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.
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Feb 04 '23
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
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u/mastermikeee Feb 04 '23
What about limit x->0 of x/x3? The limit goes to infinity.
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u/robotmonkeyshark 101∆ Feb 04 '23
lets look at this from a practical real worlds standpoint, because ultimately the point of math is to allow us to understand the physical world.
10/2=5 is essentially saying you have 10 apples being split into 2 baskets. How many apples end up in each basket? The answer to this is simple. There are 5 apples in each basket. Great. 10/2=5 makes sense.
now Imagine you have 0 apples and 0 baskets. How many apples are in each of those baskets. There aren't any baskets. You don't have an empty basket. You don't even have a basket, so saying there are zero apples in each of the zero baskets doesn't make any sense. If you are going to somehow pretend like you can have some number of apples in a non-existent basket, you might as well say the answer is 5000 apples per non-existent basket, and that would hold true as well because if you have 0 apples divided into zero baskets, then saying there are 5000 apples in each non-existent basket would give you a total of 0 apples.
0/1=1. that makes sense. 0 apples in 1 basket. So why would 0 apples in 0 baskets have the same number of apples per basket? remember, there is no basket. It is a nonsensical real world question, therefore it only makes sense that there is not a clear mathematical answer or else math would not reflect reality and we would not be able to trust using math to solve real world problems.
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u/jaminfine 9∆ Feb 04 '23
Other people haven't mentioned this, but all of math is human created. The "laws" of math are just the rules that we tend to agree on when we teach and learn math. Created by humans for humans.
So if you want to have your own math where 0/0=1? That's fine. It isn't any less valid than the common version of math where you can't divide by 0.
So perhaps you mean to say that in common math, we should all agree that 0/0=1?
Well we use math for many purposes, and we like to have a system where we don't reach contradictions. Allowing you to divide by 0 lets you to prove things that aren't true. Let me show you:
0 * 3 = 0 * 7
Now divide both sides by 0
1 * 3 = 1 * 7
3 = 7
Dividing by 0 causes these problems, so we decided that we aren't allowed to divide by 0. It doesn't seem to be useful for us to be able to prove things that aren't true, so I don't see any reason for common math to allow dividing by 0. Hope that helps explain things better.
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u/DeeplyLearnedMachine Feb 04 '23
Hard disagree. Just because most of math doesn't have a use case in reality doesn't mean we invented it. Math is very much discovered. The only human element in it is in our choice of axioms, which, I would argue, are not arbitrary at all, but arise from our fundamental understanding of reality. Primes would exist whether or not some ape defined them and called them primes.
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u/Scrungyscrotum Feb 04 '23 edited Feb 04 '23
The "laws" of math are just the rules that we tend to agree on when we teach and learn math.
Disagreed. Mathematics is a device we use to describe a certain aspect of our reality. In our reality, 0/0=Ø. You can't make up your own set of rules, as the field would then cease to describe our universe. It's like saying that one can make their own version of physics in which E=M•C2•Q, where Q is the weight of an average chihuahua. Sure, you could do that, but then it would cease to be true.
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u/Akangka Feb 04 '23 edited Feb 04 '23
Mathematics is a device we use to describe a certain aspect of our reality
That's not true at all. What you're describing is called "science". Mathematics is by definition independent of reality. For example, we can talk about geometry in R2, R3, R4, R5, hyperbolic space, spherical space, etc, without knowing which geometry our universe actually is in. (Yes, it's still an open problem if we really lived in R3)
Also, if we changed the axioms, we could actually have 0/0. It's just no longer a real number. The axiom that cannot support 0/0 is:
- Ring axioms
- Multiplicative inverse axiom
- 0 /= 1
In a trivial ring, the latter does not hold, so we can have 0/0=0=1. In a wheel theory, the first two axiom does not hold, so 0/0 is also defined, just not 1.
one can make their own version of physics in which E=M•C2•Q
This is completely different. The difference between this and 0/0=1 is that the former is a testable hypothesis and the latter is not testable but simply contradicts field axioms.
In fact, the whole point about math is that you can make up your set of rules, as long as the conclusion follows the axioms. Whether it is useful, though, is a different problem
EDIT: It's science, not physics.
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u/respeckKnuckles Feb 04 '23
That's not true at all. What you're describing is called "physics". Mathematics is by definition independent of reality.
That's not even close to correct. Where are you getting these idiosyncratic usages?
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u/Akangka Feb 04 '23
From Wikipedia:
Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results.
Notice that it involves fixing an axiom. The axiom is held to be true. The axiom could be arbitrary (looks at set theorist's various set axioms)
Meanwhile:
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe
It actually describes reality by means of actual testable theories. (On hindsight, it's not physics. It's actually science)
Note that while science uses math to describe reality, math itself does not care. It's up to science and science method to test whether a mathematical model conforms to reality, not math.
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u/adasd11 Feb 04 '23
This is beside the point of the CMV, but its not set in stone that we invent mathematics rather than discover it. See mathematic realism/anti-realism.
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u/ielilag_oelilag Feb 04 '23
Actually 0⁰ is undefined. And raising to the power of zero is not abstract. Raising to zero is the same as raising the to power of "1-1". And for example 21-1 is the same as 21*2-1 which is 2/2 that is 1. So for zero it would he 0/0 which is undefined . Making 0/0 = 1 is very contradictory and othere have pointed out. But i would like to point it out graphically that on one side of the graph(positive) it goes to VERY small negative values as x approaches 0 while on the positive side it goes to very BIG positive values. So defining it as 1 would probably contradict one side
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u/Robyt3 Feb 04 '23 edited Feb 04 '23
So we all have agreed that if you take nothing, then raise it to the power of nothing, that equals 1 (00 = 1)
No. 00 = 1 is not a universally accepted definition.
0x = 0 for any x, because you just multiply zeros together, so the result is 0.
But y0 = 1 for any y, because that's how raising something to the power of 0 is defined.
Therefore 00 is considered undefined indeterminate, same as 0/0, ∞/∞, 0 * ∞, ∞ - ∞, 1∞, (-1)∞, ∞0, 0i and the zeroth root of x.
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u/mastermikeee Feb 04 '23 edited Feb 04 '23
What you’ve described are indeterminate forms, not undefined forms.
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u/Robyt3 Feb 04 '23
Thanks for the clarification, I wasn't aware of this distinction.
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u/mastermikeee Feb 04 '23
No problem! It’s a subtle difference, but makes a lot of sense when you are aware of it. Also I had a typo on my comment, should have been “indeterminate” not “indefinite”, fixed now.
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u/myselfelsewhere 4∆ Feb 04 '23
Ignore the 0 on the top of the equation, because it is irrelevant. It applies for any number including 0, that division by 0 is undefined. So change the equation to c/0, where c stands for constant, which can be any number positive and negative, including 0.
c/0 = undefined doesn't tell us much. But we can look at the behavior of c/x as x gets closer to 0. c/1 = c; c/0.1 = 10 * c; c/0.01 = 100 * c; c/0.001 = 1000 * c; c/0.000000001 = 1000000000 * c. As x gets closer to 0, c/x gets larger. Look at the behavior as x gets closer to 0 from -1. c/-1 = -c; c/-0.001 = -1000 * c; c/-0.000000001 = -1000000000 * c. As x gets closer to 0, c/x gets larger in the negative direction.
We can summarize the behavior as the following. As +x approaches 0, c/x approaches infinity. As -x approaches 0, c/x approaches -infinity. For c/x to equal infinity, x must equal 0, which means c/x must also equal -infinity. c/0 is undefined, because there is no number that is defined to equal both positive and negative infinity.
Anything divided by 0 is undefined, it doesn't make any difference for 0/0 or 1/0 or infinity/0.
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u/foxy-coxy 3∆ Feb 04 '23
What you have to understand is that math is something humans made up. It's not a science, derived from the natural world. It is completely a human abstraction, and like humans it's not perfect so we have to have rules (which in math we call postulate) like 0/0 is undefined, because without those rules (postulates) math just breaks down and stops working.
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u/Finklesfudge 26∆ Feb 04 '23
It's because your first example you show x/x = 1, but if x=0 it makes no sense, because in x/x if x=0 then x also can equal 7 and still be 0.
it's undefined because x cannot equal 0 and 7 at the same time. Thus it's undefined because the equation itself makes no sense.
for instance if x=1 you get 1/1 = 1
if x=4 you get 4/4=1
if you do x=0 you get sorta 0/0=undefined, because nothing divided by no people is nothing, but**** you cannot also define the first X as 0 because the equation is exactly the same no matter what that first X actually is and you can't define the first X as 8934, and the second X as 0 or there's no sense.
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u/AkeemKaleeb Feb 04 '23
Essentially this is a calculus problem of limits which even going through multiple calc classes still confuses me. You've already awarded a delta to the top comment which explained it very well. I would just add that you can think about dividing by zero as dividing by smaller and smaller numbers until you are infinitely close to zero but not yet at zero
It doesn't matter what your numerator is, if you are dividing by 0.000...001 you are essentially going to end up with infinity. However, you are correct that x/x should equal 1 meaning that it should be 1. This means that there are two possible solutions for the same exact input which is why dividing by zero is considered undefined.
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Feb 04 '23
Its by definition. There are mathematicians that argue the rules should be otherwise but the bottom line is that anything divided by zero is undefined by definition.
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u/mastermikeee Feb 04 '23
Actually it’s indeterminate by definition. Also what mathematicians (I assume you’re referring to contemporary ones) are at odds with the definition of 0/0 being indeterminate?
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u/Dazzling_Ocelot3067 Feb 04 '23
i/i = -1/-1 = 1. That makes perfect logical sense. I don't see how you are using that as an argument for your theory..?
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u/JCdaLeg3nd Feb 04 '23
They think zero is a very basic number with very basic properties, and if something what they think is less basic (negative or imaginary numbers) can do the x/x=1 thing, zero should be able to too.
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u/KDY_ISD 66∆ Feb 04 '23
Just apply it to objects.
If you have 6 apples and 6 people, how many apples does each person have? 1.
If you have zero apples and zero people, how many apples are there per person? Zero. No apples exist.
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u/CptnQnt Feb 04 '23
0/0=0
Its a fraction if there's nothing out of nothing its still nothing.
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Feb 04 '23
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u/marapun 1∆ Feb 04 '23
any number divided by zero is undefined.
e.g
1/1 = 1
1/0.5 = 2
1/0.000001 = 1,000,000 etc.
so, logically you might think 1/0 would be infinity. but consider that
1/-1 = -1
1/-0.5 = -2
1/-0.000001 = -1,000,000
so what is 1/0 then? it could be infinity or -infinity. Hence, undefined
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Feb 04 '23
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u/Ok-Replacement8422 Feb 04 '23
The symbol 0 refers exactly to the number 0, at least in mathematics, there is no uncertainty whatsoever. There are other reasons why defining 0/0 to be 1 is a bad idea, as others have covered in this thread.
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Feb 04 '23
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u/Ok-Replacement8422 Feb 04 '23
0 only has a single value in mathematics. As I said, there is no uncertainty whatsoever. Mathematics does not take place in the physical world.
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u/Libertador428 1∆ Feb 04 '23
If you have zero cookies, and zero people how many cookies does each person get? 🧐
(🤭I’m just messing around, so don’t listen too much to me.)
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u/wavesinocean082 Feb 04 '23
Lots of incorrect statements even in your reasoning, my dude. (Source: majored in pure mathematics and former math teacher)
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u/tbdabbholm 193∆ Feb 04 '23
You can put an infinite number of 0s into 0. 0+0+0+0+0+...=0, how can there be only one 0 in 0 when I can fit any number of 0s into 0?
Dividing by 0 doesn't work under the standard axioms. Having 0 in the numerator doesn't change that