zero divided by zero - one, or undefined?

[VirtualLarry]

just had that thought. any number divided by itself is 1 (identity), right?

but any number divided by zero is undefined, right?

which rule wins out?

Edit: apparently, it is "indeterminate".

https://en.wikipedia.org/wiki/Indeterminate_form

 

[Dude111]

Well 0/0 is 0 really (0 - NOTHING)

I have always wondered why calculators cannot properly display 0 divided by ... The answer is 0!!

5/0 - Is 0!!!!!!

There was 1 calculator I remember using in the 80s THAT DID DO IT RIGHT..... I typed in 0/0 and it said 0 (Not DIVISION BY ZERO ERROR)

 

[Hitman928]

Well 0/0 is 0 really (0 - NOTHING)

I have always wondered why calculators cannot properly display 0 divided by ... The answer is 0!!

5/0 - Is 0!!!!!!

There was 1 calculator I remember using in the 80s THAT DID DO IT RIGHT..... I typed in 0/0 and it said 0 (Not DIVISION BY ZERO ERROR)

I don't have time to explain but no, completely wrong. I'll try to post tomorrow with an explanation.

 

[Ancalagon44]

Well 0/0 is 0 really (0 - NOTHING)

I have always wondered why calculators cannot properly display 0 divided by ... The answer is 0!!

5/0 - Is 0!!!!!!

There was 1 calculator I remember using in the 80s THAT DID DO IT RIGHT..... I typed in 0/0 and it said 0 (Not DIVISION BY ZERO ERROR)

No, its not zero.

Try the reciprocal.

Lets say you divide 5 by 0, and get 0. You should be able to get 5 again, by multiplying by 5 again, right? No, you still get 0.

 

[Paul98]

Well 0/0 is 0 really (0 - NOTHING)

I have always wondered why calculators cannot properly display 0 divided by ... The answer is 0!!

5/0 - Is 0!!!!!!

There was 1 calculator I remember using in the 80s THAT DID DO IT RIGHT..... I typed in 0/0 and it said 0 (Not DIVISION BY ZERO ERROR)

That doesn't even make sense, why would 5/0 be 0?

 

[Jeff7]

0/5.
How many times does 5 go into nothing? Zero times.
0/5 = 0


5/0.
How much "nothing" can you fit into 5?

 

[Hitman928]

0/5.
How many times does 5 go into nothing? Zero times.
0/5 = 0


5/0.
How much "nothing" can you fit into 5?

Kind of going along with that, the algebraic answer to a non-zero, real number divided by 0 is undefined as has basically been explained above. The calculus answer would be that it equals infinity as the limit of the equation as you approach zero in the denominator would give you a number that approaches infinity. If you want to know more about calculus / limits, go to this website, it's really good : http://tutorial.math.lamar.edu/ .

As a basic explanation, take this quick graph I made where y = 1 / x :

When you divide 1 by x, the result from 10 down to a little below 1 is easy to predict. However, as you approach x = 0, you can see y "shoots" straight up to very high values. At x = 1E-9 (nano scale), you can see that y is already at 1 billion. As x gets smaller and smaller, y will give bigger and bigger until it's so big that we say it approaches infinity. If you have taken (pre)calculus or go through that website, it's easy to see that lim(1/x) x -> 0 = infinity. If you don't want to bother with that, then technically there is a "hole" in that graph at x = 0 because y has no value there.

edit: to the op, yes, 0 / 0 is indeterminate although again with calc this isn't always true. This is a place where our math system essentially fails. If you really dive into some math, you can find a lot of debate about stuff like this. For example, what is 0^0? Well, any non-zero real number (e.g. 5^0) is 1. But how can you get 1 from zero? So is it 1 or 0? I believe current consensus is that it is 1 though I don't claim to follow this stuff too closely. Another area you can really get into is complex numbers (numbers that include the imaginary plane). Lots of fun to be had there.

 

[Sheep221]

The reason why you can't devide by zero is very simple, you can cut pizza to 2/4/8 slices, yet you can't cut it to zero slices!

 

[SecurityTheatre]

just had that thought. any number divided by itself is 1 (identity), right?

but any number divided by zero is undefined, right?

which rule wins out?

Edit: apparently, it is "indeterminate".

https://en.wikipedia.org/wiki/Indeterminate_form

A number over zero is undefined (and/or Infinity).

It will really twist your noodle when you study limits and realize there are different "kinds" of infinity. In fact, it's possible to prove, using algebra, that one infinity is larger than the other. You can even add and multiply them together to get a different infinity.

It's all about whether or not you can compare the original definition, because infinity as a concept is vague, so depending on how it's defined, it can have different "values" from a mathematical perspective. 🙂

 

[sm625]

just had that thought. any number divided by itself is 1 (identity), right?

but any number divided by zero is undefined, right?

which rule wins out?

Edit: apparently, it is "indeterminate".

https://en.wikipedia.org/wiki/Indeterminate_form

Divide by zero wins. 0/0 cannot be evaluated because you cannot divide by zero. Period. You can prove this with the following expression:

x/x = 1

This is always true, except when x is 0. When x is 0 it instantly blows up with a divide by zero error.

 

[Hitman928]

Divide by zero wins. 0/0 cannot be evaluated because you cannot divide by zero. Period. You can prove this with the following expression:

x/x = 1

This is always true, except when x is 0. When x is 0 it instantly blows up with a divide by zero error.

You can divide by zero though. You just need a calculator smart enough to give you the right answer.
http://www.wolframalpha.com/input/?i=5/0

0/0 is a unique case, however, and has a special designation as indeterminate (not the same as undefined).
http://www.wolframalpha.com/input/?i=0/0

 

[Paperdoc]

50 years ago when I did math in university, we were told that this question had been settled by common agreement among mathematicians: the quotient 0/0 has the value 1, by definition. That's not to say there is a mathematical "proof" or series argument, or anything like that. The practitioners in the field had just made a definition for this specific anomalous situation, and agreed to use it.

Has this changed?

 

[videogames101]

A number over zero is undefined (and/or Infinity).

It will really twist your noodle when you study limits and realize there are different "kinds" of infinity. In fact, it's possible to prove, using algebra, that one infinity is larger than the other. You can even add and multiply them together to get a different infinity.

It's all about whether or not you can compare the original definition, because infinity as a concept is vague, so depending on how it's defined, it can have different "values" from a mathematical perspective. 🙂

Limits which approach infinity can be larger or smaller, but limits aren't the same as the value/concept "infinity", which is not larger or smaller than itself.

 

[Hitman928]

Limits which approach infinity can be larger or smaller, but limits aren't the same as the value/concept "infinity", which is not larger or smaller than itself.

This is not true. If you limit yourself to a scope of mathematics, it is true, but if you take in all current math, it is not.

Using set theory you can show that some infinities are bigger than others. The basic example is this, if you have one set that contains all numbers between 0 and 1, then the total amount of numbers your set contains is equal to infinity. This is true because you can always add another decimal point (e.g. 0.1, 0.01, 0.001, etc).

Now, if you have another set that contains all numbers between 0 and 100, once again, your total amount of numbers in your set is equal to infinity, however, this infinity is larger than your previous infinity because your previous set is a subset of your current set. In other words, your current set includes all the numbers of your previous set and more. This also leads to the concept of infinite infinities. Set theory is actually fairly new to mathematics (~100 years or so if I remember right) but has proven to be fairly powerful.

 

[videogames101]

This is not true. If you limit yourself to a scope of mathematics, it is true, but if you take in all current math, it is not.

Using set theory you can show that some infinities are bigger than others. The basic example is this, if you have one set that contains all numbers between 0 and 1, then the total amount of numbers your set contains is equal to infinity. This is true because you can always add another decimal point (e.g. 0.1, 0.01, 0.001, etc).

Now, if you have another set that contains all numbers between 0 and 100, once again, your total amount of numbers in your set is equal to infinity, however, this infinity is larger than your previous infinity because your previous set is a subset of your current set. In other words, your current set includes all the numbers of your previous set and more. This also leads to the concept of infinite infinities. Set theory is actually fairly new to mathematics (~100 years or so if I remember right) but has proven to be fairly powerful.

the 2 sets used as examples map one to one to each other by multiplying any element in the first set by 100 - i would argue they have the same number of elements

 

[Hitman928]

50 years ago when I did math in university, we were told that this question had been settled by common agreement among mathematicians: the quotient 0/0 has the value 1, by definition. That's not to say there is a mathematical "proof" or series argument, or anything like that. The practitioners in the field had just made a definition for this specific anomalous situation, and agreed to use it.

Has this changed?

I don't know the history of it, but currently, it is definitely not 1, but an indeterminate form in general. Perhaps you are thinking of lim(x/x) x -> 0 (the limit of x/x as x goes to zero)? This is in fact equal to one using L'Hopital's rule. However, lim(x^2 / x) x -> 0 is equal to 2, even though without limits they would both be 0 / 0. This is why 0 / 0 is called in indeterminate in general form as the answer depends on each individual form.

For a very brief, non-rigorous intro to L'Hopital's rule, it states that if your equation is equal to 0 / 0, then you can take the derivate of the numerator and denominator (multiple times if necessary) until you get a non-zero result on top or bottom (or both). This was demonstrated above. Perhaps a better example is the sinc() function.

sinc(2x) = sin(2x) / 2x . If you look at this and try to plot it, at x = 0 you get sin(0) / 0 which is 0 / 0. However, the sinc() function is certainly not 0 at 0 degrees. Use L'Hopital's rule and you get 2cos(2x) / 2 which at x = 0 is equal to 2 / 2.

 

[Hitman928]

the 2 sets used as examples map one to one to each other by multiplying any element in the first set by 100 - i would argue they have the same number of elements

There's a very big problem with your argument that directly contradicts set theory. I'll try to show why and also provide another example.

You have two sets of natural numbers (1, 2, 3, 4. . .), the first is from 1 to infinity, the second is from 2 to infinity. Both contain an infinite set of numbers, but the second excludes the number 1 and is therefore smaller than the first set.

In the first example, you saw that, hey, for every number between 1 to 100, I can take a fraction of one, multiply it, and thus get every set of numbers between 1 and 100 from my set of 0 to 1. This is somewhat true.

The main problem, though, is this. Take the number 10. In the second set [0, 100], you have the number 10. So, from the first set [0,1], you take 0.1 and map it to 10. Fine, except in the second set [0, 100], you also have the number 0.1. So what do you map from the first set to give you 0.1 which is contained in the second? There will always be numbers in the second set that you cannot map from the first set without breaking the set. Hence, the infinite set of numbers from the second set is larger than the first.

 

[Wizlem]

The main problem, though, is this. Take the number 10. In the second set [0, 100], you have the number 10. So, from the first set [0,1], you take 0.1 and map it to 10. Fine, except in the second set [0, 100], you also have the number 0.1. So what do you map from the first set to give you 0.1 which is contained in the second? There will always be numbers in the second set that you cannot map from the first set without breaking the set. Hence, the infinite set of numbers from the second set is larger than the first.

The mapping function is y = 10x so 0.01 maps to 0.1. The classic example from set theory of two sets having the same cardinality is the natural numbers and the rational numbers. These sets map one to one in a slightly less obvious way than your examples but they have the same cardinality which is what you should say instead of "total amount of numbers your set contains." The different sizes of infinity in set theory is usually presented using the real numbers and power sets.

 

[Hitman928]

The mapping function is y = 10x so 0.01 maps to 0.1. The classic example from set theory of two sets having the same cardinality is the natural numbers and the rational numbers. These sets map one to one in a slightly less obvious way than your examples but they have the same cardinality which is what you should say instead of "total amount of numbers your set contains." The different sizes of infinity in set theory is usually presented using the real numbers and power sets.

Whether the mapping is 100x (which is what it would have to be in this example as your bounds are 1 to 100) or 10x, it doesn't matter, the same issue exists with the argument that every number within the [0,1] set is already mapped directly to the same numbers in the [0,100] set as [0,1] is a subset of [0,100], therefore you will not be able to map (1,100] as the numbers you would use to map to those elements have already been mapped to cover the subset.

As far as proper names and formal arguments, I was trying to avoid such as this is not a math forum and contains people from many different backgrounds, so if I start using words like cardinality rather than describing the word, how many people would know what that means? As I said in the beginning, I was just trying to give very basic, non-rigorous examples so that pretty much anyone could at least understand the concept without needing much, if any, formal math education.

 

[disappoint]

0/5.
How many times does 5 go into nothing? Zero times.
0/5 = 0


5/0.
How much "nothing" can you fit into 5?

You can fit an infinite number of nothings into 5. 😉

 

[DrPizza]

I don't know the history of it, but currently, it is definitely not 1, but an indeterminate form in general. Perhaps you are thinking of lim(x/x) x -> 0 (the limit of x/x as x goes to zero)? This is in fact equal to one using L'Hopital's rule. However, lim(x^2 / x) x -> 0 is equal to 2, even though without limits they would both be 0 / 0. This is why 0 / 0 is called in indeterminate in general form as the answer depends on each individual form.

For a very brief, non-rigorous intro to L'Hopital's rule, it states that if your equation is equal to 0 / 0, then you can take the derivate of the numerator and denominator (multiple times if necessary) until you get a non-zero result on top or bottom (or both). This was demonstrated above. Perhaps a better example is the sinc() function.

sinc(2x) = sin(2x) / 2x . If you look at this and try to plot it, at x = 0 you get sin(0) / 0 which is 0 / 0. However, the sinc() function is certainly not 0 at 0 degrees. Use L'Hopital's rule and you get 2cos(2x) / 2 which at x = 0 is equal to 2 / 2.

The limit of x²/x, as x -> 0 is 0, not 2. L'Hopitals rule is fairly easy to "prove." Let's say the limit, as x -> a of f(x)/g(x) evaluates to 0 over 0 by substitution. We can look at the local linear approximation of f(x) at a, which is f(a) + f'(a)(x-a), likewise the local linear approximation of g(x) at a is g(a)+g'(x-a). Since f(a) and g(a) both evaluate to 0, as x -> a, f(x)/g(x) -> f'(a)(x-a) over g'(a)(x-a). The (x-a)'s cancel, bam.

Whether the mapping is 100x (which is what it would have to be in this example as your bounds are 1 to 100) or 10x, it doesn't matter, the same issue exists with the argument that every number within the [0,1] set is already mapped directly to the same numbers in the [0,100] set as [0,1] is a subset of [0,100], therefore you will not be able to map (1,100] as the numbers you would use to map to those elements have already been mapped to cover the subset.

As far as proper names and formal arguments, I was trying to avoid such as this is not a math forum and contains people from many different backgrounds, so if I start using words like cardinality rather than describing the word, how many people would know what that means? As I said in the beginning, I was just trying to give very basic, non-rigorous examples so that pretty much anyone could at least understand the concept without needing much, if any, formal math education.

I saw your previous post as well as this. The size of the set of numbers from 0 to 2 is the same as the size of the set of the numbers from 0 to 100. There's an exact 1 to 1 mapping between the two sets. For any number in your set, if you divide it by 50, it corresponds exactly to one number in the set of numbers from 0 to 2. You cannot find any element of the set of numbers from 0 to 100 which does not correspond to a distinct element of the set of numbers from 0 to 2.

 

[Hitman928]

The limit of x²/x, as x -> 0 is 0, not 2..

Yep, you are right, brain fart there.

I saw your previous post as well as this. The size of the set of numbers from 0 to 2 is the same as the size of the set of the numbers from 0 to 100. There's an exact 1 to 1 mapping between the two sets. For any number in your set, if you divide it by 50, it corresponds exactly to one number in the set of numbers from 0 to 2. You cannot find any element of the set of numbers from 0 to 100 which does not correspond to a distinct element of the set of numbers from 0 to 2.

The first set was 0 to 1, but regardless, the issue is in the distinct. I understand that we are comparing both infinities as being uncountable and by thus they are seen as equal as they are uncountable, but going beyond that into the more formal ranking of infinities using the Aleph number, I think you could make a very strong argument as to why they two sets would not have the same cardinality.

Anyway, I'll let it go, I know the concept of Aleph numbers is still debated amongst mathematicians and I don't claim to be one at that level.

 

[DeathRayLoveMachine]

Whether the mapping is 100x (which is what it would have to be in this example as your bounds are 1 to 100) or 10x, it doesn't matter, the same issue exists with the argument that every number within the [0,1] set is already mapped directly to the same numbers in the [0,100] set as [0,1] is a subset of [0,100], therefore you will not be able to map (1,100] as the numbers you would use to map to those elements have already been mapped to cover the subset.

No, the reason I can't (easily) create a mapping from [0,1] to (1,100] is that one set is a closed subset of the reals and the other is an open subset of the reals. But I can really easily inject the set [0,100] into the set [0,1], thereby proving that |[0,1]| >= |[0,100]|. In fact, given any two real numbers x and y with x != y, I can inject the entire real number line into [x,y].

This leads directly to the counter-intuitive conclusion that any continuous subset of the real numbers has the same cardinality as the entire real number line.

As far as proper names and formal arguments, I was trying to avoid such as this is not a math forum and contains people from many different backgrounds, so if I start using words like cardinality rather than describing the word, how many people would know what that means? As I said in the beginning, I was just trying to give very basic, non-rigorous examples so that pretty much anyone could at least understand the concept without needing much, if any, formal math education.

Don't front, kid. If you have a rigorous argument to show us, then show us. Plenty of people on this forum can understand formal math.

 

[DeathRayLoveMachine]

50 years ago when I did math in university, we were told that this question had been settled by common agreement among mathematicians: the quotient 0/0 has the value 1, by definition. That's not to say there is a mathematical "proof" or series argument, or anything like that. The practitioners in the field had just made a definition for this specific anomalous situation, and agreed to use it.

Has this changed?

That sounds bizarre. What was the specific situation? (ETA: was it something like this?)

When I took real analysis, we constructed a lot of fields/rings/algebras that were extensions of the real numbers, and many of them allowed division by zero. But the result of dividing a real number by zero is not itself a real number, division by zero does not have an inverse operation, and the whole concept of dividing by zero is pretty much useless anyway.

The kicker to all of this is that the result of division by zero isn't computed or proved; you just sort of declare it to exist. You make it an axiom that division by zero has a result, and that's that.

 

[DeathRayLoveMachine]

The reason why you can't devide by zero is very simple, you can cut pizza to 2/4/8 slices, yet you can't cut it to zero slices!

If I divide a number by 3/4, I get back 4/3 of the number I started with. I can't divide a pizza into 3/4 of a slice, though. Does that mean it's impossible to divide by 3/4?