Is the answer to zero divided by zero infinity?
It's a discussion I just heard on BBC radio 2
100+ Answers
Strictly speaking you could. The '/' sign is just a symbol and if you really wanted to you could do math with the definition 0/0=1. But that's not a good idea for several reasons which is why there's a (very) strong consensus to keep it undefined:
Reason 1: It doesn't advance our understanding of mathematics in any meaningful way. As other people have pointed out it could just as easily be 2 or 0 or 17 - this seems arbitrary instead of elegant. On the other hand it destroys some beautiful relationships between multiplication and division (a*(b/c) =/= (a*b)/c) which takes math a step backward.
Reason 2: It doesn't help you with mathematical proofs. The reason we have division in the first place is to be a unique inverse to multiplication. Any time you divide within a proof you will still have to be careful about 0/0 because you can easily end up with the a*(b/c) vs. (a*b)/c case rendering your proof invalid. You'll actually have to be even more careful, verifying that things like (a*b)*c = a*(b*c) won't get you into trouble if one of a,b,c is the result of a division.
Reason 3: It doesn't help you solve any problems in practice. Most real-life problems have units attached to them but units can land you into trouble here. For example Question A is "If 0 people have 0 dollars total, how much money is there per person?" and Question B is "If 0 people have 0 cents total, how much money is there per person?". Assuming 0/0=1 the answer to Question A is 1 dollar and the answer to Question B is 1 cent - very different answers. In both cases you have the same physical situation, no people with no American currency, so you should get the same answer. With this definition you'd either have to check for 0/0 in problems anyway or get strange results that don't correspond to the physical world.
Reason 1: It doesn't advance our understanding of mathematics in any meaningful way. As other people have pointed out it could just as easily be 2 or 0 or 17 - this seems arbitrary instead of elegant. On the other hand it destroys some beautiful relationships between multiplication and division (a*(b/c) =/= (a*b)/c) which takes math a step backward.
Reason 2: It doesn't help you with mathematical proofs. The reason we have division in the first place is to be a unique inverse to multiplication. Any time you divide within a proof you will still have to be careful about 0/0 because you can easily end up with the a*(b/c) vs. (a*b)/c case rendering your proof invalid. You'll actually have to be even more careful, verifying that things like (a*b)*c = a*(b*c) won't get you into trouble if one of a,b,c is the result of a division.
Reason 3: It doesn't help you solve any problems in practice. Most real-life problems have units attached to them but units can land you into trouble here. For example Question A is "If 0 people have 0 dollars total, how much money is there per person?" and Question B is "If 0 people have 0 cents total, how much money is there per person?". Assuming 0/0=1 the answer to Question A is 1 dollar and the answer to Question B is 1 cent - very different answers. In both cases you have the same physical situation, no people with no American currency, so you should get the same answer. With this definition you'd either have to check for 0/0 in problems anyway or get strange results that don't correspond to the physical world.
Consider the definition of division. For arbitrary real numbers x, y, and z, if
[math]\frac{x}{y} = z[/math]
then
[math]x = y \times z[/math]
for nonzero y. But if we let x and y = 0, then:
[math]0 = 0 \times z[/math]
This holds for any z: 0, 42, [math]\sqrt{-1}[/math], and so on. This means that [math]\frac{0}{0}[/math] is undefined.
Strictly speaking, no algebraic expression can be [math]\infty[/math] or [math]-\infty[/math], which are values that are defined to be the absolute upper and absolute lower bounds of all real numbers. When an expression with a free variable, like [math]1/x[/math], is said to be infinity, it usually means that the expression grows larger and larger as its free variable is brought closer to some focal point. For example, [math]1/x[/math] gets larger and larger when the variable [math]x[/math] is positive and is swept closer and closer to zero. In this case, the "limit" of the expression as x approaches zero from the right is said to be infinity ([math]\infty[/math]) because [math]1/x[/math] grows without bounds. Similarly, if [math]x[/math] is assumed to be negative and is brought closer and closer to zero, the expression [math]1/x[/math] will grow in the negative direction. So that limit is said to be [math]-\infty[/math]. The expression [math]1/0[/math] is still said to be "undefined" because it makes no sense (i.e., nothing multiplied by zero is equal to 1) and moreover could be interpreted to be positive infinity or negative infinity depending on how the zero got in the denominator.
A critical difference between limits that evaluate to [math]1/0[/math] and limits that evaluate to [math]0/0[/math] is that we have absolutely no intuition about what the latter case does. In the former case, we know that the expression that led to [math]1/0[/math] either exploded to positive or negative infinity. However, [math]0/0[/math] might have resulted from an expression whose limit is either finite or infinite. For example, as [math]x[/math] approaches 0, the expression:
It's actually a frequent problem on the hairier edges of physics, and there's a whole set of techniques called "renormalization" to rephrase the problems in terms that don't involve undefined quantities. These techniques are very sensitive to the precise formulation of the problem, which is why some recent scientific results (like faster-than-light neutrinos or variable values of alpha) are so puzzling: they make the renormalization techniques impossible, and you end up having to throw out all of physics!
[math]\frac{x}{y} = z[/math]
then
[math]x = y \times z[/math]
for nonzero y. But if we let x and y = 0, then:
[math]0 = 0 \times z[/math]
This holds for any z: 0, 42, [math]\sqrt{-1}[/math], and so on. This means that [math]\frac{0}{0}[/math] is undefined.
Strictly speaking, no algebraic expression can be [math]\infty[/math] or [math]-\infty[/math], which are values that are defined to be the absolute upper and absolute lower bounds of all real numbers. When an expression with a free variable, like [math]1/x[/math], is said to be infinity, it usually means that the expression grows larger and larger as its free variable is brought closer to some focal point. For example, [math]1/x[/math] gets larger and larger when the variable [math]x[/math] is positive and is swept closer and closer to zero. In this case, the "limit" of the expression as x approaches zero from the right is said to be infinity ([math]\infty[/math]) because [math]1/x[/math] grows without bounds. Similarly, if [math]x[/math] is assumed to be negative and is brought closer and closer to zero, the expression [math]1/x[/math] will grow in the negative direction. So that limit is said to be [math]-\infty[/math]. The expression [math]1/0[/math] is still said to be "undefined" because it makes no sense (i.e., nothing multiplied by zero is equal to 1) and moreover could be interpreted to be positive infinity or negative infinity depending on how the zero got in the denominator.
A critical difference between limits that evaluate to [math]1/0[/math] and limits that evaluate to [math]0/0[/math] is that we have absolutely no intuition about what the latter case does. In the former case, we know that the expression that led to [math]1/0[/math] either exploded to positive or negative infinity. However, [math]0/0[/math] might have resulted from an expression whose limit is either finite or infinite. For example, as [math]x[/math] approaches 0, the expression:
- [math]x/x[/math] approaches 1
- [math]4x/x[/math] approaches 4
- [math]x/(4x)[/math] approaches 1/4
- [math]x^2/x[/math] approaches 0
- [math]x/x^2[/math] approaches [math]\infty[/math] or [math]-\infty[/math]
It's actually a frequent problem on the hairier edges of physics, and there's a whole set of techniques called "renormalization" to rephrase the problems in terms that don't involve undefined quantities. These techniques are very sensitive to the precise formulation of the problem, which is why some recent scientific results (like faster-than-light neutrinos or variable values of alpha) are so puzzling: they make the renormalization techniques impossible, and you end up having to throw out all of physics!
What is a-b?
Spend some time thinking about it. Now, how do you explain a-b to someone who knows only how to add? You'll have to explain it only in terms of 'addition', right? How do you do that? Think...
You say, a-b is "that number, which when added to b, gives a". That is, to find a-b, you solve the following problem:
"Find c such that b+c = a"
Now switch to this question:
What is a/b?
Again, spend some time thinking about it. But you won't think much, because by now, you know exactly what I mean! How do you explain a/b to someone only in terms of 'multiplication'? How do you do that?
You say, a/b is "that number, which when multiplied to b, gives a". That is, to find a/b, you solve the following problem:
"Find c such that b x c = a"
Let's now answer the question:
What is 0/0?
Using the recently established definition of division, how do we explain 0/0 in terms of multiplication? We say, 0/0 is "that number, which when multiplied to 0, gives 0".
That is, to find 0/0, we have to "find c such that 0 x c = 0". What do you think the answer should be?
You're right, any number!
Even our query was wrong. It should have been: "What are those numbers, which when multiplied to 0, give 0"?
So, 0/0 is not really one number, it represents many numbers. Pick a number, and that's 0/0. It's like a blank cheque on which you can enter any amount. But how can you have multiple values for a mathematical constant (which 0/0 is supposed to be)? Is it allowed? It becomes a constant variable! Paradoxical, right?
Now, same is not the case with other mathematical constants, like 1, -19, 25000, 3.93, 3+2i, etc. Each number has a unique value. That is why they are called "determinate", since they have single, unambiguous, deterministic values.
But 0/0 doesn't have such a value. It could be anything; it cannot be determined. It's "indeterminate". And such an entity is dangerous for maths; and we do exactly what we do with dangerous people in the society:
So, 0/0 is disallowed in maths.
Image credit:
http://www.123rf.com/photo_16105...
EDIT 1:
Someone below was curious about the difference between 0/0 and infinity (1/0). To understand the difference, just try to apply that definition of division to 1/0. Find a number that when multiplied to 0 gives 1. You'd be amazed to find that there exists no such number!
So, 0/0 exists, but has too many values (all values), but 1/0 doesn't exist at all. It's only an abstraction. As far as maths is concerned, both are dead! You can say that 0/0 dies of overeating, and 1/0 dies of starvation!
EDIT 2:
Someone else in the comments suggested the "divide a into b parts" definition. This explanation is only true when b is a positive nonzero integer, i.e., when b lies in {1,2,3,...}.
It has no meaning for many kinds of b's. Can you:
divide 4 into -2 parts?
Or 8 into 0.5 parts?
Or 6 into 4/3 parts?
Or 2 into 2+3i parts?
Or 9 into √2 parts?
None of these statements make sense. You have to see beyond that trivial definition of "divide a into b parts". In the same way, dividing into 0 parts doesn't make sense. It's nothing special.
It was also suggested that 0/0 is imaginative. That's right. 0/0 is a concept, not a number. It's a concept that's been disallowed for use in formal mathematics. Maths is full of rules. For example, can you tell me what is the value of "7+%4!-6/*"? No one can, because it's nonsensical, and is disallowed in maths.
Spend some time thinking about it. Now, how do you explain a-b to someone who knows only how to add? You'll have to explain it only in terms of 'addition', right? How do you do that? Think...
You say, a-b is "that number, which when added to b, gives a". That is, to find a-b, you solve the following problem:
"Find c such that b+c = a"
Now switch to this question:
What is a/b?
Again, spend some time thinking about it. But you won't think much, because by now, you know exactly what I mean! How do you explain a/b to someone only in terms of 'multiplication'? How do you do that?
You say, a/b is "that number, which when multiplied to b, gives a". That is, to find a/b, you solve the following problem:
"Find c such that b x c = a"
Let's now answer the question:
What is 0/0?
Using the recently established definition of division, how do we explain 0/0 in terms of multiplication? We say, 0/0 is "that number, which when multiplied to 0, gives 0".
That is, to find 0/0, we have to "find c such that 0 x c = 0". What do you think the answer should be?
You're right, any number!
Even our query was wrong. It should have been: "What are those numbers, which when multiplied to 0, give 0"?
So, 0/0 is not really one number, it represents many numbers. Pick a number, and that's 0/0. It's like a blank cheque on which you can enter any amount. But how can you have multiple values for a mathematical constant (which 0/0 is supposed to be)? Is it allowed? It becomes a constant variable! Paradoxical, right?
Now, same is not the case with other mathematical constants, like 1, -19, 25000, 3.93, 3+2i, etc. Each number has a unique value. That is why they are called "determinate", since they have single, unambiguous, deterministic values.
But 0/0 doesn't have such a value. It could be anything; it cannot be determined. It's "indeterminate". And such an entity is dangerous for maths; and we do exactly what we do with dangerous people in the society:
So, 0/0 is disallowed in maths.
Image credit:
http://www.123rf.com/photo_16105...
EDIT 1:
Someone below was curious about the difference between 0/0 and infinity (1/0). To understand the difference, just try to apply that definition of division to 1/0. Find a number that when multiplied to 0 gives 1. You'd be amazed to find that there exists no such number!
So, 0/0 exists, but has too many values (all values), but 1/0 doesn't exist at all. It's only an abstraction. As far as maths is concerned, both are dead! You can say that 0/0 dies of overeating, and 1/0 dies of starvation!
EDIT 2:
Someone else in the comments suggested the "divide a into b parts" definition. This explanation is only true when b is a positive nonzero integer, i.e., when b lies in {1,2,3,...}.
It has no meaning for many kinds of b's. Can you:
divide 4 into -2 parts?
Or 8 into 0.5 parts?
Or 6 into 4/3 parts?
Or 2 into 2+3i parts?
Or 9 into √2 parts?
None of these statements make sense. You have to see beyond that trivial definition of "divide a into b parts". In the same way, dividing into 0 parts doesn't make sense. It's nothing special.
It was also suggested that 0/0 is imaginative. That's right. 0/0 is a concept, not a number. It's a concept that's been disallowed for use in formal mathematics. Maths is full of rules. For example, can you tell me what is the value of "7+%4!-6/*"? No one can, because it's nonsensical, and is disallowed in maths.
I think this is a nice way of explaining the problem:
Consider the possible values of
Clearly we could have chosen any numbers other than 6 and 11,
so we get the weird conclusion that
can equal ANYTHING at all!!
Actually, we say that
is INDETERMINATE.
Now this quantity keeps popping up in our theory when trying to find gradients, so we have to learn how to DETERMINE what
approaches as h approaches 0.
