0/0 and 1/0 different or the same?

[Mr Indeterminate]

I've had division by zero represented to me two ways.

The first way is that all division by zero is undefined. Thus, 0/0, 1/0 and 22687/0 are all the same. All are undefined.

The second way is that division of a non-zero and zero is different to zero divided by zero. In that 0/0 is classed as being indeterminate rather than undefined as it is in all other cases. i.e https://www.khanacademy.org/video/undefined-and-indeterminate

Which is correct? If the second is correct, how would one best describe the difference between indeterminate and undefined?

 

[fresh_42]

I've had division by zero represented to me two ways.

The first way is that all division by zero is undefined. Thus, 0/0, 1/0 and 22687/0 are all the same. All are undefined.

Correct.

The second way is that division of a non-zero and zero is different to zero divided by zero. In that 0/0 is classed as being indeterminate rather than undefined as it is in all other cases. i.e https://www.khanacademy.org/video/undefined-and-indeterminate

What is mathematically the difference? It might be a very, very sloppy way to distinguish limits like $\lim_{x \rightarrow 0}\frac{1}{x}$ and $\lim_{x \rightarrow 0}\frac{x^c}{x}$. I definitely don't recommend such a view. It makes things unnecessarily complicated and ambiguous. The correct terms are the different types of singularities, not "undefined" and "indeterminate". Semantic yak in my ears.

Which is correct? If the second is correct, how would one best describe the difference between indeterminate and undefined?

The short answer is: Zero isn't part of the multiplicative group and the question about its multiplicative inverse is thus meaningless. The division by zero is in my opinion plain nonsense - in any case.

 

[Mr Indeterminate]

Hi Fresh 42,

Please do excuse the late reply, I've been crazy busy! Also, thank you in pointing out which one is correct!

While you have provided a reasoning, would it be possible to frame that reasoning against that Khan Academy video?

I'm having difficulty seeing an issue with the rationale Salman provides :S

You can skip to 4:10 of the video to save a bit of time.

 

[Buffu]

Hi Fresh 42,

Please do excuse the late reply, I've been crazy busy! Also, thank you in pointing out which one is correct!

While you have provided a reasoning, would it be possible to frame that reasoning against that Khan Academy video?

I'm having difficulty seeing an issue with the rationale Salman provides :S

You can skip to 4:10 of the video to save a bit of time.

He starts with $\dfrac{\color{magenta}{0}}{\color{blue}{0}} = K$, then he multiplies with $\color{blue} 0$ to both sides and get $\color{magenta}0 = K \color{blue}{0}$. He canceled out both blue zeroes essentially saying that $\dfrac{\color{blue} 0 }{\color{blue} 0 } = 1$.

So the question now he why did not $\dfrac{\color{magenta}0}{\color{blue} 0} = 1 = K$ ? If he defines $0/0 = 1$.

 

[Vanadium 50]

Thus, 0/0, 1/0 and 22687/0 are all the same. All are undefined.

The fact that they are all undefined does not mean they are equal. Is fribnitz the same as globbiwobby?

 

[mathwonk]

to me a/b means a number that multiplies b into a. so x = 0/0 means x.0 = 0. this is true for all numbers, which is why some people say 0/0 is indeterminate, i.e. not determined by the defining phrase. on the other hand 1/0 = x means that x.0 = 1, a rather difficult challenge for any finite number.

since however in the realm of polynomials we have z which equals zero at the origin, and we have 1/z which has limit infinity at the origin, and since (1/z).z = 1, some people consider 1/0 to be like 1/z at the origin, hence "equal" to infinity.

 

[Mark44]

to me a/b means a number that multiplies b into a.

It took me a little while to parse this sentence, trying to reconcile a/b with "that multiplies b into a." Clearer would be "a/b is a number k, so that k *b = a."

so x = 0/0 means x.0 = 0. this is true for all numbers, which is why some people say 0/0 is indeterminate, i.e. not determined by the defining phrase. on the other hand 1/0 = x means that x.0 = 1, a rather difficult challenge for any finite number.

since however in the realm of polynomials we have z which equals zero at the origin, and we have 1/z which has limit infinity at the origin

If z is restricted to positive numbers only, then yes, $\lim_{z \to 0^+}\frac 1 z = \infty$. Without that restriction, the limit doesn't exist in any sense.

, and since (1/z).z = 1, some people consider 1/0 to be like 1/z at the origin, hence "equal" to infinity.

 

[Stephen Tashi]

. i.e https://www.khanacademy.org/video/undefined-and-indeterminate

That video does not deal with a topic that is part of any particular system of mathematics - i.e. it is not a question about the real number system , it is not a question about limits of functions etc.

The video deals with what might be called "meta-mathematics". It discusses how a philosopher might reason when trying to extend a system of mathematics to handle divsion by zero.

The distinction is analogous to the difference between 1) Discussing how an event in a baseball game is handled by the rules of baseball - versus- 2) Discussion how baseball would be affected by changing or adding rules to the game of baseball.

There is no axiom or definition within the mathematics of the real numbers that says "'fribnitz' is undefined" or that says "'1/0' is undefined". So when you say that

I've had division by zero represented to me two ways.

you can't claim that division by zero has been represented within any standard mathematical presentation of the real numbers. (You can claim that division by zero has been discussed in various ways by people talking philosophically and informally.)

If we step outside the scope of the definitions and axioms of the real numbers, we can ask "What would happen if we try to change the axioms and definitions by adding a definition for '1/0' ? That's what the video is doing.

 

[mathwonk]

"If z is restricted to positive numbers only, then yes, lim, z→0+, of 1/z=∞. Without that restriction, the limit doesn't exist in any sense."

sorry, i guess i tend to think of complex numbers, where it is more natural to compactify by adding only one point at infinity, which of course you can also do for the reals if you like. in that sense then 1/z does go to "infinity", if you define infinity properly, but you are right that is not the usual one variable calculus definition. thanks.

 

[PeterO]

I've had division by zero represented to me two ways.

The first way is that all division by zero is undefined. Thus, 0/0, 1/0 and 22687/0 are all the same. All are undefined.

The second way is that division of a non-zero and zero is different to zero divided by zero. In that 0/0 is classed as being indeterminate rather than undefined as it is in all other cases. i.e https://www.khanacademy.org/video/undefined-and-indeterminate

Which is correct? If the second is correct, how would one best describe the difference between indeterminate and undefined?

I am worried by your statement
"The first way is that all division by zero is undefined. Thus, 0/0, 1/0 and 22687/0 are all the same. All are undefined."

The 1st and last sentences say the same thing - but the middle sentence I find a worry.

In the modern world of refugees, I can demonstrate my concern. If a refugee camp is full of people who have no proof of identity (their identity is undefined), that does not mean they are all the same. Some may be African, some from the Middle East, some form the Far East. Yes they are all refugees with no proof of identity - but that just shows their identity is undefined.

Similarly, I agree 0/0, 1/0 and 22687/0 as all undefined - but it is a long bow to thus claim they are the same, since saying they are the same strongly implies they have the same value. Better to just say "undefined".

 

[Mr Indeterminate]

Thank you all for your input!

The video does start off with a hypothetical philosopher. Thus as Stephen Tashi suggests, interpreting the video as a presentation of "meta-mathematics" makes quite a bit of sense. However, it is near the end that a problem emerges.

Consider the below transcript portion from the video, beginning at 7:04:

"Now we see a bit nuance here. One divided by zero… you just couldn't define it, it leads to direct contradictions. Zero divided by zero… it could be anything, you can't determine it, and so that's why, when you do higher level math and you'll often see this when you take a calculus course, we see that zero divided by zero is indeterminate"

Thus Salman effectively states, he has provided the reason as to why higher levels of math will refer to 0/0 as having an indeterminate result, rather than just being undefined.

Is this incorrect or does it have a basis?

 

[Stephen Tashi]

Thus Salman effectively states, he has provided the reason as to why higher levels of math will refer to 0/0 as having an indeterminate result, rather than just being undefined.

Is this incorrect or does it have a basis?

By what standard do you wish to judge correctness?

As mentioned before, within the assumptions and definitions for the real numbers, there are no theorems or axioms that say "0/0 is undefined because...". There is no definition that defines what it means for a number to be "indeterminate".

To talk about "0/0", we are not talking about a number. We are talking about a string of symbols. What set of definitions and axioms are we using to talk about strings of symbols?

It is possible to talk about strings of symbols in a rigorous manner. For example, one can study computer programming languages formally. However, the video is not doing this. The video is an informal discussion of the problems that a person would face in trying to extend the definitions of the real numbers so that "0/0" would be defined to be a number. As an informal discussion, I agree with the ideas in the video. However, the video isn't giving a formal proof of anything. It also isn't giving a formal definition of what it means for a string of symbols to be "indeterminate".

There are definitions in mathematics that use the word "indeterminate" in various contexts. Mathematical definitions, when written precisely, do not define single nouns or adjectives. Mathematical definitions define statements, i.e. complete sentences.

In a calculus book, some authors might define the statement:
"The expression $lim_{x \rightarrow a} \frac{f(x)}{g(x)}$ has indeterminate form 0/0"
to mean the statement:
" $\lim_{x \rightarrow a} f(x) = 0$ and $lim_{x \rightarrow a} g(x) = 0$".

Such a definition does not define the symbols "0/0" to be a number. (it also does not define the adjective "indeterminate" as an isolated concept.)

For example, the expression $lim_{x \rightarrow 1} \frac{ 2(x-1)}{x-1}$ has indeterminate form 0/0, but the limit is equal to 2. So, in the context of calculus, "the expression has indeterminate form 0/0" is not the same as saying "the limit denoted by the expression does not exist" or "the limit denoted by the expression is not equal to any particular number".

 

[Mark44]

In a calculus book, some authors might define the statement:
The expression $lim_{x \rightarrow a} \frac{f(x)}{g(x)}$ has indeterminate form 0/0"
to mean the statement:
" $\lim_{x \rightarrow a} f(x) = 0$ and $lim_{x \rightarrow a} g(x) = 0$".

Such a definition does not define the symbols "0/0" to be a number.

Many textbook authors write this as $[\frac 0 0]$, with brackets, to emphasize that this is an indeterminate form, rather than a number, and do the same for the other indeterminate forms such as $[\infty - \infty]$, $[\frac \infty \infty]$, $[1^{\infty}]$, and a few others.

 

[mathwonk]

with respect to post #9, one might think of the projective line as the compactification of the real line with only one point added at infinity, to obtain a circle, rather than a closed interval.

 

[mathwonk]

i think the point of your recent remarks is that although the number 0/0 alone cannot be specified to have a particular value x just by using the definition x.0 = 0, still it is often true that expressions which look as ifn they would be merely 0/0 turn out to have more information allowing them to be evaluated.

e.g. every derivative has this form, i.e. the limit of (f(x)-f(a))/(x-a) as x-->a. This was indeed the origin of the scepticism with which Newton's definition was viewed by people such as Bishop Berkeley who could not get past the confusion that arises by letting both top and bottom become zero before trying to make sense of the limit of the quotient.

 

[Mr Indeterminate]

By what standard do you wish to judge correctness?

I'm going to go with a broad standard and say that the video should provide the intended audience with the correct view on the matter.

It is my expectation that the bulk of the intended audience would be led to think:
(1) 1/0 has an undefined result because there isn't a value for k that could make k.0 = 1
(2) 0/0 has an indeterminate result because any value for k could make k.0 = 0

By this broad definition of correctness, is the video correct?

 

[jbriggs444]

I'm going to go with a broad standard and say that the video should provide the intended audience with the correct view on the matter.

It is my expectation that the bulk of the intended audience would be led to think:
(1) 1/0 has an undefined result because there isn't a value for k that could make k.0 = 1
(2) 0/0 has an indeterminate result because any value for k could make k.0 = 0

By this broad definition of correctness, is the video correct?

That interpretation seems to treat the division operation as something that takes a pair of numbers in the front side and always spits something out the back side. It then casts mathematicians in the role of trying to determine based on the force of reason what that something could be.

In the case of 1/0, the claim would be that the result is undefined. In the case of 0/0, the claim would be that the result is indeterminate.

The mathematical view is that the division operation is only defined when applied to pairs of numbers when the second number is non-zero. It does not make sense to talk about the value of the result that is obtained otherwise. Nor does it make sense to ask whether the non-existent result is "undefined" or "indeterminate". There is no result.

In short, the video may be correct, but that interpretation is not.

 

[Mr Indeterminate]

That interpretation seems to treat the division operation as something that takes a pair of numbers in the front side and always spits something out the back side.

The video explicitly assumes X ÷ Y x Y = X always, regardless of what values X and Y happen to be.

Using that assumption, the video approaches the division of a non-zero by zero, through defining X = Non-Zero, Y = 0 and Non-Zero ÷ 0 = K. In this instance, it is concluded that K could not be equal to any value, for K x 0 = 0 regardless of what value K assumes. Thus a Non-Zero ÷ 0 is undefined.

Contrasting the above, the video then approaches the division of zero by zero, through defining X = 0, Y = 0 and 0 ÷ 0 = K. In this instance, it is concluded that K could be equal to any value, for K x 0 = 0 regardless of what value K assumes. Thus 0 ÷ 0 has an indeterminate result, which is distinctly different from an undefined result.

When I read your post JBriggs444, you state a mathematical view, which I interpret as being different to the video's assumption. In that X ÷ Y x Y = X so long as Y is not equal to zero.

Have I interpreted the mathematical view correctly?

 

[Stephen Tashi]

I'm going to go with a broad standard and say that the video should provide the intended audience with the correct view on the matter.

It is my expectation that the bulk of the intended audience would be led to think:
(1) 1/0 has an undefined result because there isn't a value for k that could make k.0 = 1
(2) 0/0 has an indeterminate result because any value for k could make k.0 = 0

By this broad definition of correctness, is the video correct?

If the intended audience wants an informal explanation, I'll say the video is correct.

The video is not offering a mathematical proof.

 

[Mr Indeterminate]

If the intended audience wants an informal explanation, I'll say the video is correct.
The video is not offering a mathematical proof.

While it doesn't offer a mathematical proof, how should one view 0/0? Is in undefined or is its result indeterminate?

 

[Stephen Tashi]

While it doesn't offer a mathematical proof, how should one view 0/0? Is in undefined or is its result indeterminate?

If you interpret the symbol "0/0" as denoting the division of the number 0 by the number 0, the symbol does not define a number.

Whether you should call the symbol "0/0" "indeterminate" or not depends on what you mean by "indeterminate". If your definition of "indeterminate" is that an "indeterminate" symbol denotes the entire set of real numbers, the symbol "0/0" does not denote that set of numbers.

As mentioned before, the symbol "0/0" might be used in the context of describing the "form" of a ratio of limits. In that context "0/0" might be called an "indeterminate form".