your a bit fast here buddy, x/0 is not 0 its infinite

unless when x = 0, then its a wash (or something)

If you have an infinite amount of 0 oranges then you have x oranges, right?

fyp

I guess you are gonna play out some mathematical trick with this so i prefer to not be on record for allowing something that involves an undefined infinite to becoming a condenced number x.

0/0 is Indeterminate*. X/0 is Undefined.

dr.math


* X Beers/X little white pills is also indeterminate.

So the equation x = x is indeterminate, and the equation x = x + 1 is undefined?

1=1. 0=0. Are those equations or statements? Or an axiom? 3=4 seems indeterminate.

I'm lost as to the second equation being relevant.

Every number is a solution to the equations x = 0/0 and x = x, while no number is a solution to the equations x = 1/0 and x = x + 1, is the connection.

I'm fairly sure that modern math decrees that both 0/0 and 1/0 are undefined, although that's not a convention I like much. Some equations have no solutions, while others have more than one solution is all.

When we say 0/0 is undefined, we're saying that this combination of symbols is without meaning. It doesn't say anything about a solution to an equation.

When we write 0/0 as an indeterminate form, we're talking about the result of a limit where the naive calculation leads to an expression of the form 0/0, not that 0/0 is the actual limit.

I used to think that 0/0 ought to be defined as something, but I no longer live in that dream.

Except that it is meaningful to ask how many zeros make zero.

Is it "meaningful" if different people can answer the same question with different answers?

Also, you're choosing only one of many potential interpretations of expressions of the form a/b. Consistency across a spectrum of perspectives would be useful for something like this.

Yes.

What are the main interpretations, other than the number of b's that make a?

This is something where I would expect you'll find much disagreement. Non-uniqueness of answers to questions like these leads to a lack of well-defined terms and objects.

One area where you lose meaning is in the establishment of the equivalence relation that defines what it means for fractions to be equal.

The standard definition of the equivalence of the fractions a/b and c/d is that ad=bc.

So you would (unsurprisingly) get that 0/0 is equal to all other fractions. However, then you would lose transitivity because while 0/0 = 1/2 and 0/0 = 1/3, you do not have that 1/2 = 1/3, so this would require a substantial redefinition of the equal sign where it no longer has transitivity. And this is a huge problem because we implicitly use transitivity of equations throughout even even the most elementary levels of algebra.

This isn't a violation of the concept of a fraction in particular, but more of a violation of the concept of a number in general. We want a = b and b = c to mean that a = c when we have numbers. Defining 0/0 to be all numbers would violate that.

That's fine, because I don't think it's all that important anyway. Have it as undefined. Although I would argue that, pedagogically, this tends to give the concept an unnecessary mystique.

Fair point. The definition could be amended, of course, into something less clean and therefore perhaps with its own mystique.

Interesting. I'd say that trying to define 0/0 is what creates mystique. You now have this one symbol that's kind of all numbers at the same time, but with special rules. Very mysterious.

I understand (and teach) that dividing by zero is bad because it leads to inconsistent results, and math doesn't do well with inconsistency. In my head, that's a down-to-earth practical understanding of it.

It doesn't necessarily lead to inconsistent results if one is careful and amends other definitions accordingly, which is arguably more trouble than it's worth. Division by zero would be a multivalued function.

In my dream, things are normally undefined because to define them would involve a category error, not because defining them causes inconvenience. Am I labouring under another delusion?

I think Ted Chiang wrote a story about this.

reflexive-symmetric-transitive-properties

Without quoting anyone above;

1= 1/1

A/B = C (for all real numbers) [this can only be true if other operations also consistently apply)



0/0 = 0 if true will chaos rule the math world?


0/0=1 if true will chaos rule the math world ?

I share a small fascination with 0/0 and other undefined thingamajigs. 0/0 can also be expressed as "don't ****ing know" or "math doesn't go there."

Possibly. There are lots and lots of ways of looking at it for which division by zero doesn't work, and only a few for which they do.

For example, if division by zero were considered meaningful, then the binary operation we call "division" (div(a,b) = a/b) would not just be a multi-valued function for a=b=0, but it would be undefined for where a != 0 and b = 0. So you're expanding the domain by a single point, and that single additional point takes this perfectly nice function and turns it into something else. To me, that seems like a bit of a category error because the thing you're adding in the extension of the function is completely unlike everything else having to do with that function. It just feels very misplaced.

When looking at it from an algebraic perspective, you can ask what 0^{-1} (zero inverse) is, and you find that it cannot be a real number, which again makes it completely unlike every other numbers, since the multiplicative inverse of every other real number is a real number.

I also think that division by zero leads to problems for other number systems, like the surreal numbers, for exactly the same reason it fails in the real numbers.

Having said that, one context in which division by zero (the x/0 where x != 0) makes sense as "infinity" is by thinking about slopes of lines, as long as we think of infinity as a one point compactification of the real line (by bringing the real line into a circle with infinity as the connection point between the positive and negative numbers on the "opposite" side of zero). Now when we vary slopes continuously, we get a nice smooth (and meaningful) function from lines to the slopes and infinity would work perfectly well (as long as you bring both positive and negative infinity together into a single point instead of leaving them far away from each other). But this example doesn't tell you anything about 0/0.

Sorry, "multi-valued function" was the wrong term for me to use here, as in this case a/b would have to be the empty set.

There is a definition out there somewhere of a "multi-function" from X to Y as a function from X to the power space of Y, which can include the empty set. Y (and X) is a topological space, and its power space consists of the set of all its closed subsets together with one of three topologies: lower, upper, and Vietoris. Google has bits and pieces on this, but I'm struggling to find a good reference (not a good sign, I agree).

I think I agree that adding a point at infinity is okay topologically, although it plays havoc with the algebra, of course.

Both, I shouldn't wonder. You either have to take the standard line that 0/0 is undefined, or define it as the set of all numbers (in whatever number field you're talking about, e.g. the real line), which is the position I am trying to defend (somewhat as devil's advocate).

Well maybe "undefined" and "the set of all numbers" means the same thing. When you define something you are really limiting it, from being everything to being something smaller than everything.

E.g from the starting point a definition may go something like this: "something", then you can say "something that has magnitude", and then further say "something that has magnitude and direction".

The more you define it, the smaller it gets, or the less it contains. So something undefined is something that is really everything, or here "the set of all numbers".

So x=0/0 is strong enough defined so we know that it can only be something from the set of all numbers, we know its not gonna be a "duck" or something totally random. But x is too weak defined for us to be more specific than the set of all numbers.