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The stuff on wheels is interesting! I found the paper by Carlstrom (http://www2.math.su.se/reports/2001/11/ http://www2.math.su.se/reports/2001/11/) and it allu

The stuff on wheels is interesting! I found the paper by Carlstrom (http://www2.math.su.se/reports/2001/11/ http://www2.math.su.se/reports/2001/11/) and it alludes to applications, though I didn't see any specifics about applications (to computer science, at least). Thank you for the pointer.

gizmo686 10y ago
If you avoid division by 0, wheels revert back to normal fractions [0]. This should mean that you can use them as a drop in replacement for a rational number datatype. Doing so should allow you to defer checking for division by zero; possible moving the check outside of a tight loop. Granted, this should also be doable as an optimization of normal rational arithmetic. The only use cases I can think of for wheels amount to them being a principled way of adding NaN to the number system. Of course, if history is anything to go by, a hundred years from now someone may look back on this comment the same way we look back on people calling sqrt(i) "imaginary". [0] At least under the explicit construction presented.
- nonbel 10y ago
  This conversation may have moved well above my head, but I mean start with 0/0 = "undefined". I don't see why that should be the case. x = 0/0 x*0 = 0 1*0 = 0 2*0 = 0 ... From this we see that it isn't really that x is undefinable, rather it can be any value at all. There is apparently no issue with an equation having two equally valid solutions (eg quadratic formula), so at what point are there too many?
  - bikenaga 10y ago
    I think I see what you're asking. I think the answer is: Before we start talking, you have to tell me what all the rules are. So when you say "start with 0/0 = 'undefined'", what is the ambient number system? The proof I gave earlier showed the number system can't be a ring - so it's not the real numbers, the integers, the rationals ... at least not by the standard definitions of those number systems. When you do math, you don't make up the rules as you go. As far as the example you gave goes, you start by assuming 0/0 is defined. But if you're trying to show 0/0 is defined you're assuming what you want to prove. The logic isn't correct. Note that giving 0/0 the name "x" doesn't do anything. Simply naming something doesn't establish any fact. It just makes "x" shorthand for "0/0". Anyway, observing that 0/0 * 0 = 0 but also 1 * 0 = 0, 2 * 0 = 0, and so on doesn't establish any necessary connection between "0/0" and 1, 2, ... You wouldn't conclude from "1 * 0 = 0" and "2 * 0 = 0" that "1 = 2", or that "1 could be 2", for instance. So nothing has happened. But what could happen? Remember that you started by assuming that "0/0" was defined. "Assume" in math means you've assumed it's true. In that case, you're done, right? Its "definedness" isn't probabilistic. And if starting with that assumption you did find out something true, it doesn't follow that the assumption is "independently" true. (The truth of "if P, then Q" and the truth of "Q" do not together imply the truth of "P".) You might want to look at the post on wheels higher up this thread. It shows what you could do - namely, use a different set of rules. This may be more than you wanted to know ...

gizmo686 10y ago

If you avoid division by 0, wheels revert back to normal fractions [0]. This should mean that you can use them as a drop in replacement for a rational number datatype. Doing so should allow you to defer checking for division by zero; possible moving the check outside of a tight loop. Granted, this should also be doable as an optimization of normal rational arithmetic. The only use cases I can think of for wheels amount to them being a principled way of adding NaN to the number system. Of course, if history is anything to go by, a hundred years from now someone may look back on this comment the same way we look back on people calling sqrt(i) "imaginary". [0] At least under the explicit construction presented.

nonbel 10y ago
This conversation may have moved well above my head, but I mean start with 0/0 = "undefined". I don't see why that should be the case. x = 0/0 x*0 = 0 1*0 = 0 2*0 = 0 ... From this we see that it isn't really that x is undefinable, rather it can be any value at all. There is apparently no issue with an equation having two equally valid solutions (eg quadratic formula), so at what point are there too many?
- bikenaga 10y ago
  I think I see what you're asking. I think the answer is: Before we start talking, you have to tell me what all the rules are. So when you say "start with 0/0 = 'undefined'", what is the ambient number system? The proof I gave earlier showed the number system can't be a ring - so it's not the real numbers, the integers, the rationals ... at least not by the standard definitions of those number systems. When you do math, you don't make up the rules as you go. As far as the example you gave goes, you start by assuming 0/0 is defined. But if you're trying to show 0/0 is defined you're assuming what you want to prove. The logic isn't correct. Note that giving 0/0 the name "x" doesn't do anything. Simply naming something doesn't establish any fact. It just makes "x" shorthand for "0/0". Anyway, observing that 0/0 * 0 = 0 but also 1 * 0 = 0, 2 * 0 = 0, and so on doesn't establish any necessary connection between "0/0" and 1, 2, ... You wouldn't conclude from "1 * 0 = 0" and "2 * 0 = 0" that "1 = 2", or that "1 could be 2", for instance. So nothing has happened. But what could happen? Remember that you started by assuming that "0/0" was defined. "Assume" in math means you've assumed it's true. In that case, you're done, right? Its "definedness" isn't probabilistic. And if starting with that assumption you did find out something true, it doesn't follow that the assumption is "independently" true. (The truth of "if P, then Q" and the truth of "Q" do not together imply the truth of "P".) You might want to look at the post on wheels higher up this thread. It shows what you could do - namely, use a different set of rules. This may be more than you wanted to know ...