Wikipedia:Reference desk/Archives/Mathematics/2016 February 26

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February 26

0/0

What is the value of 0/0 ?--86.187.166.218 (talk) 03:08, 26 February 2016 (UTC)

See division by zero. In almost all cases you're likely to come across, the answer is "the expression does not represent a value" (more succinctly, "undefined"). But there are some very unusual contexts where that might not be true, and the linked article probably discusses them. --Trovatore (talk) 03:10, 26 February 2016 (UTC)

Actually, you might spend a lot of time on that article without finding the exception. The direct link is wheel theory. It's not very important. Hmm, that last statement might be a bit out of line. Let me rephrase it. I don't know of any applications of wheel theory, nor do I know of any ongoing research in the area. --Trovatore (talk) 03:12, 26 February 2016 (UTC)

Hm, here [1] is someone claiming that wheel theory is useful for modeling certain aspects of computation in real-world computers. No references are given, but the claim does have a certain sensibility and reasonableness to it, whether or not it's true. SemanticMantis (talk) 20:25, 26 February 2016 (UTC)

In the case of a batter with at least one plate appearance, but no at bats or hits, 0/0 can be rendered as "---". ←Baseball Bugs ^{What's up, Doc?} carrots→ 03:49, 26 February 2016 (UTC)

Similarly, if you don't go anywhere in your car, but use no gas, your mileage is 0/0 miles per gallon. Bubba73 ^{You talkin' to me?} 04:26, 26 February 2016 (UTC)

However, note that as X and Y approach zero, X/Y may very well be defined. For example, if X = 2n, and Y = n, then X/Y = 2, even as n approaches zero. (Although what happens when n actually reaches zero is open to debate.) StuRat (talk) 04:18, 26 February 2016 (UTC)

The trick is to approach 0 but never reach it. For example, if you halve the distance each time, you'll never get there. ←Baseball Bugs ^{What's up, Doc?} carrots→ 06:13, 26 February 2016 (UTC)

You will if you halve the time each time too. See what is usually called Zeno's paradox, although the article on the subject calls it the dichotomy paradox. --69.159.61.172 (talk) 07:17, 26 February 2016 (UTC)

Where did StuRat mention "time"? ←Baseball Bugs ^{What's up, Doc?} carrots→ 08:02, 26 February 2016 (UTC)

Yes. 0/0 is one of the seven indeterminate forms. (Or more properly, one of the seven indeterminate form is denoted 0/0.) -- ToE 15:06, 26 February 2016 (UTC)

In the context of computer floating-point calculations, 0/0 is NaN (not a number). -- ToE 15:12, 26 February 2016 (UTC)

Right, and the fact that many languages have rather complicated support for dealing with NaNs leads me to think maybe someone has used some wheel theory as part of their programming language design. SemanticMantis (talk) 20:27, 26 February 2016 (UTC)

Floating point in essentially all modern programming languages is based on the IEEE standard. It's hard to find information about wheels in a web search, but I don't think the design of IEEE floating point was motivated by wheel theory. Most of the complexity of the standard is an attempt to minimize the impact of rounding errors. ±∞ exist so that overflowing exponents have something to "round" to, and ±0 exist so that the signs of small nonzero quantities aren't lost to underflow (which matters for functions that have discontinuities at 0, like the principal branches of some complex functions). The associative and distributive laws fail because of rounding, and optimizing compilers normally don't use them (they do use them to optimize integer arithmetic). NaN means "I have no idea what the answer should be" and is designed to infect all later calculations so that the final result will be wrong in an obvious way. NaNs do behave very much like a wheel's ⊥, but otherwise they seem quite different. -- BenRG (talk) 21:55, 26 February 2016 (UTC)