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> correctly state the undefined nature of 1/0 Erm, when we say it's "undefined" we mean it literally -- standard mathematical systems of arithmetic do not defi
by repsilat 8y ago
> correctly state the undefined nature of 1/0
Erm, when we say it's "undefined" we mean it literally -- standard mathematical systems of arithmetic do not define a value for that division.
If you make another system of arithmetic you can define it how you want and be consistent with "regular maths" for the operations in which things are defined. It's an extension.
> We have "NaN" for a reason
Funnily enough, IEEE754 defines 1/0 as positive infinity, not Nan. But none of these things are "the truth" in any reasonable sense of the word, just "useful systems".
Defining it as 0 at least means floating point numbers are (presumably) closed under arithmetic operations, which could be handy.
- tobinfricke 8y ago> Defining it as 0 at least means floating point numbers are (presumably) closed under arithmetic operations, which could be handy. They already are closed. Floating point includes Inf and Nan for that reason. :-)
- repsilat 8y agoAh, of course. Still, this definition gets you to something stricter still -- the "things that normal people think of as numbers" are closed under those operations. (Depending on what happens for overflow and 0/0, I guess.)
- chess19 8y agoStandard mathematical systems of arithmetic do not permit a reasonable definition of 1/0. Mathematicians define equality of fractions by stating that a/b = c/d if and only if a·d = b·c. This means that if we define 1/0 = 1 then 0 = 1. To be fair, this is perfectly consistent, except everything in our system is equal to zero.
- chess19 8y agoThe story of what happens if we defined, for example, 1/0 = 0 is a little more complicated. I will include it because it is sufficiently interesting. Equality of fractions is defined differently when zero divisors are allowed in the denominator. A zero divisor is a non-zero number that can multiply with another non-zero number to get zero. For example, if we work mod 12 then 3 is a zero divisor because 3·4 = 0. If we want to allow zero or zero divisors in our denominators then we say that a/b = c/d if and only if there is some value s such that s·(a·d - b·c) = 0 where s is anything allowable in the denominator. If we are working with the integers, including this s term does nothing because s has to be something that can be a denominator and we only allow non-zero denominators. So, even if we define 1/0 = 0 then literally every fraction would be equal to every other fraction. These conventions can be broken (like, for example, addition of floating point numbers is not associative as pointed out by other comments) but it is definitely not "natural". In other fields of mathematics, like measure theory, it is possible to define things like "zero times infinity is zero" which is traditionally undefined but is a convenient shortcut and does not break anything that people working in measure theory care about. For more: https://en.wikipedia.org/wiki/Localization_of_a_ring#For_general_commutative_rings https://en.wikipedia.org/wiki/Localization_of_a_ring#For_gen...
- a_wild_dandan 8y ago> Mathematicians define equality of fractions by stating that a/b = c/d if and only if a·d = b·c. This means that if we define 1/0 = 1 then 0 = 1. That's not implied, so far as I can tell. a / b = 1 / 0, thus a * d = b * c => 1 * d = 0 * c => d = 0 So all you can say is that d = 0, or at most that c / 0 = 0. Is there some extra step you're taking?
- boyce 8y agoSo if 1·d = 0·c, write 1 in terms of the other elements and then evaluate.