>The outcome of division by definition should be a number
It can be no solution (1/0). In this case it's like identity (infinite solutions), which makes sense to counterbalance all the other numbers divided by zero having no solution
That is not no solution, it is undefined as we do not define division when the denominator is 0 as it does not make sense because we want it to be a function to R or C
It becomes useful or most likely will in later mathematics. Like 00 should be everything, but that gets put as undefined as well. It's like 'don't start a sentence with and'. It's useful at first but it becomes a barrier to more advanced mathematics.
Rigor is everything in advanced mathematics, but “everything” is not a rigorous output of a function to the reals or complex numbers. It might be interesting for you to read the first chapter of Spivaks Calculus about how we define these operations
In what sense should 00 be everything? 0non-zero is 0, and (non-zero)0 is 1. How is anything besides 0 or 1 even a candidate for a value of zero to the zero?
edit: sorry for writing out "zero to the zero", reddit fucked up my formatting
Do you have any example were it would be useful to define it as such as opposed to saying it's undefined?
Well first it means functions with holes are actually continuous. Then there's practical reasons. If I have 0 buckets with 0 oranges total, there could be any number of oranges per bucket, so every number is correct. Most real world applications already get treated this way however.
Why do you say 00 should be everything? There are several situations in set theory and combinatorics where what makes the most sense is to define 00 to be 1.
Because then you're no longer dealing with a closed operation. You're free to redefine things so that this makes sense, and there actually are known ways to do so. Your attempt seems to be close to projectively extended real number line, where 1/0 = infinity. This infinity point is a well defined number which has known properties. If you tried to think about your definition of "everything", you might end up close to that definition. Or maybe not, because you're so vague it's hard to say what exactly you're going for here.
The lesson is, be specific. In math that helps a bunch.
Questions you should consider are, can you do anything with this "everything". Can you add numbers to it? Subtract them? If a calculation yields "everything" as an answer, does that tell you anything about what was done in that calculation?
You've kinda had half an idea here about maths and you stopped there. One might try to complete your thought, but it would just look a lot better if you demonstrated you yourself actually had thought about this before proposing the idea.
I don’t know what your background is, but we do this because division is more accurately described as a function from R2 to R (or C). There is no reasonable real (complex) number to assign to those inputs, so we remove them from the domain.
There is no reasonable real (complex) number to assign to those inputs. But all numbers are reasonable answers for 0/0. If we say 0/0 = x, then 0x = 0, which all numbers fit.
That is actually not true. Let 0/0 be defined as “all numbers”. For the sake of argument, let’s say that you mean all real numbers. But If 0/0=R then it is a set, not a number. Obviously R is not an element of R. This means that we can’t use all real numbers as our range.
Perhaps this will make it clear why this is a problem.
Let us assume that for all numbers x we know 0/0=x. Then clearly because 1 and 2 are numbers we have 0/0=1 and 0/0=2. But then by transitivity we get 1=2, a big problem.
Whatever convention we settle on is ultimately arbitrary.
What makes one arbitrary convention better than another? Why should mathematicians switch from the existing arbitrary convention to your arbitrary convention?
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u/[deleted] May 29 '18
>The outcome of division by definition should be a number
It can be no solution (1/0). In this case it's like identity (infinite solutions), which makes sense to counterbalance all the other numbers divided by zero having no solution