math-is-hard

Woohyun’s determination to be perfect at everything is unbearably attractive

math.stackexchange.com
Why is Division by Zero Undefined? A (Mostly) Honest Answer from MSE

“Why can’t you divide by zero?” 

It’s a question ringing out across elementary school classrooms nationwide about the same time every year, and it’s good, a couple times, for derailing a middle school math class as well.

Here is a very well-written, well-intentioned page that gives multiple perspecives to try to explain this phenomenon to explain to a general audience, but they’re all a little hollow. Following is a list of my paraphrasing of the answers and a small rejoinder:

  • You can’t do that division because you can’t find an answer (what if we just haven’t looked hard enough?)
  • Physical reasoning says there is no answer (physical reasoning says there are no imaginary numbers either, and look how that worked out)
  • Infinity isn’t a number, so there’s no answer. If it were a number, you can’t tell whether it’s positive or negative (who says ±∞ are different?)
  • [*] There’s no good way to get “everything to work”, even if you allow infinity (This is really the “right” answer, but as presented it’s hella sketch. What is ‘everything’? and, again, are you sure that you’re trying hard enough?)
  • [this answer addresses 0/0 specifically, which is really a different issue]
  • Limits are a thing! Different physical reasoning says it doesn’t work, and infinity doesn’t work either, just trust me (no, I can’t do this; it’s unsportsmanlike, shooting fish in a barrel)
  • The answer cannot be a [complex] number [with explanation], but infinity is not a useful answer either because… look at all those indeterminate forms! (“… um, sorry, what was I saying?”)
  • It’s not possible, because something something field axioms. (This answer, again, is “right” but it just defers the work to a bunch of (assumed new) algebraic terminology)
  • [another 0/0 answer]

[*] FWIW, this was the explanation I usually got in school, and even then I was unsatisfied for these two reasons. Sadly, my teachers were not able to explain the fundamentals of algebra to me at the time… my memory isn’t too great, so I will admit that it is entirely possible that they tried; if they did, it didn’t work.

Moreover, from a more mature mathematical perspective, all this discussion misses an important point. You CAN do something that looks a lot like dividing by zero… if you are working in the zero ring. 

——

The link produces an explanation that traces the difficulty of dividing by zero back to the fact that functions are single-valued, and therefore division is not a function (unless the codomain has size 1, as with the zero ring). Hence, if you demand “only one answer”, then your demand is impossible. Moreover, it does all this in an extremely weak setting: division is allowed to take place over any magma, which does not even have to be associative (!)

The question is really impossible to answer if you don’t admit that math is based on foundations which are essentially decisions. They are extremely well-motivated decisions, but they are decisions nonetheless. The MSE answer definitely makes decisions. The decision to use a magma is a pretty weak one but it’s not entirely trivial (it means that multiplication must be closed). Also, what should division mean in a magma? What is a zero in a magma?

Biased I may be, but let it not be said that I am unfair. Here is a cynical take on the MSE answer:

  • 0/0 doesn’t work [with explanation] and let’s just ignore other possibilities, okay?

The answer does actually give you the tools to explain why you can’t divide anything else by zero as well. With that proof in hand, it deals with infinity indirectly. However, it does make the decision that we define the product of infinity and zero to be zero; otherwise it is possible for one number to be divisible by zero. However, if multiplication is associative we can determine this number must be infinity; it is still not possible to divide any finite numbers by zero.

So, yes, the argument is incomplete, and the details are tricky. But there is much less to do, and I would argue much easier things to do, than in the other “can’t get everything to work” answers.

I’d like to think of a way to bring the notation “down to earth” because, of course, this answer really is unsatisfying without a lot of mathematical maturity. However, I think the idea has value because I find the typical answers to this question very shaky, at best. As a more alarmist take, you could even reasonably make the case that the answers are damaging to a student’s understanding about the very nature of mathematics— even more troubling when you realize that this is one of the first truly “mathematical” questions that a student would recognize as math!

——

Speaking of “unsatisfying without a lot of mathematical maturity”… any of the cat theory folks want to provide an explanation in even greater generality? It takes me a lot of work to make any sense of categorical language, and sometimes I can’t manage it, but I’d definitely try for this problem.

i think its kinda of silly how tv shows and movies portray the “smart kid” as someone who can do these ridiculously hard math problems that arent realistic for the child’s age. like, theyre ten?? like if they know how to divide big numbers or name every single president of the united states, then theyre already a smart cookie. it just shows children that they have to be able to do college level bullshit to be considered above average???? and why do they always have them show their intelligence through math?? why do they have to be good at EVERYTHING scholarly??? like some one could be excel in history but be poor in other classes and still be the smartest guy around. why do yall gotta go around lowkey putting standards on children like that??

fallautumns asked:

I was studying for my maths exam tomorrow, and by the end of the whole thing.... my brain was so fried that I had to Google the definition of exponent.

Ugh.  Math can be rough.  I have to deal with statistics, but it’s not really then same thing.  With math, you feel like you should be all

But, then you actually just end up like

Good luck on your exam tomorrow!  Get some rest and let all that information consolidate to your long term memory.  I’m sure you’ll rock that test!