Yeah I think showing the pattern is helpful, but I like the version where instead of showing the (under dispute) 1 term, you show the division pattern, like in this video https://www.youtube.com/watch?v=X32dce7_D48
Essentially the argument is that x^(n-1) = x^n / x applies in every other case, so why not apply it for x^0 also?
For me it was also helpful to point out that you could define x^0 = 0 (and in fact, some fields leave 0^0 undefined), but it would cause all kinds of other problems you’d have to account for and make arithmetic a lot harder and less useful.
Technically, you could define 2+2=5, but that would cause a lot of problems everywhere. :D
I agree that sometimes the definitions are motivated by “how do you intend to use that, and which definition will make your job easier?”
But I think that situations with two meaningfully different answers are rare. I imagine there is often either one good answer, or one good answer that brings some problems along (such as introducing a new type of number) so some people choose to leave it undefined. For example:
1⁄0 is either undefined, or we introduce some notion of “infinity”
0⁄0 is probably better left undefined, or some kind of “error” value
square root of −1 is either undefined, or we introduce complex numbers
a sum of zero numbers is = 0
a product of zero numbers is = 1
a union of zero sets is = empty
an intersection of zero sets is either undefined, or the entire universe (if such thing exists)
0! = 1
x^0 = 1 …maybe unless x = 0, in which case one might reasonable argue also for 0
parallel lines either do not intersect, or they “intersect at infinity”
infinity minus one is either undefined, or we introduce surreal numbers
Maybe I just selected examples that support my point. Feel free to add examples to contrary.
Seems to me that 0^0 is the only case where one could successfully argue for both “0^0=1” and “0^0=0″ depending on whether you see it as a limit for “x^0” or a limit for “0^x”. (Even here I feel that the former choice is somehow better, because it also extends to negative x.)
(In set theory, there is a disagreement about which axioms to use, and it seems to be resolved as “agree to disagree” but I am not an expert.)
Yeah I think showing the pattern is helpful, but I like the version where instead of showing the (under dispute) 1 term, you show the division pattern, like in this video https://www.youtube.com/watch?v=X32dce7_D48
Essentially the argument is that x^(n-1) = x^n / x applies in every other case, so why not apply it for x^0 also?
For me it was also helpful to point out that you could define x^0 = 0 (and in fact, some fields leave 0^0 undefined), but it would cause all kinds of other problems you’d have to account for and make arithmetic a lot harder and less useful.
Technically, you could define 2+2=5, but that would cause a lot of problems everywhere. :D
I agree that sometimes the definitions are motivated by “how do you intend to use that, and which definition will make your job easier?”
But I think that situations with two meaningfully different answers are rare. I imagine there is often either one good answer, or one good answer that brings some problems along (such as introducing a new type of number) so some people choose to leave it undefined. For example:
1⁄0 is either undefined, or we introduce some notion of “infinity”
0⁄0 is probably better left undefined, or some kind of “error” value
square root of −1 is either undefined, or we introduce complex numbers
a sum of zero numbers is = 0
a product of zero numbers is = 1
a union of zero sets is = empty
an intersection of zero sets is either undefined, or the entire universe (if such thing exists)
0! = 1
x^0 = 1 …maybe unless x = 0, in which case one might reasonable argue also for 0
parallel lines either do not intersect, or they “intersect at infinity”
infinity minus one is either undefined, or we introduce surreal numbers
Maybe I just selected examples that support my point. Feel free to add examples to contrary.
Seems to me that 0^0 is the only case where one could successfully argue for both “0^0=1” and “0^0=0″ depending on whether you see it as a limit for “x^0” or a limit for “0^x”. (Even here I feel that the former choice is somehow better, because it also extends to negative x.)
(In set theory, there is a disagreement about which axioms to use, and it seems to be resolved as “agree to disagree” but I am not an expert.)