One of my favorite math moments was actually a teacher explaining that some aspects of math are a social construct. Specifically, I was confused about why x^0 = 1 and not 0. There’s a lot of good reasons for this, but rather than starting with those, he pointed out that you could define it that way if you wanted to, and mathemeticians have just agreed on the 1 definition (and then he went into how the 1 definition is much more useful).
I remember getting the same answer to the same question at school and being quite surprised.
If I was asked the question now I would show the interested student this pattern, and ask them what makes sense for the last term.
x^3 = 1*x*x*x
x^2 = 1*x*x
x^1 = 1*x
x^0 = ?
(if they are still unconvinced invite them to approach 0 from the other side)
x^(-3) = ((1/x)/x)/x [up to brackets = 1/x/x/x]
x^(-2) = (1/x)/x [1/x/x]
x^(-1) = 1/x
In some ways the negative powers (if the student already knows those) are cleaner, because the implicit factor of 1 is made explicit in fractions. Although the need for brackets makes the pattern less elegant.
I think the main reason the x^0 = 0 definition is awful is that it means you can no longer rely on the fact that
x^n * x^m = x^(n+m)
(because for n=0 the left hand side is zero, but the right hand side is x^m, which may or may not be zero). Although I think that (depending on the age of the student) this might not be the best explanation.
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.)
The one I like is that for cardinal (counting) numbers, x^y counts the number of functions from a set of y elements to one with x elements. This is very foundational to how powers are defined in the first place, and in many ways even more foundational than addition and multiplication.
If y is empty (regardless of x), then there is exactly one such: the empty function. So x^0 = 1 for all cardinal numbers x including x=0 and all the natural numbers, but also all the infinite cardinalities as well.
From there you can extend to integers, rational numbers, reals, and so on.
It’s been a while since I did much math, but I thought that was the one that counterintuitively equals 1. Whereas 0^1=1 just seems like it would create an unwelcome exception to the x^1=x rule.
One of my favorite math moments was actually a teacher explaining that some aspects of math are a social construct. Specifically, I was confused about why x^0 = 1 and not 0. There’s a lot of good reasons for this, but rather than starting with those, he pointed out that you could define it that way if you wanted to, and mathemeticians have just agreed on the 1 definition (and then he went into how the 1 definition is much more useful).
I remember getting the same answer to the same question at school and being quite surprised.
If I was asked the question now I would show the interested student this pattern, and ask them what makes sense for the last term.
x^3 = 1*x*x*x
x^2 = 1*x*x
x^1 = 1*x
x^0 = ?
(if they are still unconvinced invite them to approach 0 from the other side)
x^(-3) = ((1/x)/x)/x [up to brackets = 1/x/x/x]
x^(-2) = (1/x)/x [1/x/x]
x^(-1) = 1/x
In some ways the negative powers (if the student already knows those) are cleaner, because the implicit factor of 1 is made explicit in fractions. Although the need for brackets makes the pattern less elegant.
I think the main reason the x^0 = 0 definition is awful is that it means you can no longer rely on the fact that
x^n * x^m = x^(n+m)
(because for n=0 the left hand side is zero, but the right hand side is x^m, which may or may not be zero). Although I think that (depending on the age of the student) this might not be the best explanation.
Well, when you put it like this, the conclusion feels irresistible!
But it also feels like an artificial move. Not something the student would come up with, unless they already understand it.
I would probably go with:
x^3 / x = x^2
x^2 / x = x^1
x^1 / x = …complete the pattern, then contemplate on what “x^1 / x” is
I do exactly what you describe with my students, but sadly with extremely limited results.
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.)
The one I like is that for cardinal (counting) numbers, x^y counts the number of functions from a set of y elements to one with x elements. This is very foundational to how powers are defined in the first place, and in many ways even more foundational than addition and multiplication.
If y is empty (regardless of x), then there is exactly one such: the empty function. So x^0 = 1 for all cardinal numbers x including x=0 and all the natural numbers, but also all the infinite cardinalities as well.
From there you can extend to integers, rational numbers, reals, and so on.
Of course, from a pedagogical point of view it may be hard to explain why the “empty function” is actually a function.
Just to check, did you here mean 0^0 ?
It’s been a while since I did much math, but I thought that was the one that counterintuitively equals 1. Whereas 0^1=1 just seems like it would create an unwelcome exception to the x^1=x rule.
Er yeah, I’ll edit. Thanks!