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General Science / Re: What's 0^0 ?
« on: 09/11/2021 14:07:33 »
Hi again.
If you really believed that the value at x=0 is significantly distinguished from values we obtain when x is approximately 0 but not 0, then continuity is certainly not what you're going to have. No matter how close we get to x=0 we should be prepared to accept that the value of the expression could be very different when x is precisely 0.
In which case, why are you so willing to lose continuity for 0x as x→0 ?
0^1 = 0, 0^0.1 = 0, 0^0.01 = 0, ....., 0^(any small positive rational) = 0, ......
If there's any hope for continuity then 0^0 = 0.
Don't get me wrong, there's no reason why continuity of these expressions should be important but we do need to ask the following question: Is it reasonable to imply that continuity is important in some situations and some expressions like the x^(xc) argument but then choose to ignore continuity in some other expressions like 0x ?
0x isn't an amazingly complicated expression or some example I have deliberately cooked up to get another limit. It just does look like one of the sensible expressions we might consider when we're trying to determine 00. What reason would there be for specifically ignoring that expression and prioritising continuity in an expression like x^(xc)?
There are plenty of more unusual expressions that we can "cook-up" (deliberately construct) to show other limits. Do we need some criteria to decide which expressions are more important than others?
Anyway, arguing for a value of 00 based on limits and continuity of apparently well defined expressions involving only exponents that are already well defined (so that will be rational exponents for example) doesn't usually get very far. It just shows that there doesn't have to be a unique value AND/OR that attempting to maintain continuity was not a suitable assumption or approach to tackle the problem. It isn't possible to maintain continuity for every simple expression involving exponents by assigning a single unique value for 00.
You do seem to have moved on from just using simple ideas based on continuity to something using complex number - but this post is already too long, so I'll stop here for a moment.
Best Wishes.
I meant x^(cx)Thanks, Hamdani. I can see this. So this is using some idea of continuity, the equation holds as you take a limit as x → 0.
Try a smaller number for x, say 10-100. The equation applies for both positive and negative value of x, provided that its absolute value is much smaller than absolute value of c. Hence there's no reason to exclude only when x is exactly 0.
We need to distinguish between exact 0 and approximate 0.But that's not what we are doing when we use limits or continuity ideas to suggest a value for 00. We are trying to make the behaviour at x=0 as much like the behaviour for other values of x as we possibly can. Specifically, we are keeping the value of the expression within some small difference of the value of the expression when x is just off 0 and we are confident that we can make this error as small as we want provided we take x close enough to 0.
If you really believed that the value at x=0 is significantly distinguished from values we obtain when x is approximately 0 but not 0, then continuity is certainly not what you're going to have. No matter how close we get to x=0 we should be prepared to accept that the value of the expression could be very different when x is precisely 0.
(From Eternal student) ....is continuity important in anything?OK. So continuity is of some importance to you and it was used in your argument x^(xc) = 1 when x→0.
Hamdani Yusuf replied: It simplifies things, which improves efficiency. Efficiency is a universal instrumental goal.
In which case, why are you so willing to lose continuity for 0x as x→0 ?
0^1 = 0, 0^0.1 = 0, 0^0.01 = 0, ....., 0^(any small positive rational) = 0, ......
If there's any hope for continuity then 0^0 = 0.
Don't get me wrong, there's no reason why continuity of these expressions should be important but we do need to ask the following question: Is it reasonable to imply that continuity is important in some situations and some expressions like the x^(xc) argument but then choose to ignore continuity in some other expressions like 0x ?
0x isn't an amazingly complicated expression or some example I have deliberately cooked up to get another limit. It just does look like one of the sensible expressions we might consider when we're trying to determine 00. What reason would there be for specifically ignoring that expression and prioritising continuity in an expression like x^(xc)?
There are plenty of more unusual expressions that we can "cook-up" (deliberately construct) to show other limits. Do we need some criteria to decide which expressions are more important than others?
Anyway, arguing for a value of 00 based on limits and continuity of apparently well defined expressions involving only exponents that are already well defined (so that will be rational exponents for example) doesn't usually get very far. It just shows that there doesn't have to be a unique value AND/OR that attempting to maintain continuity was not a suitable assumption or approach to tackle the problem. It isn't possible to maintain continuity for every simple expression involving exponents by assigning a single unique value for 00.
You do seem to have moved on from just using simple ideas based on continuity to something using complex number - but this post is already too long, so I'll stop here for a moment.
Best Wishes.