Ok folks… I got a math question:
Why is it that a minus times a minus equal a positive???
IE: -1 x -1 = 1
Thx!
-N8r
Well, no matter what you define 0[sup]0[/sup] as, there will be a host of functions which are not semicontinuous at 0, as Jabba’s x[sup]x[/sup] is if 0[sup]0[/sup] = 1, and
William_Ashbless’s 0[sup]x[/sup] is not. So a limit argument is not very compelling.
William–
Looks to me like you’ve almost got the proof you’re after that 0^0 is indeterminate like 0/0. Take your line of reasoning a little further and:
0 log(0) = 0 x -infinity = 0 x -1/0 = 0/0
But I’m not entirely sure if this kind of algebra with infinities is possible. Can anyone confirm?
Chronos–
“In general, there are two sorts of problems you can get with algebraic operations. Some problems, such as arcsin(2) or sqrt(-4), make no sense in the real numbers, but do make sense in the complex numbers (note, however, that there are multiple complex solutions in each of these cases). All of the other ‘undefined’ things in algebra are essentially division by zero problems.”
Are you including linear algebra in that statement? Because I’ve just learned that the product of, say, a 6x8 and a 3x3 matrix is undefined for reasons that have nothing do do with division by zero and permit no complex solution. Try doing the multiplication and you run out of rows/columns in one of the matrices midway through…
Almost done with it - EXCELLENT book.
See panamajack’s proof in this thread.
No, that’s not possible. -infinity is not equal to -1/0.
**
Chronos was talking about basic algebra, not the linear type.
I would consider that more due to a lack of a notational definition than actually being mathematically undefined. Kind of like how the following is undefined, because it doesn’t make any sense:
-23 × + + 4 × = )) 6.0.00
Chronos: as I suggested in my post to this thread, your decision that the “other” type of undefined result is one that kind that we need complex numbers for is a purely arbitrary one. One may just as well say that 1 - 2 is undefined unless we invent negative numbers, or that 1/2 is undefined unless we invent fractions and that these are the “other” type of undefined result.
No - better to say that basically everything in mathematics is undefined until we decide to define it. And that there has never been any compelling reason nor any obvious way to define division by zero so it is left undefined.
Don’t think “why does 1/0 have no definition?” Instead the question is “why does 1/2 have a definition?” And the answer to that question is: “because it is useful”.
pan