121–130 of 136 posts
http://en.wikipedia.org/wiki/Transreal_arithmetic#Transreal_...
We had this guy as a lecturer. Whether nullity exists or not (though James's argument is that it's as valid as j), and I have to admit his arithmetic does make some things simpler.
Technically, 0^0 is an indeterminate form and has no specific solution. Accurate but unhelpful. Practically, 0^0 highlights the issue that most of us don't have a good conceptual model for what exponents really do. How would you explain to a 10-year old why 3^0 = 1 beyond "it's necessary to make the algebra of powers work out". I use an "expand-o-tron" analogy http://betterexplained.com/articles/understanding-exponen…
7^2=7 * 7=49
7^2/7^2=7 * 7/ 7 * 7=49/49=7^(2-2)=7^0=1
But 0^0 was never intuitive to me.
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>most of us don't have a good conceptual model for what exponents really do Instead of matching math to real world objects (1= one banana, 2 = two bananas, 1+2 = 3 bananas etc. ) and building up to exponentiation, multiplication etc. thereby introducing all sorts of paradoxes, group theory dodges all that and treats the whole thing as a very consistent rule-based system. Things fall into place quickly once the rules…
You're right, apart from the fact that real exponentiation does not really belong to group theory -- or even abstract algebra, for that matter.
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By "exponentiation rules" I mean algebraic equalities, like a^x * a^y = a^(x+y). Most of them work no matter if you define 0^0 = 1 or 0, but some of them are cleaner with 0^0 = 1. It's also consistent with cardinal and ordinal exponentiation (look it up). "Approaching along x axis" is not algebraic notion, it's analytic one. 0^0 makes no less sense than, say, -e^(i pi). They're both 1 because we define them like this…
> Also, mathematicians agree in this , seriously. Go and ask one. Mathematician here; we do not. See http://math.stackexchange.com/questions/11150/zero-to-zero-p... . More precisely, as Arturo Magidin points out at http://math.stackexchange.com/questions/11150/zero-to-zero-p... , if we view exponentiation in the 'discrete setting', then $0^0$ must be $1$; whereas, if we view it in the continuous setting, there is sim…
Thinking back on my math education, part of the difference in viewpoint may be the first time I rigorously met the continuous-domain exponential.
This was in real analysis. Exponentiation is defined first for positive integer exponents, and then for rational exponents. All elementary. Then it's extended to real-valued exponents by taking the limits of rational numbers, and appealing to continuity.
I just looked, this is exercise 6 in chapter 1 of baby Rudin.
So, because the notion of limit and continuity is embedded in this definition of the exponential function, it's natural to "approach" (groan) 0^0 as a special case, because the conditions of this definition (continuity) don't hold.
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Mathematicians don't argue about what an expression "really is" (or at least, real mathematics doesn't involve this). They define functions and use axioms to prove theories about them. "No really" . Mathematics just isn't concerned with this stuff. Sometimes infinity it defined as single point making the real number compact, sometimes a "positive infinity" and a "negative infinity" are defined. Sometimes you add poin…
Are you sure you're replying to right post? I ask, because nothing I see in yours can be seen as reply to mine -- it actually repeats my point.
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You're right, apart from the fact that real exponentiation does not really belong to group theory -- or even abstract algebra, for that matter.
The notion of a^n definitely belongs to group theory, whether you call that n an exponent or an annihilator or a period is a matter of some contention. See: http://mathoverflow.net/questions/44393/notation-exponent-of... http://mathoverflow.net/questions/32116/exponent-of-a-group
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I successfully managed to explain 3^0 to 10 year olds (as a recovering high school math teacher) as: 3^2 = 9 3^1 = 3 (divide 9 by 3) 3^0 = 1 (divide 3 by 3) 3^-1 = 1/3 (divide 1 by 3) etc This can logically be explained as n^0=1 for all real numbers. Unfortunately this doesn't really handle 0^0 but fortunately 10 year olds are rarely that difficult.
This explanation works pretty well on adults: Just include the multiplicative identity (1) in the expansion. 3^3 = 3*3*3*1 = 27 3^2 = 3*3*1 = 9 3^1 = 3*1 = 3 3^0 = 1 = 1 and likewise: 0^3 = 0*0*0*1 = 0 0^2 = 0*0*1 = 0 0^1 = 0*1 = 0 0^0 = 1 = 1 I haven't yet tried this on an actual 10 year old, though.
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> "why don't we just agree that 0^0 = 1" Because sometimes it's better not to. Sometimes it's inconsistent with our definitions. Just like sometimes we agree that you can't divide by zero, and sometimes we agree that you can. Sometimes infinity is an actual value (say, in the extended reals), and sometimes it's just a symbol for "unbounded". Sometimes we agree that you can't take the square root of a negative number,…
> Because sometimes it's better not to. Sometimes it's inconsistent with our definitions. I'd love to see even one example of 0^0=1 being inconsistent with a definition. The closest I've ever seen is that it bothers people that for reasons of their own had their hearts set on (x,y) -> x^y having no discontinuities...
Perhaps it's more precise to say "Because sometimes it's better not to. Sometimes there is no canonical choice that follows from our definitions, and it doesn't help to assign an arbitrary value that doesn't help solve any related problems."
What is "x" equal to? In general, I mean, not in the context of any equation like "x+1=2". You could say "x=7 in the study of free variables over integers when no other constraints are given", and that is completely consistent with the rest of mathematics, and yet would not be particularly useful and introduces an ugly (philosophical weasel word, yes) asymmetry in the theory (I'd say it introduces a gauge invariance (https://secure.wikimedia.org/wikipedia/en/wiki/Gauge_theory), but I'm really not qualified to discuss that in a rigorous way.)
Someone like Scott Aaronson could put this claim on more solid footing, but I would state that, intuitively, "assigning a value to an indeterminate form leads to a more complex definition of a mathematical system" in some formal complexity-theory sense.