Talk:Exponentiation/Archive 2012

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solve

4^x+4^1/x=18 — Preceding unsigned comment added by 49.244.51.160 (talk) 04:23, 16 September 2011 (UTC)

This page is for discussing improvements to the article. You can post questions about maths to Wikipedia:Reference desk/Mathematics but they would like to see you have tried to solve it yourself first so you should say what approach you have tried or else say why or how you come across the problem if it is not one that was set to you. For instance have you tried it out with a few numbers to see how the left hand behaves? Also a title like 'Exponentiaion problem' for instance would be appreciated there rather than just 'solve'. Dmcq (talk) 09:43, 16 September 2011 (UTC)

I assume the author of the o.p. was somehow proud to actually know the solution (which, as mentioned, is not that difficult if one proceeds by trial and error, testing x=1, then x=2, ...). OTOH, (s)he probably meant 4^(1/x) and not 4^1/x = 4/x... (even though, admittedly, this does not change that much in the case at hand...) — MFH:Talk 23:43, 16 May 2012 (UTC)

Request for new image

The current image under section Exponentiation#Zero to the zero power shows a 3D plot of z=|x|^y, x,y∈ℝ. My impression is that the only purpose of for plotting the absolute value is to give a mirror image for a different perspective. I think that this is potentially confusing and adds very little, though the image is in many respects very good. Would some soul be able to replace this with a similar image showing only x≥0?

Secondly, three of the four green curves through (x,y)=(0,0) have no formula that is evident from inspection of the curves. It might be worthwhile replacing these with curves of constant y/x, where their form is illustrated perhaps by a plane hanging from the respective curve to the plane z=0. This will more clearly illustrate the limit as approached from any fixed direction other than along x=0. — Quondum^☏✎ 09:26, 14 February 2012 (UTC)

If you click on the image, you will get information on who originally uploaded the image. Probably the best would be to leave a note on that user's talk page. Isheden (talk) 10:21, 14 February 2012 (UTC)

I've done so (and received an enthusiastic response). — Quondum^☏✎ 14:39, 14 February 2012 (UTC)

0^0 again (not a rant, but I think something is wrong)

In the section listing the 'pros' of having 0^0 = 1, the final point given is that it satisfies differentiating x^n, saying 'the power rule is not valid for n = 1 at x = 0 unless 0^0 = 1.' This would mean d/dx (0) = 1, which is definitely not true. — Preceding unsigned comment added by 2.27.26.57 (talk) 21:48, 14 February 2012 (UTC)

Using the same argument the integral of 1/x would be 0, they're just indications. Dmcq (talk) 23:53, 14 February 2012 (UTC)

Who's baiting whom? — Quondum^☏✎ 07:09, 15 February 2012 (UTC)

Sorry yes I should have pointed that out. There is though that bit about differentiating x¹ the same as x³ say, it comes to 1x⁰ and at 0 that 1×0⁰. It really is just reiterating about x⁰ being 1 and is redundant in the article. The point about integrating 1/x is that the usual rule applied to x^-1 would give x⁰/0, since x⁰ is the constant 1 we can take this constant 1/0 away (!) thus leaving 0 as the integral. ;-) Dmcq (talk) 07:17, 15 February 2012 (UTC)

Well, you can take an arbitary "constant" away in an indefinite integral, I guess. At least the rule for ∫xⁿdx is properly stated with an exception for n=−1. I'm not sure I agree about redundancy, if you're referring to the last bullet of Exponentiation#For discrete exponents. — Quondum^☏✎ 08:41, 15 February 2012 (UTC)

Yes that's the place. By the way I followed power rule and it gives the integral without the log and carefully hides that there might be trouble by just saying 'for natural n'. The log is given at the very end of the article under generalization. The problem there is that power rule is redirected to calculus with polynomials. I'm not sure that's a reasonable article to have in the first place or else it should be a bigger article with more about the history of calculus as the the integrals were calculated before Newton. Dmcq (talk) 08:54, 15 February 2012 (UTC)

I also found that article before I located List of integrals#Rational functions. Calculus with polynomials does seem odd, as I'd expected something closer to what you describe given that title. Also, power rule should not redirect there. Your suggestion on its talk page seems fair. — Quondum^☏✎ 09:09, 15 February 2012 (UTC)

exponentiation by squaring

I somehow disfavour writing, in that section, e.g., 3. (2^4)^2 = 2^8 = 256 because it suggests that calculation of (2^4)^2 is done via calculation of 2^8, while the idea is the converse: One calculates the required powers of 2 via squaring of the previous result, thus: 2^8 = (2^4)^2 = 16^2 = 256 — that's the way the reasoning and calculation goes, not the other way round. — MFH:Talk 23:58, 16 May 2012 (UTC)

Sounds reasonable. --Trovatore (talk) 00:45, 17 May 2012 (UTC)

Negative Exponents

Negative Exponents redirect here, but I did not find where these are defined, neither here nor at algebraic notation, which dabs me to Mathematical notation. --Pawyilee (talk) 04:36, 9 September 2012 (UTC)

PS Multiplicative inverse defines x to the negative first power as the reciprocal (inverse) of x, and I think x to the minus second power is the square, minus third the cube root, &c., and seem to recall that raising a quantity to the power of zero leaves it unchanged; however, Multiplication does not explain how these are used in the arithmetical combination of exponents. Operation (mathematics) (redirected from Algebraic operations) sends readers here to find out how to perform operations with exponents, such as happens to ±n when moved from one side of the equals mark to the other. Arithmetic, Calculation, and Calculator don't cover addition and subtraction of exponents, either. Is this a lost art? --Pawyilee (talk) 05:28, 9 September 2012 (UTC)

Negative exponents are defined in the section Exponentiation#Arbitrary_integer_exponents. Jowa fan (talk) 06:58, 9 September 2012 (UTC)

You'll see from the same section that b⁰ = 1 when b ≠ 0 (not "unchanged, or b as you surmised). I have changed the redirect Negative Exponents to point to this specific section. This is, however, not ideal: it does not deal with the more general case of negative real exponents. One has to search a bit carefully for the statement

The identities and properties shown above for integer exponents are true for positive real numbers with non-integer exponents as well.

which then allows one to draw the inference that b^−r = 1/b^r, provided that b is positive. From an editing perspective, should we make this process a little more direct for the reader? — Quondum 07:26, 9 September 2012 (UTC)

And me! It's been 50 years since I used this stuff and longer since I learned it—and when I learned it, it was explained so that I could understand it rather than as presently expounded upon in Identities and properties. --Pawyilee (talk) 17:24, 9 September 2012 (UTC)

nonnegative integer exponents

The section Positive integer exponents begins:

Formally, powers with positive integer exponents may be defined by the initial condition

$b^{1}=b$

&c

I suggest the heading be changed to Nonnegative integer exponents and the beginning to:

Powers with nonnegative integer exponents may be defined by the initial condition

$b^{0}=1$

&c

This is to emphasize that $0^{0}=1$ is a possible definition rather than a matter of religious faith. I request your comments. Bo Jacoby (talk) 08:31, 22 November 2012 (UTC).

While I personally strongly prefer this approach and feel that coping with the corresponding indeterminate form in analysis despite the function then being defined on (0,0) is a small price to pay for the wealth of benefits, I expect you will get strong resistance. [Analysis needs to cope with equivalent examples outside "primitive" functions that cannot so readily be "defined away", e.g. discontinuities in a Fourier transform.] In the end, in this context notability should probably decide which approach (or both approaches) should dominate in the article. — Quondum 13:25, 22 November 2012 (UTC)

Quondum, are you for or against the suggested change? Bo Jacoby (talk) 21:56, 22 November 2012 (UTC).

I think everything regarding 0^0 is covered perfectly well in that section, and there is no need to rehash the discussions that already led to the current state of the article. — Carl (CBM · talk) 23:29, 22 November 2012 (UTC)

Thanks Carl. I'm not trying to rehash the discussion but rather to simplify the article, that puts too much emphasis on a rather unimportant topic. Bo Jacoby (talk) 07:21, 23 November 2012 (UTC).

Unfortunately, any change of this nature will involve rehashing the discussion, largely because the topic seems to be so divisive between mathematicians (and WP editors). From the article and my memory of previous discussions, I'm convinced that the suggested change will not be accepted. If this were a textbook, I would argue for the change; as a reference (which this should be), I expect the suggested change is not appropriate. I am not putting any weight on either side of the notability argument because I do not know how prevalent the respective definitions are. I concur with CBM that the article currently does a very good job of presenting this aspect. In a nutshell: much as I'd like it, I will not support the change as suggested. — Quondum 08:09, 23 November 2012 (UTC)