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The following identities hold, provided that the base is non-zero whenever the integer exponent is not positive:
Using this property..
What's going on here.. 67.1.3.194 ( talk) 22:28, 23 April 2013 (UTC)
The article currently contains the statement
This revert removed the content
which makes a significant point (albeit not well-worded) not obviously contained in the statement quoted above. In particular, a two-variable function xy extended to 00 being defined as 1 is continuous on the set {(x,y): x > εy, x < y/ε} for any ε > 0. This could be phrased as a noteworthy and "certain condition". In particular, it may be worth mentioning that very marginally constraining the domain (as opposed to defining a path for the limit) ensures continuity, which may also be seen in the adjacent diagram. — Quondum 11:27, 14 June 2013 (UTC)
My removal of the section Exponentiation#Negative exponents with edit summary (Undoing 2 edits: not particularly coherent, and detracting from the quality.) has been reverted by the author of the section, with the edit summary (there needs to be a "negative exponents" subsection, so let's increase the quality not remove the contribution. thanks.) Since the content of this section is more than adequately covered under the section Exponentiation#Arbitrary integer exponents, it should be removed. All it contains is effectively a repetition of one statement from this latter section, and then simply adds some trivial examples. I do not want to get involved in an edit war and will leave it to other editors to preserve the GA status. — Quondum 03:43, 25 July 2013 (UTC)
I think this article would benefit from some restructuring and from more consequent usage of terminology. I would suggest developing the case of positive real base b first with increasingly general (probably up to real) exponents. After that, the case of base 0 could be treated, and finally negative bases leading to the most general case where both base and exponent may be complex. Also, the usage of the term power is sometimes not clear. For example, does "Complex power of a complex number" refer to wz being complex or to z being complex, or both? Isheden ( talk) 09:39, 25 July 2013 (UTC)
Why not to put the multiplicative identity in front of series? This would answer what is the answer in case you have no items to multiply together. You still have 1 in result of such product because here is intuitive definition of the product. Likewise, the repeated sum a×n is must be defined as 0+a+a+a+, w.r.t. to additive identity 0. This is the true meaning of the exponentiation. Why not to claim that explicitly? This would resolve a lot of confusion right away. I suppose that in continuous analysis we have something different from 0⁰. Why do we say that we have zeroes there if they are infinitesimals and not true zeroes actually? -- Javalenok ( talk) 10:43, 11 September 2013 (UTC)
I agree with Javalenok. When the exponent, n, is a nonnegative integer, the recursive definition
is indeed appropriate. Trovatore's argument about 41/2 is invalid because 1/2 is not an integer. Trovatore's talking about the real-number context is nonsense because zero is an integer as well as a real number. The present quite unsatisfactory state of the article on exponentiation does not reflect consensus but merely the fact that I don't make edit wars and Trovatore did. Dmcq's argument that exponentiation by an integer is different from exponentiation by a real refers to the computer science meaning of a 'real'= floating point number, which is different from the mathematical concept of a real number. In computer science zero may be represented in several ways, such as fixed point binary and floating point binary. Mathematically zero a single object. Bo Jacoby ( talk) 20:51, 6 December 2013 (UTC).
The discontinuity cannot be helped. The constant term of the polynomial is even if x=0. Undefining 00 has a high cost but no benefit. Bo Jacoby ( talk) 04:02, 7 December 2013 (UTC).
The overwhelming consensus is 0^0 is 1, and wikipedia should say so. Try to find a textbook that gives the derivative of x^n, while listing n=1 as a separate case! If we can't find one, it means that everyone implicitly uses 0^0=1, and it is OK for wikipedia to say so. As for the debate and the controversy, there is just as much controversy about Monty Hall's problem, and nevertheless, wikipedia simply gives the correct answer there without giving merit to persistent incorrect views. The limit argument only shows that 0^(approximate 0) is not defined. Applying a limit argument to 0^(exact 0) is not logically sound, because the continuous version of x^y is only defined after x^y is defined for integer values of y. We should not let an unsound argument interfere with presenting what is clearly (at least implicitly, with polynomial and power series representation, and many other formulas) the overwhelming consensus. Mark van Hoeij — Preceding unsigned comment added by 128.186.104.253 ( talk) 15:49, 17 December 2013 (UTC)
The obvious fact that some books don't define 00 is unimportant. No mathematician, save Trovatore, defines a function f(x) such that the definition of f(0) depends on whether 0 is 'in a real-number context'. Our article needs improvement. Bo Jacoby ( talk) 17:48, 7 December 2013 (UTC).
I don't want to drag on the debate and the edits, but I made an edit that I think is a good compromise. Rather than simply defining 0^0 to be 1, we can state (without controversy or originality) that there are formulas that require 0^0 to be interpreted as 1. The previous text is not OK because it simply says that 0^0=1 reduces the number of cases, but fails to mention that 0^0=1 is needed for the correct interpretation of these formulas. Mark van Hoeij — Preceding unsigned comment added by 71.229.28.197 ( talk) 18:39, 18 December 2013 (UTC)
Trovatore wrote: "The most common definition for real-to-real exponentiation is , and it doesn't follow from that". So, according to Trovatore et al., we must also undefine e.g. (−1)2 as ln(−1) is not defined as a real number. Bo Jacoby ( talk) 23:34, 18 December 2013 (UTC).
We have three real functions, f1 and f2 and f3 defined by
and
and
The domain of f1 is D1 = ℝ×{0,1,2,3,...}.
The domain of f2 is D2 = (ℝ×{0})∪((ℝ\{0})×{−1,−2,−3,...})
The domain of f3 is D3 = ℝ+×ℝ.
If (x,y) ∈ Di ∩ Dj then fi(x,y)=fj(x,y) for i,j=1,2,3.
Then define xy:
This definition covers the real cases of xy. I hope very much that we can eventually get rid of the ridiculous nonsense that 00 is defined or undefined depending on whether 0 is an integer or a real. Bo Jacoby ( talk) 21:51, 20 December 2013 (UTC).
Yes, 2+i0 = 2 is true. 2 > 1 is true. 1 = 1+i0 is true. So 2+i0 > 1+i0 is true. The fact that the ordering of reals is not extended to non-real complex numbers does not undefine the ordering of reals. Do you think that 2+i0 > 1+i0 is false? And why? Bo Jacoby ( talk) 21:51, 20 December 2013 (UTC).
You don't solve Clausen's paradox by making 2≠2+i0. The point is that a number is not either real or complex; a real number is both. Bo Jacoby ( talk) 20:02, 21 December 2013 (UTC).
Not every formalization is the usual one. Other formalizations are equally valid models. Real numbers can be defined as complex numbers of the form x+i0, and so on. Then we have the sequence of inclusions ℕ⊂ℤ⊂ℚ⊂ℝ⊂ℂ. Exponentiation of positive real (complex) numbers is not the same operation as exponentiation of negative or nonreal (complex) numbers. It is still ridiculous nonsense to claim that 2+i0≠2. Bo Jacoby ( talk) 21:16, 21 December 2013 (UTC).
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 2010 | Archive 2011 | Archive 2012 | Archive 2013 |
The following identities hold, provided that the base is non-zero whenever the integer exponent is not positive:
Using this property..
What's going on here.. 67.1.3.194 ( talk) 22:28, 23 April 2013 (UTC)
The article currently contains the statement
This revert removed the content
which makes a significant point (albeit not well-worded) not obviously contained in the statement quoted above. In particular, a two-variable function xy extended to 00 being defined as 1 is continuous on the set {(x,y): x > εy, x < y/ε} for any ε > 0. This could be phrased as a noteworthy and "certain condition". In particular, it may be worth mentioning that very marginally constraining the domain (as opposed to defining a path for the limit) ensures continuity, which may also be seen in the adjacent diagram. — Quondum 11:27, 14 June 2013 (UTC)
My removal of the section Exponentiation#Negative exponents with edit summary (Undoing 2 edits: not particularly coherent, and detracting from the quality.) has been reverted by the author of the section, with the edit summary (there needs to be a "negative exponents" subsection, so let's increase the quality not remove the contribution. thanks.) Since the content of this section is more than adequately covered under the section Exponentiation#Arbitrary integer exponents, it should be removed. All it contains is effectively a repetition of one statement from this latter section, and then simply adds some trivial examples. I do not want to get involved in an edit war and will leave it to other editors to preserve the GA status. — Quondum 03:43, 25 July 2013 (UTC)
I think this article would benefit from some restructuring and from more consequent usage of terminology. I would suggest developing the case of positive real base b first with increasingly general (probably up to real) exponents. After that, the case of base 0 could be treated, and finally negative bases leading to the most general case where both base and exponent may be complex. Also, the usage of the term power is sometimes not clear. For example, does "Complex power of a complex number" refer to wz being complex or to z being complex, or both? Isheden ( talk) 09:39, 25 July 2013 (UTC)
Why not to put the multiplicative identity in front of series? This would answer what is the answer in case you have no items to multiply together. You still have 1 in result of such product because here is intuitive definition of the product. Likewise, the repeated sum a×n is must be defined as 0+a+a+a+, w.r.t. to additive identity 0. This is the true meaning of the exponentiation. Why not to claim that explicitly? This would resolve a lot of confusion right away. I suppose that in continuous analysis we have something different from 0⁰. Why do we say that we have zeroes there if they are infinitesimals and not true zeroes actually? -- Javalenok ( talk) 10:43, 11 September 2013 (UTC)
I agree with Javalenok. When the exponent, n, is a nonnegative integer, the recursive definition
is indeed appropriate. Trovatore's argument about 41/2 is invalid because 1/2 is not an integer. Trovatore's talking about the real-number context is nonsense because zero is an integer as well as a real number. The present quite unsatisfactory state of the article on exponentiation does not reflect consensus but merely the fact that I don't make edit wars and Trovatore did. Dmcq's argument that exponentiation by an integer is different from exponentiation by a real refers to the computer science meaning of a 'real'= floating point number, which is different from the mathematical concept of a real number. In computer science zero may be represented in several ways, such as fixed point binary and floating point binary. Mathematically zero a single object. Bo Jacoby ( talk) 20:51, 6 December 2013 (UTC).
The discontinuity cannot be helped. The constant term of the polynomial is even if x=0. Undefining 00 has a high cost but no benefit. Bo Jacoby ( talk) 04:02, 7 December 2013 (UTC).
The overwhelming consensus is 0^0 is 1, and wikipedia should say so. Try to find a textbook that gives the derivative of x^n, while listing n=1 as a separate case! If we can't find one, it means that everyone implicitly uses 0^0=1, and it is OK for wikipedia to say so. As for the debate and the controversy, there is just as much controversy about Monty Hall's problem, and nevertheless, wikipedia simply gives the correct answer there without giving merit to persistent incorrect views. The limit argument only shows that 0^(approximate 0) is not defined. Applying a limit argument to 0^(exact 0) is not logically sound, because the continuous version of x^y is only defined after x^y is defined for integer values of y. We should not let an unsound argument interfere with presenting what is clearly (at least implicitly, with polynomial and power series representation, and many other formulas) the overwhelming consensus. Mark van Hoeij — Preceding unsigned comment added by 128.186.104.253 ( talk) 15:49, 17 December 2013 (UTC)
The obvious fact that some books don't define 00 is unimportant. No mathematician, save Trovatore, defines a function f(x) such that the definition of f(0) depends on whether 0 is 'in a real-number context'. Our article needs improvement. Bo Jacoby ( talk) 17:48, 7 December 2013 (UTC).
I don't want to drag on the debate and the edits, but I made an edit that I think is a good compromise. Rather than simply defining 0^0 to be 1, we can state (without controversy or originality) that there are formulas that require 0^0 to be interpreted as 1. The previous text is not OK because it simply says that 0^0=1 reduces the number of cases, but fails to mention that 0^0=1 is needed for the correct interpretation of these formulas. Mark van Hoeij — Preceding unsigned comment added by 71.229.28.197 ( talk) 18:39, 18 December 2013 (UTC)
Trovatore wrote: "The most common definition for real-to-real exponentiation is , and it doesn't follow from that". So, according to Trovatore et al., we must also undefine e.g. (−1)2 as ln(−1) is not defined as a real number. Bo Jacoby ( talk) 23:34, 18 December 2013 (UTC).
We have three real functions, f1 and f2 and f3 defined by
and
and
The domain of f1 is D1 = ℝ×{0,1,2,3,...}.
The domain of f2 is D2 = (ℝ×{0})∪((ℝ\{0})×{−1,−2,−3,...})
The domain of f3 is D3 = ℝ+×ℝ.
If (x,y) ∈ Di ∩ Dj then fi(x,y)=fj(x,y) for i,j=1,2,3.
Then define xy:
This definition covers the real cases of xy. I hope very much that we can eventually get rid of the ridiculous nonsense that 00 is defined or undefined depending on whether 0 is an integer or a real. Bo Jacoby ( talk) 21:51, 20 December 2013 (UTC).
Yes, 2+i0 = 2 is true. 2 > 1 is true. 1 = 1+i0 is true. So 2+i0 > 1+i0 is true. The fact that the ordering of reals is not extended to non-real complex numbers does not undefine the ordering of reals. Do you think that 2+i0 > 1+i0 is false? And why? Bo Jacoby ( talk) 21:51, 20 December 2013 (UTC).
You don't solve Clausen's paradox by making 2≠2+i0. The point is that a number is not either real or complex; a real number is both. Bo Jacoby ( talk) 20:02, 21 December 2013 (UTC).
Not every formalization is the usual one. Other formalizations are equally valid models. Real numbers can be defined as complex numbers of the form x+i0, and so on. Then we have the sequence of inclusions ℕ⊂ℤ⊂ℚ⊂ℝ⊂ℂ. Exponentiation of positive real (complex) numbers is not the same operation as exponentiation of negative or nonreal (complex) numbers. It is still ridiculous nonsense to claim that 2+i0≠2. Bo Jacoby ( talk) 21:16, 21 December 2013 (UTC).