Zero to the Zeroth Power Does Not Equal One
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after 6 votes the winner is...
Subutai
Voting Style:  Open  Point System:  7 Point  
Started:  5/5/2013  Category:  Miscellaneous  
Updated:  3 years ago  Status:  Post Voting Period  
Viewed:  1,372 times  Debate No:  33320 
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Hello. I saw your debate on whether or not 0 to the 0th power equals 1, and I would like to debate you on that topic if you are interested. Please use the first round for acceptance. Good luck.
I accept, and thank WilliamofOckham for challenging me to this debate. 

Thanks to my opponent for accepting this debate.
My basic argument is that there is no way to assign a value to the function 0^0. If we determine the limit as x approaches 0 of 0^x, we get one; if we determine the limit as x approaches 0 of x^0, we get 1  this concludes that we get two different values for the limit of x^x as x approaches 0, therefore we cannot assign it a value of one. 0 to any positive power is 0, so 0 to the power 0 should be 0. But any positive number to the power 0 is 1, so 0 to the power 0 should be 1. We can't have it both ways. Think of it another way. If the two functions (x^0 and 0^x) were placed sidebyside, there would be a discontinuity at all places. It is only reasonable to assume that the same discontinuity applies at x=0. 0^0 is undefined. I would like to thank my opponent for presenting his arguments. The fact that the form 0^{0} is an indeterminate form (in the sense that gives you no information about the limit) does not prevent us from assigning a value to the expression. It merely means that exponentiation cannot be a continuous function in any neighborhood of that value. The limit can be manipulated to this advantage. Take and solve: = = = = = = =e^{0} =1 Therefore, because =1, 0^{0}=1. There is a second proof behind this idea. The geometric series, binomial theorem, power rule, Jack Huizenga's answer, and other formulas in mathematics reuqire 0^{0}=1. If we can't prove from limit analysis what this expression equals, we can find it another way. This is an example of how the binomial theorem requires 0^{0}=1. Using this equation: After expanding the binomial power using the binomial formula and further manipulation, we get at: Pulling out both K=0: Further manipulation of the equation results in: At this point, we must consider whether 0^{0 }equals 0 or 1. Indeterminacy is not an option, since the situation is real and is required to continue the simplification. If we take 0^{0 }to equal 0, the final formula for S_{p}(N) is off by a linear constant –N. Choosing 0^{0 }to equal 1 yields the right answer. For example, for S_{5}(10), if 0^{0 }equals 0, we end up with the erroneous answer of 220,815, while is 0^{0 }equals 1, the verified result of 220,825 is determined. In general, the sum ∑_{n=0>∞} a_{n}(xc)^{n }is problematic at x=c, and this can be fixed by defining 0^{0} to equal 1, which arrives at the verified correct answer, whereas defining 0^{0} to equal 0 produces an incorrect answer. Either way, indeterminacy is not an option because the equation needs to be solved. In conclusion, the limit my opponent cites does not prove anything. Using another limit and reveal that 0^{0}can equal 1. In addition, many proven rules and theorems in mathematics require 0^{0}=1, such as the binomial theorem, which only yields a correct answer for 0 when 0^{0}=1. Overall, 0^{0}=1 for k^{j }where k and j are discrete variables. ________________________________________________________________________________________________ http://www.askamathematician.com...; http://www.quora.com... http://mathscitech.org...; 

My opponent obviously knows a lot more about math than I do, and it would be useless for me to try to continue this debate considering my opponent is more knowledgeable on the subject than I am. I will just respectfully bow out. Thanks for your time.
Intriguing. Twice in just a day that someone has conceded a debate with me. Oh well. Thanks for the debate; it was fun. 

Sorry to have wasted your time.
It's fine. I enjoyed it while it lasted. 
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6 votes have been placed for this debate. Showing 1 through 6 records.
Vote Placed by A.WitherspoonVI 3 years ago
WilliamofOckham  Subutai  Tied  

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Total points awarded:  1  6 
Reasons for voting decision: Concession
Vote Placed by Vulpes_Inculta 3 years ago
WilliamofOckham  Subutai  Tied  

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Total points awarded:  1  3 
Reasons for voting decision: Concession. Point to Pro for being honest.
Vote Placed by johnlubba 3 years ago
WilliamofOckham  Subutai  Tied  

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Total points awarded:  1  3 
Reasons for voting decision: I have no idea what this is about. but pro conceded and gets my conduct point.
Vote Placed by wiploc 3 years ago
WilliamofOckham  Subutai  Tied  

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Total points awarded:  0  3 
Reasons for voting decision: Concession.
Vote Placed by Kwhite7298 3 years ago
WilliamofOckham  Subutai  Tied  

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Reasons for voting decision: I didn't understand anything the Con wrote but pro conceeded so vote to con
Vote Placed by x2MuzioPlayer 3 years ago
WilliamofOckham  Subutai  Tied  

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Reasons for voting decision: Pro conceded.