The Instigator
WilliamofOckham
Pro (for)
Losing
3 Points
The Contender
Subutai
Con (against)
Winning
21 Points

Zero to the Zeroth Power Does Not Equal One

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Post Voting Period
The voting period for this debate has ended.
after 6 votes the winner is...
Subutai
Voting Style: Open Point System: 7 Point
Started: 5/5/2013 Category: Miscellaneous
Updated: 4 years ago Status: Post Voting Period
Viewed: 1,486 times Debate No: 33320
Debate Rounds (4)
Comments (0)
Votes (6)

 

WilliamofOckham

Pro

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.
Subutai

Con

I accept, and thank WilliamofOckham for challenging me to this debate.
Debate Round No. 1
WilliamofOckham

Pro

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 side-by-side, 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.
Subutai

Con

I would like to thank my opponent for presenting his arguments.

The fact that the form 00 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:



=

=

=

=

=

=

=e0

=1

Therefore, because =1, 00=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 00=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 00=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 00 equals 0 or 1. Indeterminacy is not an option, since the situation is real and is required to continue the simplification. If we take 00 to equal 0, the final formula for Sp(N) is off by a linear constant –N. Choosing 00 to equal 1 yields the right answer. For example, for S5(10), if 00 equals 0, we end up with the erroneous answer of 220,815, while is 00 equals 1, the verified result of 220,825 is determined.

In general, the sum n=0->∞ an(x-c)n is problematic at x=c, and this can be fixed by defining 00 to equal 1, which arrives at the verified correct answer, whereas defining 00 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 00can equal 1. In addition, many proven rules and theorems in mathematics require 00=1, such as the binomial theorem, which only yields a correct answer for 0 when 00=1.

Overall, 00=1 for kj where k and j are discrete variables.

________________________________________________________________________________________________


http://www.askamathematician.com...;
http://www.quora.com...
http://mathscitech.org...;
Debate Round No. 2
WilliamofOckham

Pro

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.
Subutai

Con

Intriguing. Twice in just a day that someone has conceded a debate with me. Oh well. Thanks for the debate; it was fun.
Debate Round No. 3
WilliamofOckham

Pro

Sorry to have wasted your time.
Subutai

Con

It's fine. I enjoyed it while it lasted.
Debate Round No. 4
No comments have been posted on this debate.
6 votes have been placed for this debate. Showing 1 through 6 records.
Vote Placed by A.WitherspoonVI 4 years ago
A.WitherspoonVI
WilliamofOckhamSubutaiTied
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Total points awarded:16 
Reasons for voting decision: Concession
Vote Placed by Vulpes_Inculta 4 years ago
Vulpes_Inculta
WilliamofOckhamSubutaiTied
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Total points awarded:13 
Reasons for voting decision: Concession. Point to Pro for being honest.
Vote Placed by johnlubba 4 years ago
johnlubba
WilliamofOckhamSubutaiTied
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Total points awarded:13 
Reasons for voting decision: I have no idea what this is about. but pro conceded and gets my conduct point.
Vote Placed by wiploc 4 years ago
wiploc
WilliamofOckhamSubutaiTied
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Reasons for voting decision: Concession.
Vote Placed by Kwhite7298 4 years ago
Kwhite7298
WilliamofOckhamSubutaiTied
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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 4 years ago
x2MuzioPlayer
WilliamofOckhamSubutaiTied
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Total points awarded:05 
Reasons for voting decision: Pro conceded.