X to the Zeroth Power Equals One
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after 13 votes the winner is...
Subutai
Voting Style:  Open  Point System:  7 Point  
Started:  4/6/2013  Category:  Science  
Updated:  3 years ago  Status:  Post Voting Period  
Viewed:  3,226 times  Debate No:  32170 
Debate Rounds (3)
Comments (26)
Votes (13)
This is just for a relaxing debate. The resolution is that X^{0}=1 for all real numbers. 24 hours to post arguments, 4000 character limit, no acceptance restrictions. 0 is a real number. 0^{0} is not equal to 1 (it is undefined). X^{0} is only equal to 1 for X not equal to 0; the resolution is false. The limit of 0^{X} as X approaches 0 is 0 The limit of X^{0} as X approaches 0 is 1 Both tend towards 0^{0}; 0^{0} is undefined. 

My opponent seems to agree that X^{0}=1 for all cases except where X=0. He says that that value is undefined. However, using math, it can be shown that for that value, the equation is also equal to 1.
Think of the function X^{X }and what happens as X approaches zero: = = = = = = =e^{0 } =1 Therefore, because =1, 0^{0 }=1^{ }________________________________________________________________________________________________ Reference: http://www.askamathematician.com... lim = limit exp = e^{X } (d/dx [f(x)]) = Equals the derivative of some function, f(x).
Limits are useful in showing what functions tend towards, however they do not show what functions are equal to. For example, lim_{x>0 }0/x = 0, however this does not mean that 0/0 is equal to 0. X^{X }may tend towards 1 as X approaches 0, however it is not equal to 1. My opponent's proof involves opperations which are undefined when X = 0, therefor it does not show that X^{X }is 1 when X is 0. Consider the following steps in my opponent's proof: exp( 0*log(0) ) log(0) is undefined lim_{x>0 }log(x) = ∞ 0 * ∞ is undefined exp( log(0)/01 ) Both 0^{1}and log(0) undefined lim_{x>0} x^{1} = ∞ ∞/∞ is undefined exp( 01/ 02) Both 0^{1 }and 0^{2 }are undefined Thus exp(0^{1}/ 0^{2}) is undefined and is not equal to exp( 0 ) thus 0^{0}=/=1 (it is undefined) Even if rewritten as exp(0^{2}/ 0^{1}) this opperation still involves division by 0 which is undefined. My opponents proof shows that X^{X} tends towards 1 as X tends towards 0, however 0^{0} remains undefined as shown above. Thus, X^{0} is not 1 for all real numbers; the resolution is negated. 

It is true that x^{0 }equals 1 for every nonzero value. It is only reasonable to assume that the function equals 1 when x is zero also. Here is a graphical representation. It is a plot of z = x^{y}. The red curves (with z constant) yield different limits as (x,y) approaches (0,0): The green curves (of finite constant slope, y = ax) all yield a limit of 1. Think of it another way. Equations such as the binomial theorem require 0^{0} to equal 1 for x=1 in order for that equation to be valid. The power rule in differential calculus requires 0^{0} to equal 1 for x=0. Here is an example of how the binomial theorem requires this. Using this equation: After expanding the binomial power using the binomial theorem and with a little other algebra, we have: Pulling the k=0 terms out of both equations yields: More algebra yields: That produces: 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 220,825, while is 0^{0 }equals 1, the verified result of 220,815 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. ____________________________________________________________________________________________ http://mathscitech.org... http://en.wikipedia.org... http://www.math.hmc.edu... ∑ = sum of Binomial Theorem: Somewhat ironically, my opponent's graph from http://en.wikipedia.org... (see: In Analysis) was actually a visual demonstration of why from an analytic point of view 0^{0} is indeterminate since x^{y} is not continuous at the origin. Depending on f and g, the limit of f(t)^{g(t)} as f and g tend towards 0 can be found to be any nonnegative real number, ∞, or undefined. The fact that for f and g under certain conditions the limit is 1 does not help my opponent's case considering the many functions where this is not the case. My opponent's source http://www.math.hmc.edu... also states that 0^{0} is undefined, but if it could be defined it "should" be 1. My opponent is correct that there are contexts such as the binomial theorem where it is useful to treat 0^{0} as 1; however, there are other contexts where treating 0^{0} as 1 is problematic and leads to contradictions, such as in analysis. Depending on the context in which 0^{0} occurs, you may want 0^{0} to be 1, indeterminate, or undefined. For this reason it would be innaccurate to state that 0^{0} = 1; the resolution is negated. Thanks to Subutai for an interesting maths debate. 
13 votes have been placed for this debate. Showing 1 through 10 records.
Vote Placed by 1Historygenius 3 years ago
Subutai  Enji  Tied  

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Reasons for voting decision: Pro used reliable sources. I also felt Pro made the better case and I understood him better. I no way am I a math wiz.
Vote Placed by NurAbSal 3 years ago
Subutai  Enji  Tied  

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Reasons for voting decision: Con's major criticism of Pro's case was the problem of a limit approaching a value rather than reaching that value. This was actually a decent argument, and I applaud Con for this. However, I think Pro's assertion that the situation is not indeterminate and therefore requires a real answer won the argument, since it substantiated his assessment of the limit as the answer. Great job to both.
Vote Placed by utahjoker 3 years ago
Subutai  Enji  Tied  

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Reasons for voting decision: While I have a decent understanding of mathematics I am certainly not a mathematician so In deciding who would get my vote I kept an open mind and decided who ever made me understand their side better would win. So going down the voting categories both are tied on who did I agree with after and before because I had have to keep an open mind. Tied on Conduct because both gave good conduct in the debate.Spelling and grammar tied didn't see any real mistakes. Now comes the breaking point who have a more convincing argument reading the arguments I have to go with Subutai they had a complicated argument, but have a road map that made it easy to understand and I could see clearly what they were talking about. When it came to sources I had to give it to Subutai they had much more sources and had very reliable ones. My votes go to Pro on a overall great debate
Vote Placed by jh1234l 3 years ago
Subutai  Enji  Tied  

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Reasons for voting decision: Sources to pro for using reliable sources, arguments to con as he negated the resolution.
Vote Placed by Ragnar 3 years ago
Subutai  Enji  Tied  

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Reasons for voting decision: Close one! However Con admitted that 0 to the 0th power sometimes equals 1; which relates to the original question more than Pro's argument of always.
Vote Placed by Lizard 3 years ago
Subutai  Enji  Tied  

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Reasons for voting decision: I'm countering KingDebator, but I'm also countering DoubtingDave because he says his RFD is in the comments but I don't see it anywhere (Am I just missing it?)
Vote Placed by DoubtingDave 3 years ago
Subutai  Enji  Tied  

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Reasons for voting decision: I'm giving pro the conduct point for con's false accusation of plagarism (note that he pointed out in the sources that he received the graph from WikiPedia). See comments for full RFD
Vote Placed by ConservativePolitico 3 years ago
Subutai  Enji  Tied  

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Reasons for voting decision: Countering LibertarianWithAVoice. He used his own source to assert the resolution and didn't back up Pro's arguments in the slightest. In fact, it sounds like he didn't read the debate at all.
Vote Placed by LibertarianWithAVoice 3 years ago
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Reasons for voting decision: First off I would like to say the formula X^0=1 is a mathematical fact. http://www.calculatorsoup.com/calculators/algebra/exponent.php Go ahead and look at this resource e. It has links to other sources. Pro gets arguments because they are right. They also get sources because of their several links and their graph.
Vote Placed by KingDebater 3 years ago
Subutai  Enji  Tied  

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Reasons for voting decision: Pro's arguments were more convincing to me, and he used more sources that were more reliable.
For future reference, If I "Vote Bomb" a debate of yours would you message me so I can fix it.
The claim that "0^0 = 1 is a mathematical fact" is false and mathematicians have debated this claim since the 19th century. An online exponent calculator is hardly a sufficient argument against this and had you read the references (which you pointed out in your RFD), Wolfram Alpha states that 0^0 is undefined and that although 0^0 is sometimes defined as 1 to simplify formulas (as your source does), this is not a mathematical truth.
It is true that I didn't provide many sources. The only sources I did provide were ones which were initially used by Subutai to support his side which actually supported my side. Notably the impressive graph from wikipedia which Subutai provided in his final round actually showed why 0^0 can not be defined as 1 in analysis because it is discontinuous at the origin. If you count the sources which Subutai provided which supported my argument, then Subutai used twice as many sources as I did; however otherwise we used the same number.
 http://mathscitech.org......
 http://en.wikipedia.org......
 http://www.math.hmc.edu......
Here are Cons:
 http://en.wikipedia.org......
 http://www.math.hmc.edu...
I think you guys should read the Debate. It's clear where the Web references are. Also Wikipedia is not a reliable source. The source I provided is a calculator ( for you without one, I was trying to be nice. ) Sorry if you didn't see the links when you were reading the debate. :(
When he pointed out that 0^0 is undefined he has already won. No more arguments are required.
People are really bad at math >_<
you could call votebomb on both his and kingdebator's RFD's. LibertarianWithAVoice's on the grounds that he's voted based on a source he provided himself in his own RFD and not explaining why he thinks pro is right, and King's also based on him giving arguments points without any actual justification in terms of explaining why he thinks pro is right.
In all honesty, I also don't see why anyone would vote source points against you seeing as you turned Pro's sources, but that's probably just a difference of opinion. To me, turning sources > greater amount of sources.
As for arguing in the comments section, I was simply rephrasing what I argued. If you read my arguments, you would have seen that I proved that indeterminacy is not an option. Which is all I'm saying. Con's limit argument doesn't disprove the resolution.
I'm not removing the conduct point. It's a counter. I was wrong that you didn't source it, but that wasn't the reason why I countered to begin with.
I also find it distasteful that you're now arguing against your opponent's last round in the comments section.