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# 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: 4 years ago Status: Post Voting Period Viewed: 3,487 times Debate No: 32170
Debate Rounds (3)

 Pro This is just for a relaxing debate. The resolution is that X0=1 for all real numbers. 24 hours to post arguments, 4000 character limit, no acceptance restrictions.Report this Argument Con 0 is a real number. 00 is not equal to 1 (it is undefined). X0 is only equal to 1 for X not equal to 0; the resolution is false.00 is not equal to 1 The limit of 0X as X approaches 0 is 0 The limit of X0 as X approaches 0 is 1 Both tend towards 00; 00 is undefined.Report this Argument Pro My opponent seems to agree that X0=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 XX and what happens as X approaches zero: = = =====e0 =1 Therefore, because =1, 00 =1 ________________________________________________________________________________________________ Reference: http://www.askamathematician.com... lim = limit exp = eX (d/dx [f(x)]) = Equals the derivative of some function, f(x). Report this Argument Con Limits are useful in showing what functions tend towards, however they do not show what functions are equal to. For example, limx->0 0/x = 0, however this does not mean that 0/0 is equal to 0. XX 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 XX is 1 when X is 0. Consider the following steps in my opponent's proof:exp( 0*log(0) )log(0) is undefinedlimx->0 log(x) = -∞0 * -∞ is undefinedexp( log(0)/0-1 )Both 0-1and log(0) undefinedlimx->0 x-1 = ∞-∞/∞ is undefinedexp( 0-1/ -0-2)Both 0-1 and 0-2 are undefinedThus exp(0-1/ -0-2) is undefined and is not equal to exp( 0 ) thus 00=/=1 (it is undefined)Even if rewritten as exp(-02/ 01) this opperation still involves division by 0 which is undefined.My opponents proof shows that XX tends towards 1 as X tends towards 0, however 00 remains undefined as shown above. Thus, X0 is not 1 for all real numbers; the resolution is negated.Report this Argument Pro It is true that x0 equals 1 for every non-zero 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 = xy. 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 00 to equal 1 for x=1 in order for that equation to be valid. The power rule in differential calculus requires 00 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 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 220,825, while is 00 equals 1, the verified result of 220,815 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.____________________________________________________________________________________________http://mathscitech.org... http://en.wikipedia.org...http://www.math.hmc.edu...∑ = sum ofBinomial Theorem: Report this Argument Con 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 00 is indeterminate since xy 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 00 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 00 as 1; however, there are other contexts where treating 00 as 1 is problematic and leads to contradictions, such as in analysis. Depending on the context in which 00 occurs, you may want 00 to be 1, indeterminate, or undefined. For this reason it would be innaccurate to state that 00 = 1; the resolution is negated.Thanks to Subutai for an interesting maths debate.Report this Argument
26 comments have been posted on this debate. Showing 1 through 10 records.
Posted by LibertarianWithAVoice 4 years ago
@Enji It says (In my Link) That 0^0 Is defined as 1, it is how mathematicians describe t in (Most) instances. I didn't mean mathematical fact, and for that inaccuracy I apologize, I meant it is the assumed greater probability of 0^0=0 as I have been taught. I am sorry that I did not meet the RFD Standards, As I am still new to this site. I thought that Pros debate was stronger. Although you had good arguments ( Especially in round 2. I believe you won round 2 ). I thought when you consented that sometimes 0^0 should equal zero "My opponent is correct that there are contexts such as the binomial theorem where it is useful to treat 00 as 1; however, there are other contexts where treating 00 as 1 is problematic and leads to contradictions" Isn't a strong enough argument. ( From my point of view.) Sorry that I didn't do good enough in my original RFD. If you could message me and held me out with how I could work on it I would be very appreciative.

For future reference, If I "Vote Bomb" a debate of yours would you message me so I can fix it.
Posted by Enji 4 years ago
@ LibertarianWithAVoice: Voting on convincing arguments is based on who made the most convincing arguments; not who is right. There's nothing wrong with believing that Pro's argument from the utility of defining 0^0 as 1 is more convincing than my argument that the best definition depends on context so it would be inaccurate to conclusively claim that 0^0 = 1, however then your RFD should be that Pro's argument was more convincing than mine; not that Pro is right.

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.
Posted by LibertarianWithAVoice 4 years ago
@Enji @Ryuu I voted for Pro because he is right. There is no question. I sent that link to show my own research. If you want to see Pros Links and sources: Reference: -http://www.askamathematician.com......
- 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. :(
Posted by 4saken 4 years ago
I can't believe that Con lost.
When he pointed out that 0^0 is undefined he has already won. No more arguments are required.
People are really bad at math >_<
Posted by RyuuKyuzo 4 years ago
@Enji,
you could call vote-bomb 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.
Posted by RyuuKyuzo 4 years ago
I'm not going to squabble with you in the comments section on what was and was not argued in the debate, but it's readily apparent that you've said new things in the comments section. Con is right, so I'm retaining my vote.
Posted by Enji 4 years ago
LibertarianWithAVoice needs to provide a better RFD than a link to a website which wasn't mentioned in the debate.
Posted by Subutai 4 years ago
I see. When I was reading your vote, I came across, "...pro didn't source where he got the graph from", I thought I should have cleared that up.

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.
Posted by RyuuKyuzo 4 years ago
@Subutai

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.
Posted by Subutai 4 years ago
The number of mappings from the empty set (plugging in x=0 and y=1) to the empty set is 0^0. It has to be 1.
13 votes have been placed for this debate. Showing 1 through 10 records.