The discussion revolves around the concept of raising a number to the power of zero, specifically whether the result being equal to one is merely a definition or agreement among mathematicians. Participants explore the implications of this definition and its necessity in maintaining mathematical properties.
Participants express differing views on whether the definition of a number raised to the power of zero as one is simply an agreement or a necessary mathematical convention. No consensus is reached, and multiple perspectives remain present.
Some participants highlight the importance of definitions in mathematics, while others question the implications of changing those definitions. There are unresolved points regarding the nature of mathematical definitions and their foundational role in proofs.
It is according to the definition of raising a number to a power that a nonzero number to the zero power is equal to one, just like it is according to this definition that a x to the second power is x times x. It is really not an unnatural definition. We know that [itex]x^n\times x^m=x^{m+n}[/itex]. If we want this to be true then [itex]x^n\times x^0=x^n \rightarrow x^0=1[/itex]. Alternitively, powers can be defined by series. In this case [itex]b^x=e^{x\ln{b}}[/itex] andTSN79 said:We all know that a number to the power of 0 equals 1. Bit I heard this is only a definition, iow something that has just been agreed on. But how can such a thing be done? If one had agreed that it would equal 0 instead, I suppose many mathematical proofs and stuff would be different, things that can't be done suddenly would be possible. Is it really the case that this is just an agreement?
TSN79 said:We all know that a number to the power of 0 equals 1. Bit I heard this is only a definition, iow something that has just been agreed on. But how can such a thing be done? If one had agreed that it would equal 0 instead, I suppose many mathematical proofs and stuff would be different, things that can't be done suddenly would be possible. Is it really the case that this is just an agreement?
Excellent explanation thank you. However N+0=N not N+0=0.HallsofIvy said:Drop the word "just" and I will agree with you. In mathematics, everything in mathematics is a definition but not "just" a definition! As Leonhard Euler said, it is not "an unnatural definition".
From the simplest case, xn with n a positive integer, where we can think of xn as meaning "x multiplied by itself n times" (which is, after all, "just" a definition), we find that xnxm= xn+m because "n times" followed by "m times" is the same as "n+ m times".
If we want that very nice property to continue to be true even if m is 0, we must have xnx0= xn+ 0. But n+0= 0 so we have xnx0= xn. As long as x is not 0, we can divide both sides by xn and get x0= 1.
That is why we define x0 to be 1 (and only define it for x not equal to 0).
HallsofIvy said:Drop the word "just" and I will agree with you. In mathematics, everything in mathematics is a definition but not "just" a definition! As Leonhard Euler said, it is not "an unnatural definition".
From the simplest case, xn with n a positive integer, where we can think of xn as meaning "x multiplied by itself n times" (which is, after all, "just" a definition), we find that xnxm= xn+m because "n times" followed by "m times" is the same as "n+ m times".
If we want that very nice property to continue to be true even if m is 0, we must have xnx0= xn+ 0. But n+0= 0 so we have xnx0= xn. As long as x is not 0, we can divide both sides by xn and get x0= 1.
That is why we define x0 to be 1 (and only define it for x not equal to 0).
That's like saying 0*0 = 0 and 0*n = 0 (n= 1,2,3...) then 0 = 1,2,3...debeng said:i was confused if 1^0 = 1 and 1^n = 1 (n= 1,2,3...)
then o = 1,2,3...
That works also.StatusX said:This is the example I usually think of when this comes up: 1=xn/xn=xn-n=x0.