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.
The concept behind "The Power of 0" is that zero, or nothing, has the potential to be powerful when it is acknowledged and agreed upon by a group of individuals. It is not just a number, but an agreement among people that gives it power.
This concept can be applied to various real life situations, such as in relationships, organizations, or even society as a whole. It suggests that when people come together and agree upon something, even if it seems insignificant or empty, it can have a significant impact and create a powerful force for change.
While there may not be direct scientific evidence for the concept of "The Power of 0," there are studies and theories that suggest the power of group dynamics and the impact of social agreements. For example, the bystander effect and herd mentality are both phenomena that demonstrate the power of group influence on individual behavior.
Yes, the power of zero can be harnessed for both positive and negative outcomes. It ultimately depends on the intentions and actions of the individuals who are in agreement. For example, a group of people can use their collective power to make positive changes in the world, but they can also use it to cause harm and chaos.
Individuals can make use of "The Power of 0" by being more aware of the agreements they make with others, whether it be in personal relationships or in larger societal structures. By recognizing and consciously participating in these agreements, individuals can have a greater understanding and influence on the power dynamics at play in their daily lives.