Understanding the Controversy: Is 0^0 Really Equal to 0?

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Discussion Overview

The discussion centers around the mathematical expression 0^0 and whether it can be equated to 0. Participants explore various mathematical approaches and reasoning related to the validity of the expression, including limits and algebraic manipulations.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a series of equations suggesting that solving x^x = x leads to the conclusion that 0^0 = 0, arguing that raising nothing to nothing results in nothing.
  • Several participants question the validity of the initial equations, particularly the transition from x^[1/x] = x to x[x^(1/x-1) - 1] = x, expressing confusion about the algebra involved.
  • Another participant points out that substituting x = 0 leads to an invalid operation (1/0), challenging the logic of the argument.
  • Some participants assert that the expression 0^0 is indeterminate, with one noting that limits approaching 0 yield different results (e.g., lim_{x → 0} x^x = 1 and lim_{x → 0} 0^x = 0).
  • Another participant argues that the expression could be defined as 1 for convenience in certain contexts, but emphasizes that it is generally not determined.
  • A participant claims that 0^0 could be considered undefined, as it presents conflicting interpretations based on the properties of exponents.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the value of 0^0, with no consensus reached on whether it should be defined as 0, 1, or left undefined. The discussion remains unresolved.

Contextual Notes

Participants highlight limitations in the algebraic manipulations presented, particularly regarding the handling of indeterminate forms and the assumptions made in the reasoning. The discussion reflects a variety of interpretations and applications of mathematical principles related to exponents.

mathelord
a friend [no longer a user of the forum]showed me this and i felt i should as well show this to you all
let x^x=x
x^[1/x]=x
x^[1/x]-x=0
x[x^(1/x-1)-1]=x
so x is 0 and 1.
which evntually gives 0^0 as 0
in sloving with anither method,he also got -1 as x.
how true is the topic 0^0=0.
he explained that if you have nothing and you raise it to nothing you eventually get nothing.
 
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mathelord said:
...
x^[1/x]-x=0
x[x^(1/x-1)-1]=x
...
How did you get from the first step to the second ?
 
If you have x = 0, you get 1/0 which is invalid.
I also fail to see how you got from the first to the second equation.
 
I don't actually see how the first equation is valid , never mind the second one.

Could somebody please explain ?
 
It's valid because he wants to solve x^x = x, for x.
 
mathelord said:
a friend [no longer a user of the forum]showed me this and i felt i should as well show this to you all
let x^x=x
x^[1/x]=x
x^[1/x]-x=0
x[x^(1/x-1)-1]=x
so x is 0 and 1.
which evntually gives 0^0 as 0
in sloving with anither method,he also got -1 as x.
how true is the topic 0^0=0.
he explained that if you have nothing and you raise it to nothing you eventually get nothing.

mathelord, this is all pretty trivial. What is it that's confusing you? If you "have nothing and you raise it to nothing", then you don't have anything to exponentiate! You don't "eventually" get nothing, it was always nothing.
 
Last edited:
benjamincarson said:
mathelord, this is all pretty trivial...
Factoring x from 0 and getting x is some pretty nontrivial algebra. :smile:
 
hypermorphism said:
Factoring x from 0 and getting x is some pretty nontrivial algebra. :smile:
I don't see how he was "factoring x from 0 ". Anyway, It wouldn't be hard to construct a proof that shows that the equation [tex]x^{x}=x[/tex] is only valid for 0 and 1. So...
[tex]0^{0}=0[/tex]
[tex]1^{1}=1[/tex]

Fiddle-dee-do
 
Well:

[tex]\lim_{x \rightarrow 0} x^x = 1[/tex]

[tex]\lim_{x \rightarrow 0} 0^x = 0[/tex]

Some times 00 = 1 is defined as for usefulness, but in general its not determined.
 
  • #10
benjamincarson said:
I don't see how he was "factoring x from 0 "...
Look at his fourth step.
benjamincarson said:
It wouldn't be hard to construct a proof that shows that the equation LaTeX graphic is being generated. Reload this page in a moment. is only valid for 0 and 1...
It would be impossible. 00 is an indeterminate form. See here and here.
 
  • #11
x^0=1 any number raised to zeor power equals 1. x*x*x*x...*x zeor amount of times equals 1 because 1 is the null value in multiplication. a number times himself zero times is equal to 1.
0^x=0 zero raised to any power equals zero. 0*0*0*0*0*...0=0 because of multiplication property. no matter how many zeors you have...you still have zero.

0^0=? Well...is it 1 becasue it is raised to zeor parts? or is it 0 becasue the 0 is raised to a power? Can't be both, but it can be none. answer: undefined.
 

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