My simple proof of x^0 = 1

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  • #26
Mute
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One can define exponentials in terms of the family of standard functions and it falls naturally that anything to the power zero is one. Exponentials are defined as follows:

[tex] n^x = e^{x\ln{n}} [/tex]

Now put x = 0 and see what happens.
Even with technicalities about proving a^0 = 1 versus defining it, your argument there is circular. You want to "prove" that [itex]a^0 = 1[/itex], and then proceed to do so by writing it as [itex]e^{x\ln a}[/itex], and then set x = 0 and say, "Oh, e^0 = 1, so a^0 = 1", but you didn't prove that "e^0 = 1"! You used the result you were trying to prove in order to prove it!
 
  • #27
Kurdt
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I didn't say I was trying to prove anything. Its merely the standard definition of an exponential function and shows how numbers to the power zero are defined to be 1. And yes I did not explain other definitions such as that of the exponential function, but I was taking it as a well established result.
 
  • #28
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Can you prove that [tex]\frac{a^n}{a^m} = a^{n-m}[/tex] when [tex]n = m[/tex]? Usually the proof is only valid when they aren't equal and then people define [tex]x^0=1[/tex] such that the property remains valid for [tex]n = m[/tex].
perfect example of what will happen if people study more than actually required
 
  • #29
arildno
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For a decreasing sequence of integers as exponents, we might use the following definition of exponentiation to prove the statement:
[tex]a^{n-1}=\frac{a^{n}}{a}, a\neq{0}, a^{1}=a, n\leq{1}[/tex]

This definition is sufficient to prove that several properties we would like exponentiation to have actually hold.
 
  • #30
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a^0 = 1

FIRST: We use a one of the laws of exponents which is (a^m/a^n) = 1.

Let m=n,then (a^n/a^n) = a^(n-n) = a^0, where a is not equal to zero.

In short, (a^n/a^n) = a^0.

SECOND: There's a theorem that any nonzero number divided by itself is equal to 1.

That is, (a^n)/(a^n) = 1.


Combining the first and second by transitive property, we have a^0 = 1.

:)
 
  • #31
[tex]1 = \frac{a^n}{a^n} = a^{n-n} = a^0[/tex]


What could be more simpler than that?
I think this actually proves it the previous once doesn't really prove it, because you'd need the information above for it. Besides this is such a simple proof and anyone can understand it.
 

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