Number to the Zero Power Equals One?

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Arjuna
 
Reply Sat 2 Jan, 2010 04:57 pm
@ACB,
ACB;116402 wrote:
Agreed, except that multiplication by 0 makes sense. If I have no boxes of 12 pencils, I have 0 x 12 = 0 pencils. If I have 12 boxes, each containing no pencils, I have 12 x 0 = 0 pencils.
Yea that's true. But you also have 0 x 144,321 pencils. That's the funny thing about 0. If 12 x 0 = 0, then does 12 = 0/0? See what I mean?
 
Owen phil
 
Reply Sat 2 Jan, 2010 06:06 pm
@Arjuna,
Arjuna;116477 wrote:
Yea that's true. But you also have 0 x 144,321 pencils. That's the funny thing about 0. If 12 x 0 = 0, then does 12 = 0/0? See what I mean?


x/x = 1, except in the case of x=0.
0/0=1, is false.

x/y =df (the z: x=y*z).

1/0 = (the z: 1=0*z). And, 1=0*z is true for no number z.
0/0 = (the z: 0=0*z). And, 0=0*z is true for all numbers z.

Division by 0 does not produce a unique number.

1/0 and 0/0 do not exist, even though they are defined.

There is no unique number that x/0 is, for all numbers x.
 
Zetetic11235
 
Reply Sun 3 Jan, 2010 04:42 pm
@Owen phil,
I still think that iterating the root function and looking at the limit gives the best feel for why x^0=1. Consider the square root functio. It has a fixed point at x=1. If we iterate the function on a value of X, it will approach that fixed point, that is; looking at the trend as we repeat the root function:

5.^(1/2)=2.23607

In[10]:= 2.23606797749979`^(1/2)

Out[10]= 1.49535

In[11]:= 1.4953487812212205`^(1/2)

Out[11]= 1.22284

In[12]:= 1.2228445449938519`^(1/2)

Out[12]= 1.10582

In[13]:= 1.1058230170302352`^(1/2)

Out[13]= 1.05158

In[14]:= 1.0515811984959769`^(1/2)

Out[14]= 1.02547

In[15]:= 1.0254663322098765`^(1/2)

Out[15]= 1.01265

In[17]:= (1.0126531154397722`)^(1/2)

Out[17]= 1.00631

And we have essentially applied the square root function recursively to 5, and our final value is 5^(1/256)=5^(.0039). Now, that exponent will only shrink as we apply the square root function again and again to the last output. So as the exponent gets very small we see that the value goes to 1, that is X^((1/2)^n)=1 as n gets large or equivalently as the exponent gets very small. This is the case for a starting point at X<1 too.

Now, in reality, there are infinitely many ways to show that x^0=1 is the only sensible value in the context of classical mathematics. Pick anyone you like, they are all logically equivalent.

To whoever said that mathematics doesn't deal with infinity; Here are ten reasons for why you are dead wrong:
Series (mathematics) - Wikipedia, the free encyclopedia
Infinitary combinatorics - Wikipedia, the free encyclopedia
Set theory - Wikipedia, the free encyclopedia
Aleph number - Wikipedia, the free encyclopedia
Algebraically closed field - Wikipedia, the free encyclopedia
Group theory - Wikipedia, the free encyclopedia
Infinite-dimensional holomorphy - Wikipedia, the free encyclopedia
Fixed point theorems in infinite-dimensional spaces - Wikipedia, the free encyclopedia
Affine Lie algebra - Wikipedia, the free encyclopedia
Infinitary logic - Wikipedia, the free encyclopedia
 
 

 
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