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Why exactly does
, and not 0, or something else?Offline
Because of the rule:
If n = m, then x^n/x^m will be 1... so x^(n - m) = x^0 must be 1 also.
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Yes, but what is 'm' and what is 'n' (in integer form)?
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m and n are any integers in this case.
But we could say they are both real numbers and the rule still applies.
The point is, x^0 = 1 is consistent with this rule and all the other rules of exponents. If it was defined as anything else, there would be inconsistency!
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Because of the rule:
If n = m, then x^n/x^m will be 1... so x^(n - m) = x^0 must be 1 also.
To be stated in a more clear fashion:
But what is a number divded by itself?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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Also, can you guess what:
Is? I'll give you a hint, it ain't pi.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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you can also see it in a pattern
looking at this pattern, or any other exponential function, looking at the patern, a^0 will always be 1 (apart from when the base is 0, in which case its a bit of an obscurity, and we let it be 1 by definition)
The Beginning Of All Things To End.
The End Of All Things To Come.
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Exactly, luca, as shown at the bottom of this page: Exponents
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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Suppose you do the algebra
2[sup]3[/sup]/(2*4)
and it is convenient and fluent for you to do 3-1-2=0 and 2^0=1
instead of having to write out 2[sup]3[/sup]/itself=1
Calculations like the former could Always replace the way similar to the latter Except when 0/0
So I hold that 0^0 should be undefined.
X'(y-Xβ)=0
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By the way, I don't quite agree with Ricky's corroboration in post 6
It is like advocating - Let's define it 1, for the continuity sake!
If that makes sense, how about we define 0[sup]0[/sup] as 0 (for the continuity of 0[sup]x[/sup]), or as 1 (for the continuity of x[sup]0[/sup])?
So the algebra reason should be the essential one, and you can refuse it so long as you would like use the second approach in post 9 all the time.
I know you mean no x multiplied doesn't make sense. But x^0 makes sense sometimes. Still you have the right to deny it.
Last edited by George,Y (2006-11-04 23:00:21)
X'(y-Xβ)=0
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0[sup]0[/sup]? I covered that one at the bottom of ANOTHER page: Laws of Exponents
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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ok guys how about this :
Write 0.7165 as e raised to a power.
Live Love Life
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On that Laws of exponents page, you say that 0^-1, for example, is undefined. And quite right because that would be 1/0 and so you have a division by 0 in there which is nonsense.
But you say that 0^0 is "indeterminate". Does that mean that there are different kinds of undefined numbers?
Why did the vector cross the road?
It wanted to be normal.
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ok guys how about this :
Write 0.7165 as e raised to a power.
0.7165=e[sup]ln0.7165[/sup]
You see- they say such a number is well defined even without writing it out
X'(y-Xβ)=0
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0[sup]0[/sup]? I covered that one at the bottom of ANOTHER page: Laws of Exponents
Yes you did!;) Vast coverage!
X'(y-Xβ)=0
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But you say that 0^0 is "indeterminate". Does that mean that there are different kinds of undefined numbers?
Ahh, the difference between "undefined" and "indeterminate" ...
0[sup]0[/sup] is not "undefined" is it? The problem is that there are two definitions!
But 1/0 is undefined, but not indeterminate
And then 0/0 is both undefined and indeterminate.
I think
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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0/0 is 1 right?
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Nice challenge.
Typically they would argue
0.1/0.1=1
0.01/0.01=1
0.001/0.001=1
......
Hence 0/0 could be defined as 1 when in the function x/x.
But I find this reason tricky and would refuse it.
X'(y-Xβ)=0
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re: the difference between "undefined" and "indeterminate" -- i think undefined means strictly "there is no meaningful way to define the symbol or operation", so we leave it undefined, 'illegal'. The term 'Indeterminate' is applied only to limit situations, ie, when we have a function like (x-1)/(x^2 - 1) and x approaches 1, the function's value approaches an indeterminate value -- not undefined, but *we are not able to determine the value being approached*.
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Nice challenge.
Typically they would argue
0.1/0.1=1
0.01/0.01=1
0.001/0.001=1
......
Hence 0/0 could be defined as 1 when in the function x/x.But I find this reason tricky and would refuse it.
That is a good reason, believe it or not, with the wrong conclusion. But you have to make it a good reason first. And to make it a good reason you do the following:
0.2/0.1 = 2
0.02 / 0.01 = 2
0.002/0.001= 2
......
Hence, we have arrived at two different ways to define 0/0, and thus, we say it is indeterminate. And we could see from this pattern that we can make f(x) / g(x) approach 0/0, but equal any real number we wish.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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George,Y wrote:Nice challenge.
Typically they would argue
0.1/0.1=1
0.01/0.01=1
0.001/0.001=1
......
Hence 0/0 could be defined as 1 when in the function x/x.But I find this reason tricky and would refuse it.
That is a good reason, believe it or not, with the wrong conclusion. But you have to make it a good reason first. And to make it a good reason you do the following:
0.2/0.1 = 2
0.02 / 0.01 = 2
0.002/0.001= 2
......Hence, we have arrived at two different ways to define 0/0, and thus, we say it is indeterminate. And we could see from this pattern that we can make f(x) / g(x) approach 0/0, but equal any real number we wish.
I have said when in the function x/x, haven't I?
X'(y-Xβ)=0
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x/x approaches 1 as x approaches 0, which would make 0/0 = 1
x^2/x approaches x as x approaches 0, which would make 0/0 = x
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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I have said when in the function x/x, haven't I?
Yes, that's what you said and I generalized it. When talking about defining numbers, you can't talk about a single function. That's just ridiculous.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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