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#1 2006-11-04 01:15:15

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

To the power of 0

Why exactly does

, and not 0, or something else?

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#2 2006-11-04 02:36:56

polylog
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Registered: 2006-09-28
Posts: 162

Re: To the power of 0

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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#3 2006-11-04 02:39:53

Toast
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Registered: 2006-10-08
Posts: 1,321

Re: To the power of 0

Yes, but what is 'm' and what is 'n' (in integer form)?

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#4 2006-11-04 03:26:44

polylog
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Registered: 2006-09-28
Posts: 162

Re: To the power of 0

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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#5 2006-11-04 05:42:57

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: To the power of 0

polylog wrote:

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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#6 2006-11-04 05:45:43

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: To the power of 0

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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#7 2006-11-04 06:32:48

luca-deltodesco
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Registered: 2006-05-05
Posts: 1,470

Re: To the power of 0

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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#8 2006-11-04 09:44:46

MathsIsFun
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Registered: 2005-01-21
Posts: 7,711

Re: To the power of 0

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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#9 2006-11-04 22:29:47

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: To the power of 0

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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#10 2006-11-04 22:37:27

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: To the power of 0

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])?eek

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.touched

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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#11 2006-11-05 00:24:45

MathsIsFun
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Registered: 2005-01-21
Posts: 7,711

Re: To the power of 0

0[sup]0[/sup]?  I covered that one at the bottom of ANOTHER page: Laws of Exponents

wink


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#12 2006-11-05 05:02:45

Neha
Member
Registered: 2006-10-11
Posts: 173

Re: To the power of 0

ok guys how about this :
Write 0.7165 as e raised to a power.


Live Love Life

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#13 2006-11-05 06:17:04

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: To the power of 0

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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#14 2006-11-05 16:27:01

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: To the power of 0

Neha wrote:

ok guys how about this :
Write 0.7165 as e raised to a power.

0.7165=e[sup]ln0.7165[/sup] dizzy

You see- they say such a number is well defined even without writing it out


X'(y-Xβ)=0

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#15 2006-11-05 16:31:07

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: To the power of 0

MathsIsFun wrote:

0[sup]0[/sup]?  I covered that one at the bottom of ANOTHER page: Laws of Exponents

wink

Yes you did!;) Vast coverage!


X'(y-Xβ)=0

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#16 2006-11-05 18:29:10

MathsIsFun
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Registered: 2005-01-21
Posts: 7,711

Re: To the power of 0

mathsyperson wrote:

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 smile

And then 0/0 is both undefined and indeterminate.

I think dizzy


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#17 2006-11-05 19:17:31

Toast
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Registered: 2006-10-08
Posts: 1,321

Re: To the power of 0

0/0 is 1 right?

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#18 2006-11-06 15:25:53

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: To the power of 0

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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#19 2006-11-06 15:47:08

polylog
Member
Registered: 2006-09-28
Posts: 162

Re: To the power of 0

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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#20 2006-11-07 12:03:15

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: To the power of 0

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.


"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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#21 2006-11-07 14:04:56

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: To the power of 0

Ricky wrote:
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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#22 2006-11-07 18:45:16

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,711

Re: To the power of 0

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
dizzy


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#23 2006-11-07 20:30:23

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: To the power of 0

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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