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#11 bonanova

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Posted 24 February 2008 - 07:52 AM

0/0 is not 1, it's indeterminate because 0 technically isnt a real number, it's more of a lack of number.

Zero is [technically] not a real number?

The concept of zero was problematic long after natural numbers were codified.
"How can nothing be something?" people would sagely ask.
But multiplication had its identity element [unity] and addition needed one as well.
Here's how Wikipedia puts it:

Zero (0) is the least non-negative integer.
The natural number following zero is one and no natural number precedes zero.
Zero may or may not be considered a natural number,
but it is a whole number
and hence a rational number
and a real number
(as well as an algebraic number
and a complex number).

That's a fairly impressive set of credentials.
So, yes, Virginia, there is a zero, and it's not technically disenfranchised.

Hints:
Think of the score of a baseball game before the first pitch. Technically not a real score?
Think of the temperature [Celsius] of ice water. Technically not a real temperature?
Think of the number of ways that bonanova is perfect. He's technically not imperfect? [I can call several credible witnesses. ;) ]
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#12 Duh Puck

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Posted 24 February 2008 - 03:55 PM

So, yes, Virginia, there is a zero, and it's not technically disenfranchised.

Hints:
Think of the score of a baseball game before the first pitch. Technically not a real score?
Think of the temperature [Celsius] of ice water. Technically not a real temperature?
Think of the number of ways that bonanova is perfect. He's technically not imperfect? [I can call several credible witnesses. ;) ]


Who's Virginia?

I suspect the confusion may stem from the viewpoint of what a number is. As can be seen from the 25 different definitions for the noun, it's clearly not an easy thing to pin down. It seems to me there are two primary ways of viewing the number 0, each of which would lead you to a different answer to the question posed in the OP.

1. Zero is the natural number used to represent the absence of a quantity, such as bonanova's first suggestion of a baseball score of 0. We recognize this as a meaningful value, one which can factor into statistics (say the game ended 1-0; you can still factor the losing team's score into an avg points/game for the season). However, we also recognize the limitations of this non-value. A statistician would view any formula dealing with a division by this number (e.g., hits/points) as being indeterminate.

According to this use of zero, 0^0 would have to be indeterminate.

2. Zero as a real number in a series, such as the example of temperature. Consider a formula like the combined gas law: [pressure]x[volume]/[temp]=[constant]. Is a temperature of 0 degrees Celsius indeterminate? Of course not. However, if you switch it to Kelvin, in which zero can be described as the absence of heat, then you're back to the previous definition.

According to this use of zero, 0^0 = 1 would be a much more logical and useful answer.
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#13 bonanova

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Posted 27 February 2008 - 08:27 AM

Who's Virginia?


Introducing ... Virginia. I thought everyone knew ... :mellow:
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#14 Duh Puck

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Posted 27 February 2008 - 04:32 PM

Introducing ... Virginia. I thought everyone knew ... :mellow:

Thanks. I did read that before, a long time ago. Just forgot. Reviewing it was almost enough to renew my belief in Santa Claus, but then I got all tripped up by Richard Dawkins. It's so hard to figure these things out.
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#15 phil

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Posted 12 December 2008 - 12:15 AM

THE RULE IS THAT ANYTHING TO THE POWER OF ZERO IS ONE.

x2 =x2-2=x0 =1
x2

Edited by phil, 12 December 2008 - 12:17 AM.

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#16 RainThinker

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Posted 14 December 2008 - 09:49 PM

It could be 0*nothing which would still be 0

Or, 0^0, which would be 1

I guess it all depends on how you look at it.. -_-
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#17 Peterotooled

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Posted 07 March 2009 - 03:32 PM

If zero to the nth power is zero, and n to the zeroth power is one, then what is zero to the zeroth power?


Zero to the zeroth power is one! Google says so and so does Mathematica, so there!!
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#18 unreality

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Posted 07 March 2009 - 03:47 PM

2. Zero as a real number in a series, such as the example of temperature. Consider a formula like the combined gas law: [pressure]x[volume]/[temp]=[constant]. Is a temperature of 0 degrees Celsius indeterminate? Of course not. However, if you switch it to Kelvin, in which zero can be described as the absence of heat, then you're back to the previous definition.

According to this use of zero, 0^0 = 1 would be a much more logical and useful answer.



actually, no. In the Kelvin scale, 0 is the impossible temperature in that its absolute zero and cannot be reached according to physical laws (not for our lack of trying). This has a certain beauty to it because Kelvin is used for all the important equations, and dividing by Kelvin in this case can never be indeterminate because the temperature on the Kelvin scale could never be 0




THE RULE IS THAT ANYTHING TO THE POWER OF ZERO IS ONE.

x2 =x2-2=x0 =1
x2



that's the basic rule, yes, but it has one exception. I don't know how old you are or if you've taken Function Algebra yet but two functions can be equal but not equivalent. For example, take these two functions:


f(x) = x2 / x

g(x) = x


f(x) equals g(x), right? That's true, but the functions are not equivalent because the DOMAIN of f(x) is different than that of g(x). Domain(f(x)) = all real numbers EXCEPT ZERO, while any real number can be plugged into g(x)

that's the same case with your rule. The domain rejects 0 if you try to plug it into x. What is 02/02?? It's commonly agreed that 0/0 cannot be determined by itself. Your own rule says that 02/02 reduces to 00.... same with 01/01, 05/05. I'm not sure if multiple zero-by-zero divisions stack up in different ways (cuz it seems to lose communitivity) but essentially you're saying that x/x = x^0 = 1. Well think of those three things as f(x), g(x), h(x). Are they equal? Yes! Are they equivalent? No! The domain is different in x/x, but laws lead that to x^0 without reduction of domain if I'm not mistaken. But dividing out the x's in the secodn one to reach 1 does change the domain






edit ~ I've been looking around and a lot of mathematical rules depend on 0^0 = 1 for the special cases of their functions, like factorial and combinatorics and things. So can a thing be indeterminate (ie, i've always thought that 0^0 = CANNOT BE DETERMINED) but assumed to be a set value for the use of a function?

Edited by unreality, 07 March 2009 - 03:54 PM.

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#19 Romulus064

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Posted 08 March 2009 - 01:13 AM

Personally, I have my own beliefs on 0. (This is not backed up by anyone)

I believe that 0 has 3 distinct "flavors"

0^n=

0 if n>0
1 if n=0
undef if n<0

I base this on the idea that indeterminable operations can be undone.
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#20 d3k3

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Posted 09 March 2009 - 08:32 PM

First, note that a0 is only defined to be 1, because lim_x->0 ax = 1 (a <>0). However, the expression cannot be evaluated directly for any a. This can be seen by trying to invert the expression: if ax = 1 when x = 0, then 11/x should equal a, which, of course, it does not, even in the lim_x->0.

Second, the limit is one-sided at a=0.
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