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

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

Indeterminate

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0^0 = 0/0

so it's whatever you think 0/0 is. This isnt really a riddle...

yes there are laws like the ones you stated in your OP as 'quick laws' to help people out such as yourself who don't understand what setting something to the zeroeth power actually means (sorry lol, but you should look it up or type it into your calculator )

there are other 'quick laws'... all with exceptions. If they didn't, 0/0 would be 1, judging by the 'quick laws'. And maybe it is. But as Aatif said, it is indeterminate

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The function f(x,y) = x^y has a discontinuity at the origin: the limit along x=0 is 0, and the limit along y=0 is 1.

There is some benefit if we define the value to be 1: for the binomial theorem valid everywhere, x^0 must = 1 for all x.

There is no comparable benefit for making the function 0^x well behaved at x=0, so much of modern thought favors defining the value as 1.

But others argue it should remain indeterminate, because x^y is in fact discontinuous at the origin.

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x0 = xa / xa

where a is any number, in the case of 0 it doesn't matter, you might as well use 1 for a (you just cant use 0)

so:

x0 = x1 / x1

x0 = x / x

00 = 0 / 0

it's simple exponents.

50 = 51 / 51 = 5 / 5 = 1

or

50 = 52 / 52 =25 / 25 = 1

etc

it's simple exponents. The number that is the actual exponent isn't a real number

53 is just 5*5*5

we only base the "quick-rule" property of 0^0 based on the fact that ALL NUMBERS EXCEPT 0 are 1 when divided by themselves

I'm not gonna speak for programming languages and stuff, they obviously are gonna make it whatever is best for programming with that language. Though if you're asking what it is at its core, it's 0/0

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x^y is well defined everywhere except the origin.

That is, [limit as x,y -> 0] x^y is not well defined - it depends on how x and y approach 0, or which one gets there first.

But that does not mean 0^0 can have any value whatsoever, the way 0/0 can:

For example,

[any_value]x=0 -> [any_value]=0/x. let x->0. [any_value]=0/0.

The 0^0 case is more well behaved.

[1] [limit x->0] x^0 = 1.

[2] [limit y->0] 0^y = 0.

Look more closely at case [1] and invoke the equation x^0 = x/x:

[limit x->0] x^0 = [limit x->0] x/x = [limit x->0] 1 = 1.

That is, x/x is well behaved [continuous] at 0; even tho it's 0/0 there, its value is unambiguously 1.

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

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0 is a number too 0 real and 0 imaginary on the complex plane (in other words, the origin point of the complex plane)

bonanova: I could also use limits to prove, with x/y, that as y approaches 0, the answer approaches infinity as it gets bigger and bigger (and thus disprove the any value agreement). So in both cases 'limits' and 'algebra' are saying different things. In both cases, they're undefined. 0/0 is undefined. 0^0 is undefined. The function of x^y is just following the trend of exponents, of which x^0 is simply expressed by the laws of exponents as x/x. Since 0/0 is not always 1, neither is 0^0. Then again, it could be blue cows ;D

Rule #1: There's an exception to every rule (including Rule #1)

lol

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It is true that x/x = 1 and will always be equal to one but there should be a limit that x <> 0 (x must not be equal to zero). If the problem given is (2x+y)/(x-1), then it should always be understood that x must not be equal to positive 1 to avoid undetermined answer.

I hope this simple info helps.

Edited by RoseOnPlayer

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Exactly. 0/0 isnt always 1 (though it can be). 0^0 is equal to 0/0, so the same applies. Bonanova was saying that 0^0 is a special case of 0/0 only if the current meaning of 0/0 was 1, however I am disputing that. He is using the function created by x^y as evidence, but a wedding ring doesn't make you married. Being married makes you have the wedding ring. The laws of exponents are the laws of exponents, nothing else as far as I know. 0^0 = 0/0. Though maybe bonanova has some more proof showing that 0^0 is only a special case of 0/0 and somehow set apart differently, I'd like to see it

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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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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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Who's Virginia?

Introducing ... Virginia. I thought everyone knew ...

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Introducing ... Virginia. I thought everyone knew ...

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

x2 =x2-2=x0 =1

x2

Edited by phil

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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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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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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

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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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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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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.

The perfect/ideal/combined gas law only makes sense if the pressure and temperature are absolute. Celsius/Fahrenheit temperatures, and gage pressures are strictly verboten.

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

-

0

(or not zero)

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There is a basic ambiguity : You cannot discuss about 0^0 without specify the related field of matematics (set theory, or elementary algebra, or topology, or...). i.e. :

"Zero to the Zero-th Power", translated from "Zéro puissance zéro"

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it makes the bionomial theorm nicer to say 0^0 = 1; though that's hardly a proof.

but math is there for convience as much as anything.

for example, if we define some scientific constants to be 1, it can make alot of other scientific calculations easier.

but i agree, though we say 0^0 = 1, that doesn't nessicarly mean 0^0 = 1.

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