obviously you just have to put in y=x^0 on your trusty ti-84+ calculator and observe that it is 1
Well, the limits as x approaches 0 from the positive and negative directions of 0^x are both 0. The limits as x approaches 0 from the positive and negative directions of x^0 are both 1. So the answer is really quite simple,
0 is God
It has to be undefined. Here’s why,
The exponent system regards every number as a multiple of one.
Eg. 1*2 = 2 = 2^1
1*2*2 = 4 = 2^2
1*2*2*2 = 8 = 2^3
If we get rid of the first multiple of 2, we’re just left with 1, so 2^0 = 1
The beauty of this system is that it also works in reverse.
Eg. 1/2 = 0.5 = 2^-1
1/2/2 = 0.25 = 2^-2
1/2/2/2 = 0.125 = 2^-3
Now if we look at 0, see that 0^1 = 1*0 = 0. So, 0^0 can’t = 0 because we’ve already decided that 0^1 = 0. They can’t both be 0. So its easier just to let 0^1 = 0 and to leave 0^0 undefined.
EDIT : I’m an idiot. I just realised that there’s no reason why they can’t both be 0. Well shit… i’m stumped.
Wolfram Alpha says indeterminant.
[quote]malonetd wrote:
http://mathforum.org/dr.math/faq/faq.0.to.0.power.html
Dr. Math is generally a decent reference to go to when these questions pop in your head.[/quote]
That was a good site. Thanks.
undefined, does not exist, umpossible (me fail english?)
[quote]debraD wrote:
But it is debatable.
[/quote]
Really, that should have ended the thread, IMO. Seems a quite solid explanation and presentation and – absolutely no disrespect to the posters that followed – no further comment seems to have added anything to the explanation in the link.
[quote]Bill Roberts wrote:
debraD wrote:
But it is debatable.
Really, that should have ended the thread, IMO. Seems a quite solid explanation and presentation and – absolutely no disrespect to the posters that followed – no further comment seems to have added anything to the explanation in the link.[/quote]
Really, this shouldn’t have been posted, IMO. Seems like quite obvious ass kissing and favour seeking and – absolutely no disrespect to you – this comment seems to have ALSO added nothing to the thread.
[quote]ukrainian wrote:
malonetd wrote:
http://mathforum.org/dr.math/faq/faq.0.to.0.power.html
Dr. Math is generally a decent reference to go to when these questions pop in your head.
That was a good site. Thanks. [/quote]
I agree that this is essentially correct and that you should differ to it.
This is ok too, although let me try to actually add something constructive to the explanations in these links.
The critical point to the taken from the first link that what 0^0 equals, if anything at all, really depends on what sort of mathematical problem we are considering. The exponential function, like any other function in mathematics after all, is merely a definition–we can define the exponential function however we like. Now, of course, some definitions of the exponential function are more natural or obvious then others–as the first link says, 0^0=1 is a natural and useful way to define the function in some contexts. Just exactly how you define the exponential function though is ultimately arbitrary, with more or less useful ways of doing it.
This sort of thing in mathematics really bothers some people, who assume that mathematics is ultimately some set of absolute truths. I had a long discussion with hedo about this very point not long ago… I wish I would have thought of this example then. In reality though, even seemingly basic functions like addition and instances of them like 2+2=4 are ultimately definitions that could be set out however we like. The study of various definitions of these functions and how useful they are in various contexts, to put it crudely, is what is called “modern” or “abstract” algebra. Of course, all the defining we do in algebra is guided by some set of axioms (the group, ring, field axioms) that intuitively tells us how we think, for example, addition should act.
The real number system and the familiar definitions of addition, multiplication, and the exponential that are just assumed in highschool algebra classes and calculus is just one particularly useful “number system” and set of functions that happen to correspond with a lot of our intuitions. Questions of whether or not there are other sorts of number systems and ways of understanding these functions generally don’t come up until you hit “singularities” like 0^0.
[quote]Ov3rman wrote:
It has to be undefined. Here’s why,
The exponent system regards every number as a multiple of one.
Eg. 1*2 = 2 = 2^1
1*2*2 = 4 = 2^2
1*2*2*2 = 8 = 2^3
If we get rid of the first multiple of 2, we’re just left with 1, so 2^0 = 1
The beauty of this system is that it also works in reverse.
Eg. 1/2 = 0.5 = 2^-1
1/2/2 = 0.25 = 2^-2
1/2/2/2 = 0.125 = 2^-3
Now if we look at 0, see that 0^1 = 1*0 = 0. So, 0^0 can’t = 0 because we’ve already decided that 0^1 = 0. They can’t both be 0. So its easier just to let 0^1 = 0 and to leave 0^0 undefined.
EDIT : I’m an idiot. I just realised that there’s no reason why they can’t both be 0. Well shit… i’m stumped.
[/quote]
This sort of reasoning by considering what we already know about a function doesn’t generally help us to come to any decisions about parts of a function we don’t know. As I described, in defining a function we want to consider things about that function that we want to hold and then define the function in such a way that they hold. For a crude example example, we want it to be true that (x^y)^z=x^(yz) and (x^y)(x^z)=x^(y+z). So, if we want a definition of the exponential where these formulas hold in all cases we see that 0^0=1 will make both these conditions true. IE, (0^0)(0^0)=11=1 and 0^(0+0)=0^0=1. You can check that if you happened to define 0^0=2, these conditions wouldn’t work. Of course, these conditions also would allow us to define 0^0=0, but there will be other sorts of conditions that one can stipulate for the exponential function that will rule this case out. My example was merely meant to show how one may set up stipulations for a function and then use those stipulations to guide the definition of the function. Also, there are of course some sorts of problems in mathematics were conditions like these aren’t critical, and the sort of stipulations the problem puts on the exponential function are such that it’s indeterminate. That’s fine too.
[quote]LittleAsianDoll wrote:
Schadenfreude8 wrote:
it’s 1, anything, no matter the number, raised to zero, is one
I have to disagree, I’ve always thought 0 was not a number.[/quote]
What is a number is a matter of definition and partly semantics. The notion of zero plays a critical role in any systematic approach to mathematics since zero is, if nothing else, the minimal element or the generator for the natural numbers {0,1,2,3,…}. Zero is also important in establishing axioms for the reals and other more complicated number systems, since zero is the additive identity element, x+0=x and 0+x=x.
Whether or not you can always substitute zero in for a variable that is suppose to range over numbers, is a question that various from context to context. For example, someone else mentioned that you can’t “divide by zero”. That’s not always true. There are certain situations where ‘x/0’ is perfectly well defined and has an answer.
[quote]JLu wrote:
Bill Roberts wrote:
debraD wrote:
But it is debatable.
Really, that should have ended the thread, IMO. Seems a quite solid explanation and presentation and – absolutely no disrespect to the posters that followed – no further comment seems to have added anything to the explanation in the link.
Really, this shouldn’t have been posted, IMO. Seems like quite obvious ass kissing and favour seeking and – absolutely no disrespect to you – this comment seems to have ALSO added nothing to the thread.[/quote]
I would kiss Debra’s ass too.
You guys are tools, just saying…
[quote]stokedporcupine8 wrote:
ukrainian wrote:
malonetd wrote:
http://mathforum.org/dr.math/faq/faq.0.to.0.power.html
Dr. Math is generally a decent reference to go to when these questions pop in your head.
That was a good site. Thanks.
I agree that this is essentially correct and that you should differ to it.
debraD wrote:
But it is debatable.
http://en.wikipedia.org/...eterminate_form
Really, that should have ended the thread, IMO. Seems a quite solid explanation and presentation and – absolutely no disrespect to the posters that followed – no further comment seems to have added anything to the explanation in the link.
This is ok too, although let me try to actually add something constructive to the explanations in these links.
The critical point to the taken from the first link that what 0^0 equals, if anything at all, really depends on what sort of mathematical problem we are considering. The exponential function, like any other function in mathematics after all, is merely a definition–we can define the exponential function however we like. Now, of course, some definitions of the exponential function are more natural or obvious then others–as the first link says, 0^0=1 is a natural and useful way to define the function in some contexts. Just exactly how you define the exponential function though is ultimately arbitrary, with more or less useful ways of doing it.
This sort of thing in mathematics really bothers some people, who assume that mathematics is ultimately some set of absolute truths. I had a long discussion with hedo about this very point not long ago… I wish I would have thought of this example then. In reality though, even seemingly basic functions like addition and instances of them like 2+2=4 are ultimately definitions that could be set out however we like. The study of various definitions of these functions and how useful they are in various contexts, to put it crudely, is what is called “modern” or “abstract” algebra. Of course, all the defining we do in algebra is guided by some set of axioms (the group, ring, field axioms) that intuitively tells us how we think, for example, addition should act.
The real number system and the familiar definitions of addition, multiplication, and the exponential that are just assumed in highschool algebra classes and calculus is just one particularly useful “number system” and set of functions that happen to correspond with a lot of our intuitions. Questions of whether or not there are other sorts of number systems and ways of understanding these functions generally don’t come up until you hit “singularities” like 0^0. [/quote]
I knew you’d eventually jump in here. Good post.
[quote]Liv92 wrote:
You guys are tools, just saying…[/quote]
Hey, I do find math interesting. I don’t understand how this would make me a tool.
stokedporcupine8, thank you for the information.
[quote]ukrainian wrote:
Liv92 wrote:
You guys are tools, just saying…
Hey, I do find math interesting. I don’t understand how this would make me a tool.
stokedporcupine8, thank you for the information. [/quote]
I think he meant we’re tools for kissing Debra’s ass 
If not though, it’s his loss and your gain. There is a wonderful and rich conceptual world to mathematics. You only get the tiniest taste of it from highschool algebra and a normal calculus series.
[quote]JLu wrote:
Wolfram Alpha says indeterminant.[/quote]
x2. indefinite answer my friend.
[quote]JLu wrote:
Bill Roberts wrote:
debraD wrote:
But it is debatable.
Really, that should have ended the thread, IMO. Seems a quite solid explanation and presentation and – absolutely no disrespect to the posters that followed – no further comment seems to have added anything to the explanation in the link.
Really, this shouldn’t have been posted, IMO. Seems like quite obvious ass kissing and favour seeking and – absolutely no disrespect to you – this comment seems to have ALSO added nothing to the thread.[/quote]
Here’s what it was intended to add:
Basically it seemed to me the thread continued on just as if the link hadn’t been posted. It sure seemed that way from many replies.
When a fine and thorough answer to the question is provided but totally ignored, and in place come many wonderings and speculations, it’s worth posting that hey, a good answer was already posted and would have ended those wonderings.
Now, some good things beyond what the article said were posted after my post, it’s true. Perhaps I should have called attention to the fact that the link would have answered most of the following posts in a different way.
[quote]stokedporcupine8 wrote:
I think he meant we’re tools for kissing Debra’s ass
[/quote]
If kissing debra’s ass (literally) means I’m a tool then oh yes yes yes I’m oh so a bloody tool
[quote]stokedporcupine8 wrote:
ukrainian wrote:
Liv92 wrote:
You guys are tools, just saying…
Hey, I do find math interesting. I don’t understand how this would make me a tool.
stokedporcupine8, thank you for the information.
I think he meant we’re tools for kissing Debra’s ass
If not though, it’s his loss and your gain. There is a wonderful and rich conceptual world to mathematics. You only get the tiniest taste of it from highschool algebra and a normal calculus series. [/quote]
I did just finish AP Calc BC, so I get to do Calc 3/Diff Eq now as a Junior. Luckily, I have always found solving math equations and just doing Calculus quite entertaining.