I have always been of the opinion that 0⁰ should be defined to be 1, but some people (like mathsyperson) insist that that 0⁰ is undefined.

To ease power series notation, 0^0 is taken to be 1 in complex analysis.

I agree with both Jane and mathsyperson, in a way.  It is perfectly fine to leave it undefined, but it is also perfectly fine to define it which ever way you like when it is useful.  Usefulness is the key.

Ricky wrote:

I agree with both Jane and mathsyperson, in a way.  It is perfectly fine to leave it undefined, but it is also perfectly fine to define it which ever way you like when it is useful.  Usefulness is the key.

I’m not convinced by Mathsy’s argument. He claims there’s a “pattern” but I don’t see any.

Last edited by JaneFairfax (2009-02-06 23:31:03)

Oof...
Point taken, that was a silly mistake.
In that case, fair enough. 0^0 =1 is intuitive.

I'm not clear on what this topic is about though.
Are we discussing what 0^0 is or what it should be?
Those two things might have entirely different answers.

Doesn't 0^0 = 0/0?

JaneFairfax wrote:

So do you now see the flaw (in both “proofs”)?

I see that there is a flaw, but I'm not sure what. If I had to guess, I'd say it's the divide by zero flaw, but which step specifically is the no-no? Is it 0^0 = 0^(1-1) step? I'm not sure why, but I think it is.

And surely 1 does equal x/x so what is the actually flaw? Like I said, I get that there is one, I just can't tell what it is specifically.

I'm not clear on what this topic is about though.
Are we discussing what 0^0 is or what it should be?
Those two things might have entirely different answers.

Looks like we got a Platonist

In the case where properties break down, I believe the Platonist has to recognize that we will never know what something is.  That is, as long as an argument can be made which contradicts what you think it should be, there is no way we can claim one as mathematically superior to the other.  You'll always be left with with a feeling out doubt.

What is complex analysis? Is it simply analyzing complex functions the Reinman Hypothesis? Or does it mean something else?

That's pretty much it.  The property of being differentiable as a complex function is much stronger than being differentiable in the real sense, and complex analysis is figuring out how much stronger it is and precisely what it gives us to play with.

simron wrote:

Google says 0^0 is 1.
Yahoo says 0^0 is 1.
Windows Live Search says 0^0 is 1.
I says 0^0 is 1.

If that's the case, I have a used car to sell you.  I have a bunch of people that will tell you it's a great deal.

Ricky wrote:

If that's the case, I have a used car to sell you.  I have a bunch of people that will tell you it's a great deal.

Sorry, I'm too young to drive. My feet don't even reach the pedals.
Actually, I've done a little thinking and changed my mind on it. 0^n is 0, so I'm inclined to believe that 0^0 is equal to 0 as well. Think about this: the definition of a^b is axaxaxa... b times. If it's raised to the 0 power, then a^0 is just this: -> <-. 0 is defined to be the absence of something, so 0^0 is 0. At least in my opinion.

Last edited by simron (2009-02-09 06:22:57)

Good point Jane...
*head explodes*
I give up.

0^0 can be defined nicely immediately after introducing the field axioms which give us multiplication and the multiplicative identity 1. 

There are a number of formulas in mathematics that are much nicer if 0^0 is defined to be 1.
You have probably seen several of them, so I won't go into them.

I don't see any reason to go up to calculus to try to make 0^0 work.  The indeterminant form
0^0 in calculus is a form that limits may tend to BUT they never actually become 0^0.  Hence using these types of limits seems inappropriate.

Let's go back and define exponentiation in a field (we will use the real number field).  The usual definition is something like:  ... x^3=x*x*x,  x^2=x*x,  x^1=x and following this pattern we would get x^0 to be a blank space.  Hence x^0 is usually defined separately as x^0=1 if x is not equal to zero.  And then 0^0 is usually said to be one or undefined.

To do it a little differently, let's define exponentiation as follows:
..., x^3=1*x*x*x.  x^2=1*x*x.  x^1=1*x.  so following this pattern x^0=1.  We don't need a separate definition for x^0.  And this definition automatically makes 0^0=1 since it doesn't matter what the x is.  Just drop the last x off of the right side of the definition for x^1=1*x.

This definition has been around a while, but it is still not widely known.
0*x=0 comes from a similar definition: ...3*x=0+x+x+x, 2*x=0+x+x, 1*x=0+x, 0*x=0.

This kind of definition works for any group.  If e is the identity for the group and # is the symbol for the operation and x is an element of the group then we have
   ... x^3=e#x#x#x,  x^2=e#x#x,  x^1=e#x,  x^0=e.
And this could be done with coefficients (like 3x, 2x, 1x, 0x above but in groups repeating the operation with a fixed element is usually represented by exponentiation).

This also works for union and intersection in topology where the identity for union is the null set and the identity for intersection is the whole space X.  This eliminates the need for vacuous arguments for unions and intersections over a null family of sets.

How we define things in mathematics is extremely important.  Also the symbolism we use can "make or break" a concept. 

Last edited by noelevans (2012-07-22 09:29:51)

I'm afraid there is a fallacy in the argument that 0^0 = 0/0.

Let y be the reciprocal of x.  Then xy=1.  Now x^0 = x^(1-1) = x^1*x^(-1) = x*y = 1 (not x/y Oops)
This works fine for non-zero x.  But when x=0 the y in the formula does not exist since zero has no reciprocal.  Hence the quantities  x^(-1) and y in the third and fourth expressions do not exist.  Hence we cannot conclude that 0^0 = 0/0.

It is not valid to apply the law of exponents x^(1-1) = x^1*x^(-1) when x=0.
So 0^0 is not equal to 0/0 via this argument.  We are then free to define 0^0 unless we can find some valid reason that it is not permissible; that is, some valid reason that it must remain undefined.  So if we want to claim that 0^0 is undefined we need to find a valid proof of this.

Last edited by noelevans (2012-07-23 10:05:31)