Zero^(0) = 1 or Not?

[nycmathguy]

Zero is a whole number.
I know that (any number)^(0) is 1.
If that is the case, why is 0^(0) not 1?

 

[MathLover1]

Zero to the power of zero, denoted by 0^0, is a mathematical expression with no agreed-upon value.
The most common possibilities are 1 or leaving the expression undefined, with justifications existing for each, depending on context.
In algebra and combinatorics, the generally agreed upon value is 0^0 = 1, but in mathematical analysis the expression is sometimes left undefined.

 

[nycmathguy]

Zero to the power of zero, denoted by 0^0, is a mathematical expression with no agreed-upon value.
The most common possibilities are 1 or leaving the expression undefined, with justifications existing for each, depending on context.
In algebra and combinatorics, the generally agreed upon value is 0^0 = 1, but in mathematical analysis the expression is sometimes left undefined.

Are you saying there is no algebraic proof for 0^(0)?

 

[MathLover1]

x>0
0^x=0^(x-0)=0^x/0^0, so
0^0=0^x/0^x=?
Possible answers:
0^0*0^x= 0^0*1, so 0^0=1
0^0=0^x/0^x=0/0, which is undefined

 

[nycmathguy]

x>0
0^x=0^(x-0)=0^x/0^0, so
0^0=0^x/0^x=?
Possible answers:
0^0*0^x= 0^0*1, so 0^0=1
0^0=0^x/0^x=0/0, which is undefined

Interesting little proof.

 

[HallsofIvy]

Another: we would like to have 0^0 defined so that both lim_{x->0} x^0 and lim_{x->0} 0^x are 0^0.

But x^0= 1 for any non-zero x so lim_{x->0} x^0= 1.
And 0^x= 0 for any non-zero x so that lim_{x->0} 0^x= 0.

So we leave 0^0 undefined.

 

[nycmathguy]

Another: we would like to have 0^0 defined so that both lim_{x->0} x^0 and lim_{x->0} 0^x are 0^0.

But x^0= 1 for any non-zero x so lim_{x->0} x^0= 1.
And 0^x= 0 for any non-zero x so that lim_{x->0} 0^x= 0.

So we leave 0^0 undefined.

I look forward to exploring the concept of zero in more detail later in my self-study time.