0^0

[benedictusk]

My math teachers say that 0 to the power of 0 is undefined, but after messing around a little, I found this:

0⁰=0⁰ Original equation

0⁰=0⁰*1 Multiply one side by one

0⁰/0⁰=1 Divide both sides by 0⁰

0⁰=1 Simplify using the division laws of exponents

 

Are my teachers incorrect or did I make a mistake?

[vin]

The term undefined itself implies that it will not obey mathematical laws..so your whole lt of equations are undefined

[mooeypoo]

Also, you can't multiply only one side without changing the meaning of the equation and making it a non equality.

 

So even if you treat the 0^0 as a variable/number, your second step should be

 

[math]0^0*1=0^0*1[/math]

And so:

 

[math]\frac{0^0}{0^0}*1=\frac{0^0}{0^0}*1[/math] Divide both sides by 0^0

 

[math]0^0*1=0^0*1[/math] Simplify using the division laws of exponents -- and go back to your first step. Hence, this exercise didn't really help much.

 

 

Moreover, when you divide both sides by 0^0 in your third step, you arbitrarily decide that 0^0/0^0 on the right side is 1 (it disappears) but on the left side it doesn't.

 

You need to be consistent.

 

~moo

[D H]

What the OP did would have been valid -- if 0⁰ was a valid expression to start with. Multiplying an expression by one (or adding zero to it) does not change the expression. Sometimes writing one (or zero) in a rather clever way can help simplify an expression greatly.

 

The problem with 0⁰ is that there is no consistent way to define it. The reason is that the expression [math]x^y[/math] takes on all values in any punctured neighborhood of the origin in [math]\mathbb R^2[/math]. Explicitly defining 0⁰ will introduce inconsistencies. Above all else, mathematics must be consistent. Introducing an inconsistency sends mathematics toward a disaster of biblical proportions. Real wrath of God type of stuff. Human sacrifice, dogs and cats living together... mass hysteria! And of course, 1=2. Not good.

 

That said, mathematicians use 0⁰=1 as a very convenient abuse of notation in many contexts. Power series, the binomial theorem, for example.

[mooeypoo]

What the OP did would have been valid -- if 0⁰ was a valid expression to start with. Multiplying an expression by one (or adding zero to it) does not change the expression. Sometimes writing one (or zero) in a rather clever way can help simplify an expression greatly.

Hm, I thought about this and I think I remember a few examples now of this done. Namely, when we wanted to get "tricks" of completing a square in a fractions' denominator, by multiplying it by its conjugate divided by the conjugate (hence, 1). So, yeah, I take it back.

 

I do have another issue with this though. Even if 0^0 was defined -- the division by 0(to the power of whatever) is undefined on its own. Yielding "infinity". Is that 'step' mathematically legal, D H ?

[D H]

There is no division by zero, mooey. The problem arises from assuming that 0⁰ is valid in the first place.

 

This is an old debate. 0⁰ is taken to be one in many contexts, zero in a few others, and indeterminate in yet other contexts. It is very handy to use 0⁰=1 in the context of power series and the binomial theorem, for example.

 

Consider a power series f(x) about some point c and the binomial expansion of [math](x+y)^n[/math].

The notional use of 0⁰=1 (strictly speaking, abuse of notation) enables us to write

 

 

[math]f(x) = \sum_{n=0}^{\infty} a_n (x-c)^n[/math]

 

[math](x+y)^n = \sum_{k=0}^n\binom n k x^{n-k}y^k[/math]

 

 

Without this convenient abuse of notation we would have to write these as

 

 

[math]f(x) = a_0 + \sum_{n=1}^{\infty} a_n (x-c)^n[/math]

 

[math](x+y)^n = x^n + y^n + \sum_{k=1}^{n-1}\binom n k x^{n-k}y^k[/math]