1 to the 0 power

The Blade Dancer

I mean, there's also like continuing a pattern, but shouldn't math be applied and not just working in theory?

The Blade Dancer

We already have a ton of exceptions in math, and including one here wouldn't be a problem. Besides, it would be possible to derive this exception easily.

debbie

@The-Rogue-Blade Good point! In practice, $x^0 = 1$ works in the sense that it's better than $x^0 = 0,$ because if $x^0 = 0,$ then we get

$$ \textcolor{red}{x^0} \times x^4 = 0 $$

$$ \textcolor{red}{0} \times x^4 = 0 $$

which is a bit strange, since $\textcolor{red}{x^0}$ would be able to have this destructive property of obliterating all other exponents.

Additionally, when we prime factorize a number, like $60,$

$$ 60 = 2^2 \times 3^1 \times 5^1 $$

we might want to be able to list them out with all primes as a sort of basis, like

$$ 60 = 2^2 \times 3^1 \times 5^1 \times \textcolor{red}{7^0} \times \textcolor{red}{11^0} \times \textcolor{red}{13^0} \ldots $$

We would only be able to do so if we define $x^0$ to equal $1.$

The Blade Dancer

Well, that's just my point of view on this. I was also about to ask why the code was weird

debbie

@The-Rogue-Blade It might be because it's a fresh post, and it takes awhile for the site to render the code...? I'm not sure, but refreshing often fixes the problem!

thomas

Interesting discussion! What do you guys think $0^0$ is?

The Blade Dancer

@thomas 0

Potato2017

@thomas It's probably undefined. Since 0^x = 0 and x^0=1 for all nonzero x, if x were to be 0, we would have 0=1, which is very, uh,

$$$$
$$$$
$$$$

let's just not think about that.

thomas

@Potato2017 Interesting! It seems like it's unclear if we can even do it at all...

Would perhaps this graph change your mind?

This is a graph of the function $x^x$. What this graph is saying is that if you look at $1^1$, $0.1^{0,1}$, $0.01^{0.01}$, ... these numbers get closer and closer to $1$. Do you think that means that $0^0$ should be $1$? Why or why not?

RZ923

@The-Blade-Dancer
I came up with another way of explaining why $1^0$ is $1$. It had something to do with Place Value.
Let’s just look at base 10.
Any number, say $1729$, can be represented with powers of 10.
For example, the $1$ represents $1000$, or $1 \times 10^3$. The $7$ represents $7 \times 10^2$, and the $2$ represents $2 \times 10^1$.
Now let’s look at the $9$. What does it represent.
Continuing our pattern, it represents $9 \times 10^0$.
So $10^0$ must be one. If it’s $0$, then we don’t have place value.
Hope that helps

The Blade Dancer

lol you have a point there