It might be a mistake to try to argue that defining 0^0=1 is the best answer or the most consistent answer. The problem is that there are conflicts. There is no such thing as more consistent, if any rule breaks, they all break. Some math rules lead you to 0^0=0, some to 0^0=1, and some to 0^0=undefined. So math doesn't have an answer for this, and what really happens is we decide that 0^0=1 by convention. There are other examples of this happening in math, and it works reasonably well. But let's not pretend it's "right", let's accept that it's a choice.
We're choosing to say that the rule x^0=1 takes precedence over the rule 0 * x = 0.
Just to play devil's advocate, I like the choice of 0 for intuition consistency, not math consistency. The positive side limit is 0, so if 0^x is defined at x=0, how about we try to make the function more continuous, instead of trying to make it discontinuous?
0^0 is also saying 'start with zero, then multiply it by itself ... Wait, on second thought, don't start'
There's nothing, because you don't multiply zero by anything. The best number we have to represent nothing is zero. Saying that not raising zero to a power is one feels like a random number.
> 0^0 is also saying 'start with zero, then multiply it by itself ... Wait, on second thought, don't start'
The same is true for any other integer: "2^0 is also saying 'start with two, then multiply it by itself ... Wait, on second thought, don't start'" You don't multiply two by anything.
The thing is, there is no argument to be had, x^0 is defined to be 1, and x^1 is defined to be x. Those agree with and make consistent lots of other rules and limits, but they are choices we made just like saying 0^0=1.
The situation is analoguous to multiplication. a X b is defined as a+a+a+... b times. If you see a problem with x^0 and x^1, you should also see a problem with a X 0 and a X 1.
So you say. If it's planly obvious, why do I get so many results for the Google search "why is x^0 1"?
I understand the multiplicative identity argument, but that's a technical explanation, not an intuitive one.
What's the intuitive reason that multiplying something zero times should equal one? If I don't multiply, then I don't have an answer, and zero's closer to nothing than one is. Why should I use the multiplicative identity if I don't multiply, why would that make sense?
Using the multiplicative identity is yet another choice, not an intrinsic property of numbers. It's a very good choice, and there are a lot of reasons why it's a good choice.
"In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity 1 (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.[1][2][3]"
> x^0 isn't defined to be 1, x^0 IS 1.
Are you sure? How can you show that? How do you define what an exponent is without defining what x^0 and x^1 are? What does it mean to suggest that x^0 is intrinsically 1? Are you absolutely certain that you just aren't so comfortable with the idea that you can't imagine other possibilities?
It all follows directly from the very definition of the exponentiation operation itself. You can go work the math yourself, I'm not here to educate illiterate people.
Hahahaha! That's an extra unfortunate choice of insult, it really doesn't look good for you. But I hope it made you feel better. Look, I'm honestly sorry if I offended you along the way, that wasn't my intent. If it was my remark about imagination, it might have been poorly worded on my part, but that wasn't directed at you personally or meant as an insult -- it's actually really difficult for all people to see how certain simple ideas were constructed, when you've known them your whole life. You know what an exponent is so well and so thoroughly, you might not have a strong grasp on how it was defined and developed through history. I don't. Another example: it's very hard to imagine life without 0, but 0 didn't always exist, the symbol 0 was given a definition, and we still haven't fully resolved how to use it in all cases.
Now, since I just quoted the definition of exponentiation itself from Wikipedia and the formal definition includes the base case b^1 = b, I think you've completely failed to make an educated point to go along with your insult. I didn't ask for you to define exponentiation so you could educate me, I asked so that you could think carefully and tell me if you can define what exponentiation is without using the base case. Were you to actually try, you may find it difficult. Or not, I might be wrong, so feel free to prove me wrong or cite a source that proves me wrong, if you want. As it stands, my takeaway for now is that your insult is a substitute for the argument you don't have, so you're forfeiting your position and handing me a walkover.
is restricted to the domain {x | x != 0}. Because e.g. when x = 0 and y = 1, the expression then contains a zero in the denominator. Division by zero is "every number, and therefore no single number" because we can use algebra to "prove" that x / 0 equals "anything we want", almost like the how the principle of explosion works.
If you wanted to prove that 0^0 really does equal 1, you would have to prove that the output of the reduction is unique.
I think of b^e as "start with 1, then loop 'multiply that by b' an e number of times". So 2^0 means "start with one, then multiply that by 2 for 0 numbers of times."
> how about we try to make the function more continuous
Look at the 3D rendering in my comment. When you look at the z=x^y surface and try to make it continuous, what comes out at 0^0 is a vertical line, aka. undefined.
You're absolutely right. And you're also wrong in the very the same way I'm wrong. ;) We decided that 0^0=1, and by 'we', I mean the mathematicians. It's easy to show why the choice 0^0=1 is a bad choice, because it breaks some rules. It's also easy to show (as the article's author did) why it's a good choice too. But, regardless, it's still the choice that has been made, and the choice can't be unmade.
It's a beautiful example, btw, thank you for contributing the visualization!
that highlights 0^y=0 in red and x^0=1 in purple. Now it's even more obvious why any single answer is insufficient: 0^0 is a vertical line, not a single point!
Hey, again, I totally agree with you. You are right, no single answer is sufficient. It was obvious before, and it's still obvious. The new lines show the specific conflict between 0^x and x^0, but there is no one answer.
Your re-iterating your point makes me wonder if you're actually hearing and understanding the part about choices and definitions trumping logic. If we define 0^0=1, then it doesn't matter what arguments we have, it doesn't matter that there are 3 or more right answers. The right answer can be, and in other parts of math, is what we define it to be, not what makes the most sense.
0^0 is defined as 1 (sometimes) because that has utility and consistency for some other expressions, formulas, operations, etc. It's not because it's right. It's not because it makes sense. It's because we decide to set 0^0=1. It's for convenience.
Now, some sources online say that 0^0 is indeterminate, not equal to one. Other sources online say that 0^0 is 1.
It might be a mistake to try to argue that defining 0^0=1 is the best answer or the most consistent answer. The problem is that there are conflicts. There is no such thing as more consistent, if any rule breaks, they all break. Some math rules lead you to 0^0=0, some to 0^0=1, and some to 0^0=undefined. So math doesn't have an answer for this, and what really happens is we decide that 0^0=1 by convention. There are other examples of this happening in math, and it works reasonably well. But let's not pretend it's "right", let's accept that it's a choice.
We're choosing to say that the rule x^0=1 takes precedence over the rule 0 * x = 0.
Just to play devil's advocate, I like the choice of 0 for intuition consistency, not math consistency. The positive side limit is 0, so if 0^x is defined at x=0, how about we try to make the function more continuous, instead of trying to make it discontinuous?
0^0 is also saying 'start with zero, then multiply it by itself ... Wait, on second thought, don't start'
There's nothing, because you don't multiply zero by anything. The best number we have to represent nothing is zero. Saying that not raising zero to a power is one feels like a random number.
But, I'm wrong. ;)