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Why is $a^0=1$ $\forall a \in Z, a\neq0$. I understand $2^4=2\cdot2\cdot2\cdot2$ How can I express $a^0$. I am serious about the practical proof of this

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marked as duplicate by Jonas Meyer, Chris Culter, apnorton, Asaf Karagila, Tunk-Fey Jul 26 '14 at 18:32

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It's by definition, it's not something you prove. – Git Gud Jul 26 '14 at 17:56
Try starting with the $n$-th root of $a$ and let $n$ tend to infinity: $\lim_{n\to\infty}a^{\frac{1}{n}}$. – David H Jul 26 '14 at 17:57
@David: I don't think this has much to do with calculus and limits. – Asaf Karagila Jul 26 '14 at 17:58
I assume that you are searching for "intuitive" explanations for schoolboys. If so, I think that Michael Hardy's answer can works. Exponentiation is iterated multiplication; thus, "going back" with repeated division, you arrive at $2^1=2$ and then, dividing again, to $2^0=2^1/2=2/2=1$. – Mauro ALLEGRANZA Jul 26 '14 at 19:01

If you want $x^n\cdot x^m=x^{n+m}$ then $x^0\cdot x^n=x^n$. This means that $x^0=1$.

Of course you are free to define it otherwise. But this gives a pretty good motivation why $x^0=1$.

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What does it practically mean? Because $2^4$ means multiplying 2 by itself 4 times – Achari S Ganesha Jul 26 '14 at 18:01
It means that multiplying $2$ by itself $0$ times gives you $1$. – Asaf Karagila Jul 26 '14 at 18:03
@AchariSGanesha Are you asking for some intuition for $n^0=1$? This isn't clear in your question. – Git Gud Jul 26 '14 at 18:04
One problem sith this argument took me a long time to appreciate: It's amazing how much lack of an actual feel for algebra can be found in students with some technical skill in its use. I've posted an answer in a quite different style. – Michael Hardy Jul 26 '14 at 18:04

$ 2^4 = 16. $

Divide by $2$ to get $2^3 = 8$.

Divide by $2$ to get $2^2=4$.

Divide by $2$ to get $2^1 = 2$.

Divide by $2$ to get $2^0 = \text{?}$

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I love this intuitive approach - it's very satisfying. Also continues naturally to powers of negative numbers. – trichoplax Jul 26 '14 at 22:35

when a nonzero number is divided by itself, the result is $1$. Therefore, for example, $$2^4\div 2^4=1$$ on the other hand, by the rule of powers, one knows that, for example, $$2^7\div 2^5=2^{7-5}=2^2$$ comparing these two notes we obtain that $$1=2^4 \div 2^4=2^0$$
hence $$2^0=1$$

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The answer is simpler than one might expect: it is so by convention.

To understand why the convention is such and not something different, look at this "puzzle": what is the most logical way to continue the sequence 16, 8, 4, 2, ... ? Clearly, the numbers are always halved, so the logical next one is 1. Now, since the sequence is $2^4, 2^3, 2^2, 2^1$, it is only logical to assign $2^0$ the value $1$.

It turns out that this choice is a good one, for example because powers obey the laws you expect them to obey, such as $a^{mn} = (a^m)^n$ and $a^{m+n}=a^m\cdot a^n$ for each $a \neq 0$ and each whole $n, m \geq 0$.

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Oh, how I despise the sentence "What is the most logical way to ...", it sends chills down my spine just to hear it being typed. :-P – Asaf Karagila Jul 26 '14 at 18:02
@AsafKaragila I guess you must really hate this, right? – Daniel Fischer Jul 26 '14 at 18:07
@alexqwx I don't know about Asaf's motivation for that comment, but if it was me, I'd say the problem is that logic is just logic, inferences rules, it's completely independent of what our (us humans) intuition tells us. Some different kind of being could expect $\pi$ to follow after $16,8,4,2$. – Git Gud Jul 26 '14 at 18:08
@Daniel: I could have lived my life without knowing that song, and now that part of my life is over. Thanks a lot. :\ – Asaf Karagila Jul 26 '14 at 18:11
OK everyone, especially the logicians present, maybe it would have been better to replace all occurrences of logical in my answer by humanly and nice-looking. I would have said natural, but I'm afraid that would unleash a bunch of angry category theorists on me :) – Dan Shved Jul 26 '14 at 18:18

\begin{align} a^0 &= a^{n-n} = a^na^{-n} = a^n/a^n = 1 \end{align}

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You can think of it as an empty product, because $1$ is the multiplicative identity.

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