Where is my thinking wrong?

Question

I have always known that $a^n=a*a*a*.....$(n times)

Then what exactly is the meaning if $a^0$ and why will it be equal to $1$?

I have checked it in the internet but everywhere the solution is based on the principle that $a^m*a^n=a^{m+n}$ and when $n=0$ it will be $a^m$ and clearly $a^0$ is equal to $1$.

But what exactly does $a^0$ mean does it mean $a*a*a*...$(zero times)?

Any help is highly appreciated.

Answer 1

You can see that in this way:

$$a^0 = a^{m - m}$$

for every value of $m$. Using the properties of powers we have:

$$a^{m-m} = \frac{a^m}{a^m} = 1$$

Because the two terms are identical so they are canceled. So

$$a^0 = 1$$

Answer 2

That's the intuition; but the powers are properly defined by recursion from $$ a^0=1,\quad a^{n+1}=a\cdot a^n $$ so $a^0=1$ by definition. It is a sound definition, because it agrees with the property $a^{m+n}=a^m\cdot a^n$ for any natural $m$ and $n$.

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Think to what you do when you have a heap of candies to count. You start from zero and take one candy at a time, uttering the corresponding number: one, two, three, and so on.

Similarly, if you have to know how many candies are in a bunch of heaps, you can count each heap and write down the number. Then you start from zero, add the first number, then the second and so on (at this stage you already know how to perform symbolic sums).

For multiplication it's the same, but you start from one! So $a^0=1$, then $a^1=a\cdot 1$, $a^2=a\cdot a^1$, and so on, each time multiplying by $a$ until you arrive at $n$ and you have your $a^n$.

Answer 3

The definition you give only applies when n is an integer greater than zero. It does not apply to n = 0 or negative n because it doesn't make sense to talk about multiplying together zero a's or a negative number of a's.

However, using the principle that you mention, we can extend this definition to zero and negative numbers. This allows us to give a meaning to a to the power of zero or negative exponents. That meaning is no longer "a certain number of a's multiplied together" but something that works consistently with the definition we already have for positive n.

Answer 4

It's defined that way (except, usually, for $a = 0$) because it's most consistent to do so. The empty product is defined to be $1$ because $1$ is the multiplicative identity, much as the empty sum is defined to be $0$ because $0$ is the additive identity. Both of these definitions allow for the pattern that arises from successive multiplication to be "extended backward" to a product (or sum) with zero terms.

The extension of exponents to zero, to negative numbers, to the rationals, to the reals, and to complex numbers, in each case continues a pattern identified in the previous, "smaller" domain. Those patterns are useful; that is why they are defined that way. Nothing stops you from defining them differently—nothing, that is, except that they generally are less useful that way.

Answer 5

You are correct in that $a$ is repeated zero times.

$a^n = a*a*a*... = 1 * a*a*a*...$

And so $a^0 = 1$ when $a$ is repeated zero times