*Post by Virgil**Post by Don H**Post by P***@aol.com**Post by Don H*1^0 = 1 ; 2^0 = 1; 3^0 = 1 ; etc.

Is this axiomatic, or is there a proof?

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It isn't a math fact --- it's a definition --- made so as to be

consistant and for convenience when writing infinite series etc.

# And so is an arbitrary definition?

What is a number n to the power x?

The x indicates how many n must be multiplied together.

Thus,

n^x = n by n (x times)

2^2 = (2, twice) = 4

2^1 = (2, once) = 2

2^0 = (2, no times) = 1 ?

- but if you multiply 2 by itself, no times, you've still 2 ?

Whereas, if you multiple 0 by 0, two times, you've still 0 ?

The standard inductive (or, if you prefer, recursive) definition of

For any real/complex number x, x^1 = x and x^(n+1) = x^n * x.

# Thanks.

However, consider two rabbits, say, Rob and Ruby, living in a Platonic

relationship (2^0), who then decide to copulate, initially without result

(2^1), but finally generating two kids (Richard, Rachel) (2^2), who all then

continue to propagate, exponentially = 2 x 2 x 2 x 2...

Likewise, two non-entities, Zero and Zilch, who, no matter how many times

they try, produce nothing - or nothing evident to our eyes.

Does it depend on what you start out with, as to what the result will be?