Why ##a^0=1##?

[sophiecentaur]

I like this definition (on Wiki article).

There's quite a high level of sophistication about rational indices. Also even for all arithmetic with rational numbers; we tell kids about 'sharing' and division but even that is a matter of going through the motions and believing you got a right answer. As a lifetime Engineer, I'm used to a black box approach to Maths - as a tool and not a religion.

 

[Mike_bb]

For example, do you know the mechanics of deriving dy/dx for y=cos(x) and where the 'limit' is involved (plus some elementary trig identities).

Yes, I do. I studied full course of Math. But your example has another type of definition. Of course, if we are talking about derivatives we should use limit conception to define. "Derivative of a function" means that we "derive some function" and thus we use limit.

 

[Andrew Mason]

If $a^0\ne 1$ then $a^0\ne -e^{\pi i}$ and the beauty of mathematics would be gone.

AM