Where's the +K key for Wolfram?
Does this mean we've resolved the OP's question about analogRead()?
We solved that in reply #1 from then on we were diverted.
I have used programming languages where ^ represented a power and in others represented an EXOR.
ChrisTenone:
C'mon my fellow pendants, ...
pedants? 
Surely he mean swingers. Anyone else want to own up?
In most contexts, reading ^ as "to the power of" is normal for me but when I first learned it is anyone's guess. I'm pretty sure it wasn't this century.
Steve
2^10 = 1024
int x = 2 ^ 10; //x = 8
See the difference? 
Since OPs Q has been answered from post #1 and the discussion goes here and there, let me put my approach too:
2^10 does not make neceserily 1024. It makes a lot of results
That conclusion comes if someone thinks: Did anybody say that discusion is for 10-base numbers?
2^10 can be in any (3 and above) -base.
In the context of “why is the range of analogRead 0-1023”, 2^10 = 1024 makes more sense than any of the other possibilities though.