Well, yes. But my point (which was admittedly not clear) was that if you need to change bases just to make it easier to add 1/3 and 1/5, then perhaps you should try adding them in a way that doesn’t depend on bases at all.
If we confine our attention to integer bases, then clearly rational numbers are the same across all bases. A rational number can, by definition, be expressed as integer/integer and regardless of the (integer) base, anything that’s an integer in one base is an integer in all (integer) bases.
Being an integer has nothing to do with the base in question. 10[sub]3[/sub] is an integer; 10[sub][symbol]p[/symbol][/sub] is not.
CurtC: Maybe, but those multiplication tables in base 60 would be a bear! Every kid would have to rote memorize what (base 10) 37 times 49 is, etc.
The kids in scribal schools in ancient Babylon, whose number system was base-60, had multiplication tables for reference (on clay tablets, natch). They doubtless memorized at least some of the tables, but if they didn’t know a particular product they could look it up.
And when they added 1/3 and 1/5, they expressed them as 20 and 12, and added them to get 32, just as we would express them as 0.333… and 0.2 and find their sum to be 0.533… The Babylonians didn’t have the base-60 equivalent of a decimal point, but that place-value system combined with a nicely divisible base like 60 definitely made their lives easier computationally.
By the way, don’t forget that pi and e are not just irrational, but transcendental. If you want an example of an “ordinary” (i.e., algebraic) irrational number, try the square root of 2 or some such.
This is a true statement.
Of course, neither p nor q need be prime, nor need they be distinct, for this statement to hold.
In general, 1/p + 1/q = (p+q)/(pq) for all nonzero integers p and q.
I never said that p+q and pq were relatively prime.