It does repeat in base (2/3 of pi):
1.1111111111....
Why not just use base pi (whatever that means), then it's just 1.
That's actually an interesting concept, using an irrational number for the base of a counting system. Off the top of my head, 1) it would prevent you from calculating numbers with infinite precision, 2) the inaccuracies would get worse as the numbers get higher (e.g. the estimate of pi8 is going to be off from pi2 sure to multiplying limited precision estimates by itself). I'd love to hear an actual math major's take on it.
Whoops, I guess for a base pi system pi would actually be 10.
But non-integer bases seem nonsense. How do the digits work?
Numerical methods would suffer as the exactitude of integer arithmetic would be lost, and the margin of error would only be magnified by iteration, but 'lazy' symbolic manipulation using irrational or transcendental bases could lead to the infinite length numbers cancelling out in the process, leading to surprising and interesting results. Eulers famous equation hints at such a possibility:
ei.Pi -1 = 0
Where e and Pi are both transcendental numbers, and i is the square root of -1. Essentially, Euler was like a magician who put some of the weirdest entities in the mathematical universe into a hat - and pulled out the simple integer 0.
Just in case anyone comes across this thread, apparently base phi (the golden ratio) counting is a result example of a counting system based on an irrational number.
I didn't want someone to tell me, technically, then it doesn't repeat. (Unless you consider 1.000000... repeating.)
(Unless you consider 1.000000... repeating.)
That is a repeating decimal. The zeros repeat indefinitely.
Mind blown!
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