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The question isn't how to approximate a given number to desired accuracy. It is whether we should be able to approximate it to desired accuracy relative to the
by btilly 7y ago
The question isn't how to approximate a given number to desired accuracy. It is whether we should be able to approximate it to desired accuracy relative to the denominator.
Getting an approximation p/q of x within 1/q is easy. Getting it within 1/q^2 takes work. Within 1/(10 q^2) is only sometimes possible.
- AnimalMuppet 7y ago> Within 1/(10 q^2) is only sometimes possible. For any q?
- btilly 7y agoFor any q? Yes. The hardest real number to approximate with rationals is (1+sqrt(5))/2. Its approximations that come closest are of the form fib(n+1)/fib(n) where fib(n) is the n'th fibonacci number. The error converges to 1/(sqrt(5) fib(n)^2). None are within 1/(10 q^2). To learn more about this, read up on continued fractions.