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Precisely. In fact, that features in many proofs of transcendence: you find too good rational approximations, which contradicts things like Roth's theorem, so
by ykonstant 4y ago
Precisely. In fact, that features in many proofs of transcendence: you find too good rational approximations, which contradicts things like Roth's theorem, so your number cannot be algebraic.
- madcaptenor 4y agoIIRC the first number proven to be transcendental was Liouville's number 0.110001000000000000000001... where the digits after the decimal place are 1 in the n!th place and 0 otherwise. This is explicitly constructed to be very close to the sequence of rational numbers 0.1, 0.11, 0.110001, ... (I expect there's nothing special about base 10 here; surely the proof works in binary as well.)