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An important example of a function to which Khinchin's theorem can be applied is the function () =, where c > 1 is a real number. For this function, the relevant series converges and so Khinchin's theorem tells us that almost every point is not ψ c {\displaystyle \psi _{c}} -approximable.
In mathematics education at the primary school level, chunking (sometimes also called the partial quotients method) is an elementary approach for solving simple division questions by repeated subtraction. It is also known as the hangman method with the addition of a line separating the divisor, dividend, and partial quotients. [1]
In cases where one or more of the b terms has more than two digits, the final quotient value b cannot be constructed simply by concatenating the digit pairs. Instead, each term, starting with b 1 , {\displaystyle b_{1},} should be multiplied by 100, and the next term added (or, if negative, subtracted).
Clearly any regular periodic continued fraction consists of restricted partial quotients, since none of the partial denominators can be greater than the largest of a 0 through a k+m. Historically, mathematicians studied periodic continued fractions before considering the more general concept of restricted partial quotients.
Using a counting argument one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers. Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded).
Forming a partial setoid from a type and a PER is analogous to forming subsets and quotients in classical set-theoretic mathematics. The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence, i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reflexive ...
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