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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]
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Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
The badly approximable numbers are precisely those with bounded partial quotients. [ 6 ] Equivalently, a number is badly approximable if and only if its Markov constant is finite and its simple continued fraction is bounded.
where P, D, and Q are integers, D > 0 is not a perfect square (but not necessarily square-free), and Q divides the quantity (for example (+) /). Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example ( 3 + 2 ) / 2 {\displaystyle (3+{\sqrt {2}})/2} ) as explained for quadratic ...
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).
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