If the difference is 0, stop otherwise find the reciprocal of the difference and repeat. To calculate a continued fraction representation of a number r, write down the integer part of r. It is customary to replace the first comma by a semicolon. Then the continued fraction representation of r is, where "…" is the continued fraction representation of 1/ f. Let i be the integer part and f the fractional part of r. As an approximation to π, is more than one hundred times more accurate than 3.1416.Ĭalculating continued fraction representations The denominators of 314/100 and 333/106 are almost the same, but the error in the approximation 314/100 is nineteen times as large as the error in 333/106. Truncating this representation yields the excellent The continued fraction representation of π begins. Truncating the decimal representation of π yields approximations such as 31415/10000 and 314/100. But clearly the best rational approximation is "1/7" itself. at various places yields approximations such as 142/1000, 14/100, and 1/10. Truncating the decimal representation of a number yields a rational approximation of that number, but not usually a very good approximation. This last property is extremely important, and is not true of the conventional decimal representation. Truncating the continued fraction representation of a number x early yields a rational approximation for x which is in a certain sense the "best possible" rational approximation (see theorem 5, corollary 1 below for a formal statement).The terms of a continued fraction will repeat if and only if it is the continued fraction representation of a quadratic irrational, that is, a real solution to a quadratic equation with integer coefficients.The continued fraction representation of an irrational number is unique.(For any rational number expressed as a continued fraction with z>1 there is a less efficient representation ending in 1, ). The continued fraction representation of any rational number is unique if it has no trailing 1.Continued fraction representations for "simple" rational numbers are short.The continued fraction representation for a number is finite if and only if the number is rational.The continued fraction representation of real numbers can be defined in this way. This is exact.ĭropping the redundant parts of the expression 4 1/(2 1/(6 1/7)) gives the abbreviated notation. But the 6 in the denominator is not correct the correct denominator is a little bit more than 6, actually 6 1/7. But the 2 in the denominator is not correct the correct denominator is a little bit more than 2, about 2 1/6, so 415/93 is approximately 4 1/(2 1/6). Actually it is a little bit more than 4, about 4 1/2. Let us consider how we might describe a number like 415/93, which is around 4.4624. Ĭontinued fraction notation is a representation for the real numbers that avoids both these problems. Where a 0 may be any integer, and each a i is an element of. Most people are familiar with the decimal representation of real numbers: The study of continued fractions is motivated by a desire to have a "mathematically pure" representation for the real numbers.įor calendars, the continued fraction can show the structure of an intercalation cycle, where the intercalary years are as evenly spaced as possible. 2 Calculating continued fraction representations.I already know they are better in regard to having the same rounding issues as is experienced in the real world, which primarily uses a base-10 numeric system. I can easily think of infinite binary fractions that do not match up with infinite decimal fractions, such as $\frac = 0.1000\ldots$ (in binary) would not be considered infinite fractions.įor context: I'm trying to determine if programming using decimal floating points is better than binary floating points because of less rounding issues due to these infinite fractions. I'm trying to understand the cardinality between the set of all infinite binary (base-2) fractions and the set of all infinite decimal (base-10) fractions.
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