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A continued fraction is an expression of the form = + + + + + where the a n (n > 0) are the partial numerators, the b n are the partial denominators, and the leading term b 0 is called the integer part of the continued fraction.
That expression is called the continued fraction representation of 415 / 93 . This can be represented by the abbreviated notation 415 / 93 = [4; 2, 6, 7]. (It is customary to replace only the first comma by a semicolon to indicate that the preceding number is the whole part.)
Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. For example, "a divided by b" can be written as: which can also be read out loud as "divide a by b" or "a over b".
In mathematics, regular continued fractions play an important role in representing real numbers, and have a rich general theory touching on a variety of topics in number theory. Moreover, generalized continued fractions have important and interesting applications in complex analysis
with ⌈ ⌉ as the smallest integer not less than x, also called the ceiling of x. By consequence, we may get, for example, three different values for the fractional part of just one x : let it be −1.3, its fractional part will be 0.7 according to the first definition, 0.3 according to the second definition, and −0.3 according to the third ...
Periodic continued fractions are in one-to-one correspondence with the real quadratic irrationals. The correspondence is explicitly provided by Minkowski's question-mark function. That article also reviews tools that make it easy to work with such continued fractions. Consider first the purely periodic part
The golden ratio was called the extreme and mean ratio by Euclid, [2] ... It is in fact the simplest form of a continued fraction, ... Dividing by interior division.
Quotition is the concept of division most used in measurement. For example, measuring the length of a table using a measuring tape involves comparing the table to the markings on the tape. This is conceptually equivalent to dividing the length of the table by a unit of length, the distance between markings.