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6 1 2 1 1 −1 4 5 9. and would be written in modern notation as 6 1 / 4 , 1 1 / 5 , and 2 − 1 / 9 (i.e., 1 8 / 9 ). The horizontal fraction bar is first attested in the work of Al-Hassār (fl. 1200), [35] a Muslim mathematician from Fez, Morocco, who specialized in Islamic inheritance jurisprudence.
Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the ...
The golden ratio φ and its negative reciprocal −φ −1 are the two roots of the quadratic polynomial x 2 − x − 1. The golden ratio's negative −φ and reciprocal φ −1 are the two roots of the quadratic polynomial x 2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer.
Every terminating decimal representation can be written as a decimal fraction, a fraction whose denominator is a power of 10 (e.g. 1.585 = 1585 / 1000 ); it may also be written as a ratio of the form k / 2 n ·5 m (e.g. 1.585 = 317 / 2 3 ·5 2 ).
13.2 2.7 0.5 24 Mexico: 11.8 0.7 0.5 25 Colombia: 10.7 2.9 0.4 26 Iran: 10.3 2.1 0.4 27 Indonesia: 9.5 0.7 0.4 28 Sweden: 8.8 1.5 0.4 29 Norway: 8.7 1.6 0.4 30 Pakistan:
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 definition, whose result can also be obtained in a straightforward way by
Now 2b 2 and a 2 cannot be equal, since the first has an odd number of factors 2 whereas the second has an even number of factors 2. Thus | 2 b 2 − a 2 | ≥ 1 . Multiplying the absolute difference | √ 2 − a / b | by b 2 ( √ 2 + a / b ) in the numerator and denominator, we get [ 17 ]
One half is the rational number that lies midway between 0 and 1 on the number line. Multiplication by one half is equivalent to division by two, or "halving"; conversely, division by one half is equivalent to multiplication by two, or "doubling".