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A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
A fixed-point representation of a fractional number is essentially an integer that is to be implicitly multiplied by a fixed scaling factor. For example, the value 1.23 can be stored in a variable as the integer value 1230 with implicit scaling factor of 1/1000 (meaning that the last 3 decimal digits are implicitly assumed to be a decimal fraction), and the value 1 230 000 can be represented ...
A vinculum can indicate the repetend of a repeating decimal value: 1 ⁄ 7 = 0. 142857 = 0.1428571428571428571... A vinculum can indicate the complex conjugate of a complex number: + ¯ = Logarithm of a number less than 1 can conveniently be represented using vinculum:
i 6 × 2 i × t i, first form 6 × 2 i × t i, second form ----- 0 3.4641016151377543863 3.4641016151377543863 1 3.2153903091734710173 3.2153903091734723496 2 3.1 596599420974940120 3.1 596599420975006733 3 3.14 60862151314012979 3.14 60862151314352708 4 3.14 27145996453136334 3.14 27145996453689225 5 3.141 8730499801259536 3.141 ...
To change a common fraction to a decimal, do a long division of the decimal representations of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator"), and round the answer to the desired accuracy. For example, to change 1 / 4 to a decimal, divide 1.00 by 4 (" 4 into 1.00 ...
Decimal numbers that have repeated patterns that continue forever can be interpreted as geometric series and thereby converted to expressions of the ratio of two integers. [4] For example, the repeated decimal fraction … can be written as the geometric series … = + + + +, where the initial term is = / and the common ratio is = /.
The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc.The nth partial sum is given by a simple formula: = = (+). This equation was known ...
In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1]. and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal.