How to calculate an n digit decimal approximation of a fraction?Say I have a number x ∈ Q which possibly cannot be writen exactly in decimal form, but at least x ∉ Z ( x is not an integer).How do I calculate the first n digits of this number x, for a given n? Also, how do I know if it has a finite or infinite amount of (nonzero) digits in decimal form?For example, if x = 1000 1001 , how do I calculate the first n = 10 digits of this number?I know this question sounds really basic, but since it is basically always done by means of a calculator or other computing device, I've never really done it before.

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How to calculate an   n digit decimal approximation of a fraction?Say I have a number   x  ∈      Q   which possibly cannot be writen exactly in decimal form, but at least   x  ∉      Z   (  x is not an integer).How do I calculate the first   n digits of this number   x, for a given   n? Also, how do I know if it has a finite or infinite amount of (nonzero) digits in decimal form?For example, if   x  =      1000    1001  , how do I calculate the first   n  =  10 digits of this number?I know this question sounds really basic, but since it is basically always done by means of a calculator or other computing device, I&#039;ve never really done it before.

upornompe · Accepted Answer

A simple algorithm is the following :Take integer part   → multiply the numerator of the remainder by   10 (in base   10)   → divide by the denominator   → repeatEach integer part obtained is one digit.Once you decide you have enough digits, use common sense to place the dot.Example :For       1000    1001  The integer part is   0 so the first digit is   0The remainder is       1000    1001  We multiply the numerator by   10 :   1000  ×  10  =  10000We divide by the denominator   1001 and take the integer part :   9So the second digit is   9The numerator of the remainder is   10000  −  9  ×  1001  =  991Multiply it by   10 and get   9910Divide by   1001, take the integer part :   9, which is our third digit...etc.So the first three digits are   0.99I let you compute the other ones !(It is basically long division, but it was so badly taught to me I didn&#039;t even understand I could just work that way without drawings or fluff, and it seems much easier now, especially for mental maths.)

Winigefx · Accepted Answer

To find if it has a terminating decimal, so has a finite number of nonzero digits, check if the denominator factors as       2    a        5    b  . If the denominator has any prime factors other than   2 or   5, the decimal will repeat infinitely. The termination will come at   max  (  a  ,  b  ) because the denominator will divide evenly into       10          max      (      a      ,      b      )

How to calculate an n digit decimal approximation of a fraction? Say I have a number x

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