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Math Help - Endless decimal representations

  1. #1
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    Endless decimal representations

    Hi.
    I'm studing floating point representation of binary numbers, and it made me notice something.
    Well, actually, I noticed it before of course, but it never occurred to me to ask about it up until now:

    There are some numbers, rational numbers, that can be presented in a finite decimal floating-point representation,
    for example:
    \frac{1}{2}=0.5

    \frac{3}{4}=0.75

    but some cannot, like:
    \frac{1}{3}=0.33333...

    \frac{5}{7}=0.714285714285...

    My question is why?
    what distinguish "infinite decimal floating-point representation" from the ones that can be written in a finite form?

    I also noticed (and correct me if I'm wrong) that when it comes to binary floating-point representation, the "problem" is much bigger:
    There are considerably much more fractions that cannot be represented in a finite series of digits, then the ones that can.
    Why does that happen?

    Thank you in advance.
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  2. #2
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    Re: Endless decimal representations

    Are you familiar with prime factorization of integers? Any integer can be expressed as a finite product of primes. Using exponents, a rational number can be expressed as a product of a finite number of prime powers (where the exponent of each prime is an integer). For example, \dfrac{1,185,921}{18,974,736,000} = \dfrac{3^4\cdot 11^4}{2^7\cdot 3^4\cdot 5^3\cdot 11^4} = 2^{-7}\cdot 3^0 \cdot 5^{-3}\cdot 11^0. When representing a fraction with decimals, if the only primes with negative exponents are for 2 and 5, the decimal expression will be finite (this is because 10 = 2\cdot 5). If any other prime has a negative exponent, it will be an infinite representation. In binary, if the only prime with a negative exponent is 2, then it will have a finite binary representation. Otherwise, it will be infinite.
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  3. #3
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    Re: Endless decimal representations

    That is absolutely fascinating.
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