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Thread: Rational numbers == infinitely repeating sequences of digits

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    Administrator lpetrich's Avatar
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    Rational numbers == infinitely repeating sequences of digits

    A feature of rational numbers is that their decimal representations always have infinitely repeating sequences of digits. This is true not only for base 10, but also for every possible base of a place system.

    Examples:

    1 = 1.000000... = 0.999999...
    1/3 = 0.333333...
    1/6 = 0.166666...
    1/7 = 0.142857142857142857...
    1/9 = 0.111111...

    In base 2:
    1/3 (11) = 0.01010101...
    1/5 (101) = 0.001100110011...

    In base 3:
    1/2 = 0.11111111...

    That is why decimal representations are usually not exactly translated into floating-point binary representations. One only has a finite number of digits available, and one has to cut off an infinite sequence of them.


    It is easy to prove that an infinite repeating sequence of digits gives a rational number.

    It is more difficult to prove that every rational number can be represented with an infinite repeating sequence of digits, but it can be done.

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    Quote Originally Posted by lpetrich View Post
    A feature of rational numbers is that their decimal representations always have infinitely repeating sequences of digits. This is true not only for base 10, but also for every possible base of a place system.

    Examples:

    1 = 1.000000... = 0.999999...
    1/3 = 0.333333...
    1/6 = 0.166666...
    1/7 = 0.142857142857142857...
    1/9 = 0.111111...

    In base 2:
    1/3 (11) = 0.01010101...
    1/5 (101) = 0.001100110011...

    In base 3:
    1/2 = 0.11111111...

    That is why decimal representations are usually not exactly translated into floating-point binary representations. One only has a finite number of digits available, and one has to cut off an infinite sequence of them.


    It is easy to prove that an infinite repeating sequence of digits gives a rational number.

    It is more difficult to prove that every rational number can be represented with an infinite repeating sequence of digits, but it can be done.
    Why is .99999999... always 1? Is there any proof? I would think that it is actually the real number closest to 1. But would it be rational?

    In broader terms, if any two real numbers are next to each other, wouldn’t the decimal expansion of the smaller one repeat forever with 9's? At least at some point?

    Which brings me to even a larger point. If numbers are infinite, why can’t we come up with two arbitrarily large numbers and divide them to come up with pi?

    SLD

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    Why does any of this matter?

    Numbers are arbitrary constructs and so are operations.

    The things that come out of these arbitrary schemes are just random noise.

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    One more point about Pi. If Pi has infinite digits, then it should have an infinite number combinations of digits. At some point wouldn’t it then be repeating the entire sequence up to that point? And for say two trillion times in a row? Granted that’s not infinite, but for all practical purposes if we discovered that point, we’d be mightily confused.

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    Quote Originally Posted by SLD View Post
    Why is .99999999... always 1? Is there any proof? I would think that it is actually the real number closest to 1. But would it be rational?
    Yes there is a proof. Let us consider n digits past the decimal point. Its value is 9/10 + 9/10^2 + ... + 9/10^n. It is rather obvious that we have a geometric series, and the sum of a geometric series is
    \sum_{k=0}^n a^k = \frac{1 - a^{n+1}}{1 - a}
    This result is easy to prove by mathematical induction.

    Applying it to this problem, I find 9/10 * (1 - 1/10^(n+1)) / (1 - 1/10) = 1 - 1/10^(n+1)

    As n -> infinity, this sum tends to 1, and thus, 0.999999... = 1.

    This proof is easily generalized to other number bases. Base-2 0.111111... = 1, base-3 0.222222... = 1, etc.

    In broader terms, if any two real numbers are next to each other, wouldn’t the decimal expansion of the smaller one repeat forever with 9's? At least at some point?
    There is no such thing as two real numbers being next to each other, because if they are different, then one can easily find a number between them, their mean.

    Which brings me to even a larger point. If numbers are infinite, why can’t we come up with two arbitrarily large numbers and divide them to come up with pi?
    No, because that it is not possible. In fact, that is what makes an irrational number irrational. Proving Pi is Irrational: a step-by-step guide to a “simple proof” requiring only high school calculus – Mind Your Decisions Proof that π is irrational

    One can get arbitrarily close to pi by using rational numbers, but one cannot get pi itself.

    It is not just pi that is irrational. The square root of 2 was discovered to be irrational almost 2,500 years ago, and there is a simple proof of it being irrational.

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    Quote Originally Posted by untermensche View Post
    Why does any of this matter?

    Numbers are arbitrary constructs and so are operations.

    The things that come out of these arbitrary schemes are just random noise.
    untermensche, why do you think that numbers are arbitrary? Do you think that 2 + 2 = 5 is just as valid as 2 + 2 = 4? The truth of 2 + 2 = 4 and the falsehood of 2 + 2 = 5 arise from the definitions of 2, 4, 5, addition, and equality of numbers.

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    Quote Originally Posted by lpetrich View Post
    Quote Originally Posted by untermensche View Post
    Why does any of this matter?

    Numbers are arbitrary constructs and so are operations.

    The things that come out of these arbitrary schemes are just random noise.
    untermensche, why do you think that numbers are arbitrary? Do you think that 2 + 2 = 5 is just as valid as 2 + 2 = 4? The truth of 2 + 2 = 4 and the falsehood of 2 + 2 = 5 arise from the definitions of 2, 4, 5, addition, and equality of numbers.
    What is arbitrary is saying 2 = 2.

    You have to look at it abstractly and totally ignore that they are not really the same thing. They are in different places. They are separate entities.

    Once 2 is defined an 4 is defined and '+' is defined and '=' is defined then 2 + 2 = 4 exists due to definition. The definitions do not allow 2 + 2 to = 5.

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    Quote Originally Posted by lpetrich View Post
    As n -> infinity, this sum tends to 1, and thus, 0.999999... = 1.
    I have some questions.

    potato) Why is .999... not called irrational form?

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    An irrational number is a number that is not a rational number, a ratio of two integers.

    Here is a proof that an infinitely repeating sequence of digits makes a rational number:

    A number N can be expressed in base B as

    N = N0 + N1*B-k1 + N2*B-k1-k2 + N2*B-k1-2k2 + N2*B-k1-3k2 + N2*B-k1-4k2 + ...

    where N0 is the integer part, N1 has at at most k1 digits, and N2 has at most k2 digits. Since we have a geometric series,

    N = N0 + N1*B-k1 + N2*B-k1 / (Bk2 - 1)

    Which is, of course, a rational number.


    The proof in the opposite direction is more difficult. It involves Euler's theorem, also called the Fermat-Euler theorem or Euler's totient theorem. It states that, for relatively prime a and n,

    aphi(n) = 1 mod n

    where phi(n) is Euler's totient function, the count of all positive integers less than n that are relatively prime to n. phi(1) = 1, phi(p) = p-1 for prime p, phi(pm) = pm-1*(p-1), and phi(a*b) = phi(a)*phi(b) when a and b are relatively prime.

    Consider a rational number N/D. Decompose D into two parts: D = D0*D1. Of these, D0 evenly divides Bm for some m, and D1 is relatively prime to B. Now use Euler's totient theorem.
    Bphi(D1) = 1 mod D1
    or
    Bd = K*D1 + 1
    where d = phi(D1)

    N/D thus becomes N/(D0*D1) = N'/D' where N' = N*(Bm/D0)*K is an integer and D' = Bm*(Bd - 1).

    By construction, this is a rational number. Furthermore, this construction gives the length of the repeat or some multiple of it.

    So (having an infinitely repeating sequence of digits) <-> (being a rational number)

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    Quote Originally Posted by SLD View Post
    One more point about Pi. If Pi has infinite digits, then it should have an infinite number combinations of digits. At some point wouldn’t it then be repeating the entire sequence up to that point? And for say two trillion times in a row? Granted that’s not infinite, but for all practical purposes if we discovered that point, we’d be mightily confused.
    Yes, it will have repeats, but it won't have form (N0).(N1)(N2)(N2)(N2)(N2)... Only rational numbers can have that form.

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