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Thread: Rational numbers == infinitely repeating sequences of digits

  1. Top | #281
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    Quote Originally Posted by Jokodo View Post

    Yes. The labels are strings, whatever they may or may not refer to. The "+" Sign. In the context of strings can only be interpreted as concatenation, and concatenation of these two strings results in the string "13".
    So boring.

    The "+" sign is not a value it is an operation sign. That is predefined in the scheme. The scheme starts with creating values that correspond to assigned labels then it is possible to define operations.

    You have not moved me one inch from my position. You have not demonstrated any greater knowledge of "3" than any novice.
    Last edited by untermensche; 02-27-2019 at 02:34 PM.

  2. Top | #282
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    Quote Originally Posted by untermensche View Post
    Quote Originally Posted by Jokodo View Post

    Yes. The labels are strings, whatever they may or may not refer to. The "+" Sign. In the context of strings can only be interpreted as concatenation, and concatenation of these two strings results in the string "13".
    So boring.

    The "+" sign is not a value it is an operation sign. That is predefined in the scheme. The scheme starts with creating values that correspond to assigned labels then it is possible to define operations.

    You have not moved me one inch from my position. You have not demonstrated any greater knowledge of "3" than any novice.
    You should try admitting to being wrong every once in a while. It makes you a better person -- and it'll make you more right in total.

  3. Top | #283
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    Quote Originally Posted by Jokodo View Post
    Quote Originally Posted by untermensche View Post
    Quote Originally Posted by Jokodo View Post

    Yes. The labels are strings, whatever they may or may not refer to. The "+" Sign. In the context of strings can only be interpreted as concatenation, and concatenation of these two strings results in the string "13".
    So boring.

    The "+" sign is not a value it is an operation sign. That is predefined in the scheme. The scheme starts with creating values that correspond to assigned labels then it is possible to define operations.

    You have not moved me one inch from my position. You have not demonstrated any greater knowledge of "3" than any novice.
    You should try admitting to being wrong every once in a while. It makes you a better person -- and it'll make you more right in total.
    OK I am wrong.

  4. Top | #284
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    Quote Originally Posted by untermensche View Post
    Quote Originally Posted by Jokodo View Post

    You should try admitting to being wrong every once in a while. It makes you a better person -- and it'll make you more right in total.
    OK I am wrong.
    You've never been so right.

  5. Top | #285
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    Quote Originally Posted by Jokodo View Post
    Quote Originally Posted by untermensche View Post
    Quote Originally Posted by Jokodo View Post

    You should try admitting to being wrong every once in a while. It makes you a better person -- and it'll make you more right in total.
    OK I am wrong.
    You've never been so right.
    What am I right about again?

  6. Top | #286
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    Quote Originally Posted by untermensche View Post
    Quote Originally Posted by Jokodo View Post
    Quote Originally Posted by untermensche View Post
    More coyness.

    How many "unit values" make up "3"?

    You are simply wrong.

    It is impossible to refer to the value by anything but the symbol or word for the symbol.

    In the scheme "3" is the value and the symbol.

    They are inseperable.
    You clearly want it to be that way, but following the definition of the natural numbers you yourself provided, this isn't so. Unless you used a very unconventional definition of "definition".

    Usually when we say x is defined as y, it implies that they can be used as drop-in replacements for one another while preserving the truth of a statement. Also, what you're saying now buys you infinite recursion.
    I gave no definition of "value".

    I gave a way to create all the values of the positive integers from one value.

    It starts by simply saying that "1" is this thing called a 1 value.

    It starts by saying there is no difference between "1" and the value "1".

    You have no point.

    And you can't talk about any value without talking about the label of the value.

    You can say three or you can say one plus one plus one.

    All you can talk about are the labels.

    There is nothing else.

    There is nothing else besides 0.3333....

    It points to itself and nothing else.
    You are arguing metaphysis. In math you define rules and identies and proceed from there.

    In a computer language like C '1' + '2' = 12 if the charters and strings are properly

    for example
    char a[10] = "1 ";b[1] = "2", c[100]; // defines two text strings
    c = strcat(a,b); // adds b to the end of a

    c = 12

    if a = 'z' and b = j' the result is c = "zj"

    c = 'a' + b ' will probably add the numerical integer ASCII character codes, I'd have to run it.

    Same with c = 'a' + 'b'.

    Without quantified values of some kind and a set of rules debate is meaningless.

    If you have excel create two text boxes and try to add them, see what happens.

    In computers characters have numbers associated with them. 1 or g are referenced as numbers. Charters.

    In typed languages you have to define what is a number and a character variable.

    '1' + '3' = '13' has no meaning as it is other than how you define it.

  7. Top | #287
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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.
    Haven't thought of repeating decimals in any base.

    1 = 1.000000... = 0.999999... This one I do not see or agree with.

    I went through this argument on another thread. The utility of using a geometric series to a fractional number is to yield the nearest approximate fraction. For example converging a decimal to the nearest fractional drill size.

    if you write it out the series produces the fraction 1/1 as an approximation to 0.999..

    Fractional approximations to repeating decimals. 0.999... does not equal 1.
    0.777... 7/9
    0.888... 8/9
    0.999... 1/1

    Rounding is context and accuracy driven. In real computations involving variables with varying number of significant digits whether to truncate 0.999... or round up to 1 depends on affect of calculation accuracIn 0.999... at what nth term does it roll over to 1?

    In 0.777... at what nth term does it roll up to say 0.8?

  8. Top | #288
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    Quote Originally Posted by steve_bank View Post
    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.
    Haven't thought of repeating decimals in any base.

    1 = 1.000000... = 0.999999... This one I do not see or agree with.

    I went through this argument on another thread. The utility of using a geometric series to a fractional number is to yield the nearest approximate fraction. For example converging a decimal to the nearest fractional drill size.

    if you write it out the series produces the fraction 1/1 as an approximation to 0.999..

    Fractional approximations to repeating decimals. 0.999... does not equal 1.
    0.777... 7/9
    0.888... 8/9
    0.999... 1/1

    Rounding is context and accuracy driven. In real computations involving variables with varying number of significant digits whether to truncate 0.999... or round up to 1 depends on affect of calculation accuracIn 0.999... at what nth term does it roll over to 1?

    In 0.777... at what nth term does it roll up to say 0.8?
    It doesn't "roll up" to 8/10. It stays exactly at 7/9 - just like 0.999..., which stays exactly at 9/9, no rounding required. The decimal system at representing rational numbers whose denominator's prime factors are a subset of the base's, not quite so great at representing others. In the case of the base 10, that's halves, quarters, fifths, eigtths, tenths, sixteenths, twentieths, twentyfifths etc., but not e.g. thirds, sixths, sevenths, nineths.

  9. Top | #289
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    that should have written: "The decimal system, like any other integer base positional notation, is great at representing rational numbers when the denominator's prime factors are a subset of the base's"
    Last edited by Jokodo; 03-01-2019 at 12:46 PM.

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