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Math Help - Finding bases from arithmetic ops

  1. #1
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    Finding bases from arithmetic ops

    Determine the base of the following operation:

    1) 302 / 20 = 12.1

    and

    2) 1234 + 5432 = 6666

    Assume that the base of the system is x.

    On the 1st problem, I would think the answer would be base 13, since the division in base 10 is 15.1. Is this correct?

    On the 2nd problem, I would think base 10, but I don't think there is enough information given, since there are no carries.
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  2. #2
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    Quote Originally Posted by aaronrj View Post

    On the 1st problem, I would think the answer would be base 13, since the division in base 10 is 15.1. Is this correct?
    You are correct to say it is base 10 but you cannot imply base 13 given simply this. The need to show it is base 13


    Quote Originally Posted by aaronrj View Post



    On the 2nd problem, I would think base 10, but I don't think there is enough information given, since there are no carries.

    By inspection you can see it is base 10.
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  3. #3
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    has no solution?

    Quote Originally Posted by pickslides View Post
    You are correct to say it is base 10 but you cannot imply base 13 given simply this. The need to show it is base 13
    By inspection you can see it is base 10.
    is the question
    302/20=12.1 proper?


    i tried to take x as the base and trace the division algebraically:
    ( number 302 implies x>3 for the digit 3 appears in some number-----inference 1)


    x + 2 + x^-1
    ----------------------------------
    2x + 0 ) 3x^2 + 2
    , :
    , :

    is what i expected to trace.
    Following,
    x
    ----------------------------------
    2x + 0 ) 3x^2 + 2
    , 2x^2
    , ---------------
    , x^2 comparing with ordinary arithmetic way,
    x^2 must be lesser than 2x ( otherwise the quotient would be 2x or more ).

    However, x^2 < 2x is true only when x=1 which is a contradiction with inference 1


    what is going wrong here?
    :
    Last edited by integerfan; September 8th 2009 at 01:58 AM. Reason: clarity
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  4. #4
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    ok

    To find the base of 302 / 20 = 12.1, could I write

    3 * x^2 + 0 * x^1 + 2 * x^0

    Then write 12.1 as

    1 * x^1 + 2 * x^0 + 1 * x^-1

    and then solve for x?

    Also, how would I show that 1234 + 5432 = 6666 is base 10?
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  5. #5
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    Quote Originally Posted by aaronrj View Post
    Determine the base of the following operation:
    1) 302 / 20 = 12.1
    ...

    Assume that the base of the system is x.

    On the 1st problem, I would think the answer would be base 13,
    Is this correct?
    ...
    in base 4 this division is correct:
    302 / 20 = 12.1
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  6. #6
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    Quote Originally Posted by integerfan View Post
    is the question
    302/20=12.1 proper?


    i tried to take x as the base and trace the division algebraically:
    ( number 302 implies x>3 for the digit 3 appears in some number-----inference 1)


    x + 2 + x^-1
    ----------------------------------
    2x + 0 ) 3x^2 + 2
    , :
    , :

    is what i expected to trace.
    Following,
    x
    ----------------------------------
    2x + 0 ) 3x^2 + 2
    , 2x^2
    , ---------------
    , x^2 comparing with ordinary arithmetic way,
    x^2 must be lesser than 2x ( otherwise the quotient would be 2x or more ).

    However, x^2 < 2x is true only when x=1 which is a contradiction with inference 1


    what is going wrong here?
    :
    The idea of a base is positional notation.
    302
    in decimal it is

    3 \cdot 10^2 + 0 \cdot 10^1 + 2 \cdot 10^0<br />

    in base x it is

     3 \cdot x^2 + 0 \cdot x^1 + 2 \cdot x^0<br />

    Try it that way.

    ( 1x^3 + 2x^2 +3x^1 + 4x^0 ) + ( 5x^3 + 4x^2 + 3x^1 + 2x^0 ) = 6x^3 + 6x^2 + 6x^1 + 6x^0

    Solve for x.

    .
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  7. #7
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by aaronrj View Post
    Determine the base of the following operation:

    2) 1234 + 5432 = 6666

    On the 2nd problem, I would think base 10, but I don't think there is enough information given, since there are no carries.
    All you can say is that the base is greater than 6. I've thought about this and I can't see how you can get any more precise than this. Anyone care to show me wrong? It's bewildering.
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