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Math Help - Conversion of bases

  1. #1
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    Conversion of bases

    So in base 10 we can represent 200 in base two as follows.

    200/2 = 100 R 0
    100/2 = 50 R 0
    50/2 = 25 R0
    25/2 = 12 R1
    12/2 = 6 R0
    6/2 = 3 R0
    3/2 = 1 R1
    1/2 = 0 R1

    So 200 in base 2 = 11001000.

    I understand how that works. But to get back we do this.

    Base 10 digits 0 1 2 3 4 5 6 7
    Base 2 digits 0 0 0 1 0 0 1 1

    where there are ones 2^3 + 2^6 + 2^7 = 200. Could someone explain how that works?

    Thanks.
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  2. #2
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    Quote Originally Posted by mark090480 View Post
    So in base 10 we can represent 200 in base two as follows.

    200/2 = 100 R 0
    100/2 = 50 R 0
    50/2 = 25 R0
    25/2 = 12 R1
    12/2 = 6 R0
    6/2 = 3 R0
    3/2 = 1 R1
    1/2 = 0 R1

    So 200 in base 2 = 11001000.

    I understand how that works. But to get back we do this.

    Base 10 digits 0 1 2 3 4 5 6 7
    Base 2 digits 0 0 0 1 0 0 1 1

    where there are ones 2^3 + 2^6 + 2^7 = 200. Could someone explain how that works?

    Thanks.
    That is precisely the meaning of "writing in basis x": you express numbers as a sum of powers of x multiplied by integers between 0 and x.

    For example, 306= 3\cdot 10^2+0\cdot 10^1+6\cdot 10^0=3\cdot 10^2+6

    That is why, thus, in base 2 you have to sum up the powers of 2 corresponding to the 1's in the expression of the number in binary.

    Tonio
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  3. #3
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    But why does doing the power of 2^x, where x is a remainder give the exact value?
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