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24 May, 06:33

How many bits does 2 to the 100 have if written in base 2?

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Answers (1)
  1. 24 May, 06:47
    0
    Not sure how to give a hint without blatantly giving the answer but ...

    Consider an n - digit number in base b.

    That is N=an-1an-2 ... a0=∑k=0akbk

    N

    =

    a

    n

    -

    1

    a

    n

    -

    2

    .

    .

    .

    .

    .

    a

    0

    =



    k

    =

    0

    a

    k

    b

    k

    Note aka

    k

    <

    b

    so we can easily show NN

    <

    b

    n

    (may have to repeat and argue inductively.

    And presumably to be n - digit than an-1≠0

    a

    n

    -

    1



    0

    so N≥bn-1

    N



    b

    n

    -

    1

    .

    So we have: every n digit number is between bn-1

    b

    n

    -

    1

    inclusively and bn

    b

    n

    exclusively. This should be blindingly obvious to us if b=10

    b

    =

    10

    .

    So ... that's a really important and fundamental result. Remember and use it.
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