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Math Help - How many digits?

  1. #1
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    How many digits?

    How many digits does the number 2^100 have? Using complicated caculator softwares it is 31 digits but how will you find out with pencil and paper
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    Hello, sundance240!

    How many digits does the number 2^100 have?
    Using complicated caculator softwares, it has 31 digits,
    but how will you find out with pencil and paper?

    We note that: .2^{10} .= .1024 . .10^3

    Hence: .2^{100} . .(10^3)^{10} .= .10^{30}

    . . which has 31 digits.

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  3. #3
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    Quote Originally Posted by sundance240 View Post
    How many digits does the number 2^100 have? Using complicated caculator softwares it is 31 digits but how will you find out with pencil and paper
    In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.
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    Quote Originally Posted by Opalg View Post
    In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... .
    Wow...
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    Quote Originally Posted by Opalg View Post
    In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.
    Wow! What a dinosaur. I suppose Napier's bones are fossils by now?

    -Dan
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    Quote Originally Posted by Opalg View Post
    In prehistoric times we used to do problems like this by using logarithms to base 10. These were used so often that everybody knew that log(2) = 0.3010... . Therefore log(2^{100}) = 100*log(2) = 30.10..., and since this is between 30 and 31 it follows that 2^{100} is a 31-digit number.
    of course, the "log" button on any reputable brand of calculator (including the generic one shipped with any version of Windows) will accomplish the same thing.
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  7. #7
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    Quote Originally Posted by topsquark View Post
    Wow! What a dinosaur. I suppose Napier's bones are fossils by now?

    -Dan
    Years "B.C."- before calculators.
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