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Dividing large numbers

  1. Mar 21, 2009 #1
    How would you Divide very large numbers without using a calculator?
    EX. [tex]\frac{125000}{299000000}[/tex]
     
  2. jcsd
  3. Mar 21, 2009 #2
    Long ago, before calculators, logarithms were used and invented for this purpose. You'd divide by subtracting logarithms and antilog the result to get the answer.
     
  4. Mar 21, 2009 #3
    One should usually first take out the obvious powers of ten, then factorize.

    e.g.

    [tex]\frac{125000}{299000000} = \frac{125}{299000}=\frac{5^3}{299\cdot 10^3} = \frac{5^3}{299\cdot (2\cdot 5)^3} = \frac{1}{299\cdot 2^3}[/tex]

    And [tex]299\cdot 8 = 3 \cdot 10^2 \cdot 8 - 8 = 24 \cdot 10^2 - 8 = 2400 - 8 = 2392[/tex],

    so that

    [tex]\frac{125000}{299000000} = \frac{1}{2392}[/tex]

    Which by hand is good enough for me.

    (This might be wrong tho, it is kinda late here)
     
  5. Mar 22, 2009 #4
    "How would you Divide very large numbers without using a calculator? "

    Long division is a correct algorithm. Are you asking whether or not there exists a faster way?
     
  6. Mar 22, 2009 #5
    Yes I am asking for a faster way.
     
  7. Mar 22, 2009 #6
    Slide rule?
     
  8. Mar 22, 2009 #7
    You could use Newton-Raphson. Computing x = 1/y for given y amounts to solving the equation:

    1/x - y = 0

    Then, Newton-Raphson yields the following recursion for the nth approximation


    x_{n+1} = x_n - (1/x_n - y)/(-1/x_n^2) =

    x_n +x_n -y x_n^2 =

    2 x_n - y x_n^2

    The iteration doesn't involve any divisions, so it is a true division algorithm. The number of correct digits doubles after each iteration, while with long division you only get one decimal at a time, so it is much faster than long division.
     
  9. Mar 22, 2009 #8
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