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EX. [tex]\frac{125000}{299000000}[/tex]

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EX. [tex]\frac{125000}{299000000}[/tex]

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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)

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Long division is a correct algorithm. Are you asking whether or not there exists a faster way?

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Yes I am asking for a faster way.

Long division is a correct algorithm. Are you asking whether or not there exists a faster way?

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Slide rule?without using a calculator?

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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.

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