How can Fourier division be used to divide large numbers without a calculator?

[Stratosphere]

How would you Divide very large numbers without using a calculator?
EX. $$\frac{125000}{299000000}$$

 

[Helios]

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.

 

[Santa1]

One should usually first take out the obvious powers of ten, then factorize.

e.g.

$$\frac{125000}{299000000} = \frac{125}{299000}=\frac{5^3}{299\cdot 10^3} = \frac{5^3}{299\cdot (2\cdot 5)^3} = \frac{1}{299\cdot 2^3}$$

And $$299\cdot 8 = 3 \cdot 10^2 \cdot 8 - 8 = 24 \cdot 10^2 - 8 = 2400 - 8 = 2392$$,

so that

$$\frac{125000}{299000000} = \frac{1}{2392}$$

Which by hand is good enough for me.

(This might be wrong tho, it is kinda late here)

 

[csprof2000]

"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?

 

[Stratosphere]

"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?

Yes I am asking for a faster way.

 

[qntty]

without using a calculator?

Slide rule?

 

[Count Iblis]

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.

 

[Count Iblis]

This is also an effective method:

http://en.wikipedia.org/wiki/Fourier_division