Dividing large numbers

Stratosphere

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

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.

[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)

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

Yes I am asking for a faster way.

qntty

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