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

In summary, there are multiple ways to divide very large numbers without using a calculator. Some methods include long division, slide rule, Newton-Raphson, and Fourier division. These methods involve various algorithms and iterations to find the answer.
  • #1
Stratosphere
373
0
How would you Divide very large numbers without using a calculator?
EX. [tex]\frac{125000}{299000000}[/tex]
 
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  • #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.
 
  • #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)
 
  • #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?
 
  • #5
csprof2000 said:
"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.
 
  • #6
Stratosphere said:
without using a calculator?

Slide rule?
 
  • #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.
 

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

What is the definition of dividing large numbers?

Dividing large numbers is the process of dividing a number that has multiple digits by another number, resulting in a quotient and possibly a remainder.

What is the general rule for dividing large numbers?

The general rule for dividing large numbers is to divide the divisor (the number being divided into) into the first digit or digits of the dividend (the number being divided) and continue dividing until the entire dividend has been divided or a remainder is left.

How do you know if a number is divisible by another number?

A number is divisible by another number if the quotient is a whole number and there is no remainder. For example, 12 is divisible by 3 because 12 divided by 3 is 4 with no remainder.

What are some strategies to make dividing large numbers easier?

Some strategies to make dividing large numbers easier include using estimation, breaking down the numbers into smaller chunks, and using multiplication to check your answer.

What should I do if there is a remainder when dividing large numbers?

If there is a remainder when dividing large numbers, you can either round the quotient up or down depending on the situation, or you can leave the remainder as a fraction or decimal. It is important to consider the context of the problem to determine the best course of action.

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