# math

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how to find the reciprocal??

you flip the reciprocal example (4 over 5 the reciprocal will be 5 over 4.

If it is a fraction, switch the numerator with the deniminator.
Example: reciprocal of 2/7 = 7/2 = 3 1/2

If it is a mixed number, write it as a fraction and turn the fraction and turn upside down.
Example: reciprocal of 2 1/2 = reciprocal of 5/2 = 2/5

If it is a whole number (or any number!) N, the reciprocal is 1/N.

If it is a non-integer in decimal form, such as 1.23, perform the opetation 1 divided by 1.23, and get 0.813...

If you need to calculate 1/a you can use the following algorithm. You start with a guess x_{0} and you calculate better and better approximations x_{1}, x_{2},...etc recursively using the formula:

x_{n+1} = 2x_{n} - a x_{n}^2

Note that there are no divisions in this equation. This algorithm will double the number of significant digits after each iteration, so it's superior to long division where you only get one more digit per step (which you have to find by trail and error, so the long division algorithm isn't completely division free itself).

Example:
We want to find the decimal expansion of 1/68. 1/50 is 0.02, 1/100 is 0.01, so let's start with an initial guess of x_{0} = 0.015.

The algorithm gives:

x_{1} = 2*x_{0} - 68*x_{0}^2 = 0.0147

x_{2} = 2*x_{1} - 68*x_{1}^2 = 0.01470588

x_{3} = 2*x_{2} - 68*x_{2}^2 =
0.014705882353

The 1/x function of my calculator gives:

1/68 = 0.0147058823529

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