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...

It looks like you already knew the answer to your own question

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).

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 =

The 1/x function of my calculator gives:

1/68 = 0.0147058823529

To find the reciprocal of a number, you can follow different methods depending on the format of the number:

1. For a fraction: Flip the numerator with the denominator. For example, the reciprocal of 4/5 is 5/4.

2. For a mixed number: Write the mixed number as a fraction and then flip the fraction. For example, the reciprocal of 2 1/2 is the reciprocal of 5/2, which is 2/5.

3. For a whole number: The reciprocal of any whole number N is 1/N. For example, the reciprocal of 3 is 1/3.

4. For a non-integer in decimal form: To find the reciprocal of a non-integer number like 1.23, calculate 1 divided by the number. For example, the reciprocal of 1.23 is approximately 0.813.

Another option to calculate the reciprocal is by using an algorithm that recursively finds better approximations of the reciprocal. One such algorithm is the Babylonian method:

1. Start with an initial guess, denoted as x₀.
2. Calculate a better approximation using the formula: x_(n+1) = 2 * x_n - a * x_n^2, where x_n is the current approximation and a is the number whose reciprocal is being calculated.
3. Repeat step 2 until you reach the desired level of accuracy or precision.

This algorithm allows for a more precise calculation of the decimal expansion of a reciprocal.

For example, to find the decimal expansion of 1/68 using the Babylonian method:
1. Start with an initial guess of x₀ = 0.015 (arbitrary guess).
2. Calculate x₁ using the formula: x₁ = 2 * x₀ - 68 * x₀^2 = 0.0147.
3. Calculate x₂ using the formula: x₂ = 2 * x₁ - 68 * x₁^2 = 0.01470588.
4. Continue this process until you reach the desired level of accuracy.

Using a calculator with a reciprocal function, you can verify the result. The reciprocal of 68 is approximately 0.0147058823529.