In Part 5 of this series, I showed that fractions of the form , , and can be converted into their decimal representations without using long division and without using a calculator.

The amazing thing is that *every* rational number can be written in one of these three forms. Therefore, after this conversion is made, then the decimal expansion can be found without a calculator.

**Case 1**. If the denominator has a prime factorization of the form , then can be rewritten in the form , where .

For example,

The step of multiplying both sides by is perhaps unusual, since we’re so accustomed to converting fractions into lowest terms and not making the numerators and denominators larger. This particular form of was chosen in order to get a power of in the denominator, thus facilitating the construction of the decimal expansion.

**Case 2.** If the denominator is neither a multiple of nor , then can be rewritten in the form .

For example,

This example wasn’t too difficult since we knew that . However, finding the smallest value of that works can be a difficult task requiring laborious trial and error.

However, we do have a couple of theorems that can assist in finding . First, since is the length of the repeating block, we are guaranteed that must be less than the denominator since, using ordinary long division, the length of the repeating block is determined by how many steps are required until we get a remainder that was seen before.

However, we can do even better than that. Using ideas from number theory, it can be proven that must be a factor of , which is the Euler toitent function or the number of integers less than that are relatively prime with . In the example above, the denominator was , and clearly, if , then . Since there are such numbers, we know that must be a factor of . In other words, must be either , , , or , thus considerably reducing the amount of guessing and checking that has to be done. (Of course, for the example above, was the least value of that worked.)

In general, if is the prime factorization of , then

For the example above, since was prime, we have .

**Case 3**. Suppose the prime factorization of the denominator both (1) contains and/or and also (2) another prime other than and . This is a mixture of Cases 1 and 2, and the fraction can be rewritten in the form .

For example, consider

Following the rule for Case 1, we should multiply by to get a in the denominator:

Next, we need to multiply by something to get a number of the form . Since is prime, every number less than is relatively prime with , so . Therefore, must be a factor of . So, must be one of , , , , , , , , and .

(Parenthetically, while we’ve still got some work to do, it’s still pretty impressive that — without doing any real work — we can reduce the choices of to these nine numbers. In that sense, the use of parallels how the Rational Root Test is used to determine possible roots of polynomials with integer coefficients.)

So let’s try to find the least value of that works.

- If , then , but is not an integer.
- If , then , but is not an integer.
- If , then , and it turns out that , an integer.

Therefore,

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