## Step 11: Using the Formula for "pure" Repeating Decimals

The trick here is to expand the decimal number as a the sum of its integer part, the non-repeating part, and whatever is left over. The whatever-is-left-over part will contain a pure repeating decimal times a power of 10. Then I use the formula for a pure repeating decimal to find the equivalent fraction f(p) = p/(10n-1) Then I substitute that fraction back into the previous expression.

Let's revisit one of the repeating decimals from Step 7.
`r = 3.45[6]`
From inspection this is an impure repeating decimal because it has a non-repeating part. It also has an integer part, which is another reason why it is not pure.

Expand this decimal as a the sum of its integer part, the non-repeating part, and whatever is left over.
`r = 3 + 0.45 + 0.00[6]`
The middle term is equal to 45/100 and the last term is equal to (1/100)*0. [6]
`r = 3 + 45/100 + (1/100)*0.[6]`
Using the formula for a pure repeating decimal, f(p) = p/(10n - 1), find 0. [6] =6/9 =2/3. Substitute this into the previous expression and get:
`r = 3 + 45/100 + (1/100)*(2/3)`
Find a common denominator for the two fractional terms, and add these together. This gives an answer in mixed fraction form.
`r = 3 + 135/300 + 2/300 = 3 + 137/300`
Convert this to an improper fraction, just to check that it is the same result as the one found in the example in Step 7.
`r = 3 + 137/300 = 900/300 + 137/300 = 1037/300`

Another example:
`r = 0.00[45]`
This decimal has zero integer part, and non-repeating part consisting of two zeros. The repeating part is not "touching" the decimal point, so this repeating decimal cannot be pure.

However it is equal to (1/100)*0. [45], and 0. [45] = 45/99 is a pure repeating decimal.
`r = (1/100)*0.[45] = (1/100)*(45/99) = 45/9900 = 1/220`
I do a lot of work using an <u>Integer Basic</u> that needs to work with decimal fractions +-32,000&bull;00000<br> and even double precision variables need to use some of the techniques you describe here.<br> <br> And some I did not know.&nbsp; I especially like the over-bar description of endless repeating fractions.<br> <br> A
I&nbsp;just noticed that, for some reason, a bunch of line-breaks were missing from certain parts of this instructable, so that text that used to look like:<br /> <blockquote> <div>equation1<br /> equation2<br /> equation3</div> </blockquote>wound up looking like:<blockquote> <div>equation1equation2equation3</div> </blockquote>That is to say, totally f-d up, and unreadable.&nbsp; I suspect this was the work of some Official Instructables robot going through and updating the markup language, which they change every so often for some reason. &nbsp;<br /> <br /> I think I&nbsp;fixed all the damage, but if I didn't please comment, I&nbsp;mean, in the unlikely event that someone besides me actually reads this 'ible.<br />
A handy way to remember the decimals in sevenths (mentioned in the 7th step) is 7 14 28 5.. Two times 7 is 14, two times 14 is 28, two times 28 is 56 but the repeat starts after the 5 so you only end up with 7 14 28 5..restart 7 14.. and so on. Any /7 remainder in decimals you only have to check what digit to start with - 2/7 -> 20/7 first digit will be 2. Then go from there, 0.285714285... Not that relevant to factoring, but kind of neat.
That is kind of neat ...7 14 28 5 ... Among the fractions {1/2, 1/3, 1/4, 1/5,1/6,1/7,1/8, 1/9} , 1/7 is the one with the weirdest decimal representation, and the hardest to remember... Thanks for the tip!
No prob. It's kind of a neat party trick (for very specific geekly parties) to quote percentages of stuff that there happens to be seven of. Aww man, twenty eight point five seven one four two percent of my meatballs are burned!
s/factoring/fraction to decimal/g
I noticed the proof in Step 10 had some kinda serious typos, places where "-n" that should have been "+n", and a "greater than" sign that should have been "greater than or equals". I decided these were serious enough to re-edit the picture containing this proof and upload it again. Hopefully that will be the last correction.
Hey everybody! Thank you to whoever it is who voted/decided this instructable was a winner for answering the "Decimal to Fraction" question. The subject of repeating decimals has always been interesting to me, especially repeating nines. One thing I forgot to mention in this instructable is the cultural phenomena of marketeers who offer their wares with prices ending in nines, e.g. \$0.99 < \$1.00 \$19.99 < \$20.00 The other day I found a beautiful 1.999 in the wild, and I decided to take a picture of this and add it to Step 10. BTW, if this ible has any really obvious errors, e.g. math errors, please point them out to me. Any other comments, positive or negative, are also welcome.