## Step 8: Improper Fractions, Proper Fractions, and Mixed Fractions.

Some of you reading this may feel a little uncomfortable calling an expression like,
`1037/300`
a fraction. I mean it has a numerator and a denominator, and it has a division operator, but still, some people don't want to call it fraction. Why? Because it has a value greater than 1, and in common usage the word "fraction" means a portion of something necessarily less than the whole; i.e. something less than 1.

Suppose for example a house painter tells you that he can paint your house for "a fraction of the price" compared to a competing house painter. That usually doesn't mean he is going to charge you more than the competitor. But if the "fraction" he had in mind was 4/3, then that's exactly what it would mean. His price would be the competing price plus 1/3 of the competing price. Or suppose there's a surgeon who wants to cut out a "fraction" of my liver to transplant it into my brother. It's not possible for that "fraction" to be 4/3, or 3/2, or any number greater than 1.

Because of this notion that in many cases the word "fraction" should mean "less than 1", and probably for other reasons, mathematicians came up with the definition of a proper fraction. A fraction is said to be proper if and only if it is strictly less than 1, which is equivalent to the numerator being strictly less than the denominator.

Example: 0. [14] = 14/99 is a proper fraction because 0. [14] = 0.141414... < 1, or equivalently, 14/99 is proper because 14 < 99

An improper fraction is any fraction greater than or equal to one, or equivalently, any fraction whose numerator is greater than or equal to its denominator.

Example: 3.125 = 25/8 is an improper fraction because 3.125 > 1, or equivalently, 25 > 8

Converting improper fractions to mixed fractions

If you don't like improper fractions, because they're so darned improper, you can always rewrite an improper fraction as a mixed fraction. A mixed fraction is the sum of an integer and a proper fraction,. The process for doing this is essentially long division.

For example: 311/99 is an improper fraction, and you can rewrite it as:

311/99 = (297+14)/99 = (3*99 + 14)/99 = 3 + 14/99

The integer part is 3. The proper fraction part is 14/99.

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.