## Step 8: Improper fractions, proper fractions, and mixed fractions.

1037/300a 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.

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