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

Picture of Improper fractions, proper fractions, and mixed fractions.
Some of you reading this may feel a little uncomfortable calling an expression like,
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.