What is the purpose of mixed numbers outside of common usage? I am wondering if there is indeed any real usage of mixed numbers such as $3 \frac{1}{2}$ meaning $3+\frac{1}{2}$ in standard Mathematics. Personally I dislike the use of such conventions in schools because it can lead to misunderstandings and misconceptions about rational numbers later on. In addition, these notations should be discouraged in areas such as competitive mathematics where it may cause confusion for the contestant.  
 A: It's a bad notation, nobody uses it, and I have no idea why anyone ever thought it was a good idea.  
Edit: This answer was a half-joke; I was being a bit hyperbolic, and I absolutely agree with LJL that it really doesn't matter at all. To be more helpful: nobody uses this notation in mathematics, because it doesn't really have a place there. There are really three common ways of expressing rational numbers, either exactly or approximately, and they each have advantages and disadvantages:  


*

*'Vulgar fractions': Certainly the most common inside of mathematics, multiplication is very easy in this form (it's just multiplication of integers), but comparison can be hard (which is larger, $\frac{43}{81}$ or $\frac{162}{235}$?) and addition is somewhat tricky (and a very common source of error in students - probably much more common than confusing mixed numbers with multiplication!). You also get to keep some of the nice behaviour of integers, c.f. "$\sqrt{2}$ is irrational".  

*Decimals (and other base expansions): Super easy to add and compare, and multiplication isn't too bad. Disadvantage in that you lose precision and some "exact" answers can be inelegant in this form. Fine for arithmetic (it's used for currency for a reason!) but almost worthless for mathematical proofs and similar things. Really comes into its own when you start discussing irrationals.

*Mixed numbers: These are sort of a compromise. You have the advantage that you don't need to lose precision in the way you do with decimal notation, but you can also compare things to first order quite easily (just by comparing the integer parts). Further comparison is just as hard as with vulgar fractions. Arithmetic is now completely ridiculous: for addition, your integer parts add nicely but your fractional parts are just as bad as if you were just using vulgar fractions (although you can estimate your solution just fine, if that's what you want). Multiplication is actually kind of hard in this notation, which is frankly impressive. I can really only imagine people using mixed numbers as a fast translation for sentences in english, e.g. "I have four and a half oranges" becomes "I have $4 \frac{1}
{2}$ oranges".

A: Mixed numbers used to be used in the U.S. stock market.  (See this link.)


These early stockbrokers looked to Europe for a model to build their system on and decided to base it on the system of Spain. This was largely due to the fact that the U.S. dollar's value had been based on the value of the Spanish real.
The real was the Spanish silver dollar and was divided into eight parts. This evolved from the method of counting on the hands, similar to the decimal system. The difference was that the decimal system used the thumbs as part of the number while this other system used the thumbs to denote the total of the four fingers. Therefore, a person would count to four on one hand and then use the thumb to indicate a total while they counted on the other hand. Two thumbs equaled eight. The real could be broken into two, four or eight parts, giving birth to the term pieces of eight.
So when the U.S. stock market began, they based the stock values on one-eighth fractions.

