4.5 Introduction and Key Insights

While it is rare for a math text to take a bold stance on almost anything, we are going to make an exception here! No one should ever use mixed number notation, especially in mathematical situations. This being said, we often see "mixed numbers" in our daily lives, most notably at gas stations. The image below shows a price of $\displaystyle 327\frac{9}{10}$ pennies, which is, of course, a mixed number.

One benefit of mixed number notation is it makes it easier to compare the quantity to whole numbers. For instance, we know that the $327\frac{9}{10}$ price is in between $327$ and $328$. When we convert this mixed number into an improper fraction, we obtain

$327\frac{9}{10} = 327 + \frac{9}{10} = \frac{3270 + 9}{10} = \frac{3279}{10}$.

The second video link below describes how to convert mixed numbers into an improper fraction. Note that while many prefer to use proper and improper fractions to notate rational numbers, these expressions make it hard to determine which improper fraction lies between consecutive whole numbers. As much as we hate to admit it, this one point supports the use of mixed numbers.

Of course, you can disfigure a lovely, beautiful, and clearly expressed improper fraction to create an ugly, inferior mixed number – there is a downgrading conversion process in this direction, too. For example, consider the improper fraction $13/3$ which we can rewrite as:

$\frac{13}{3} = \frac{12+1}{3} = \frac{12}{3} + \frac{1}{3} = 4+\frac{1}{3} = 4\frac{1}{3}$