4.4 Introduction and Key Insights

Let's say you just bought a full candy bar. Then you discover you already had a half-eaten candy bar in your pocket. How much do you have altogether? You might write that you have $1+\frac{1}{2}$ candy bars, which some math teachers would call a mixed number. It contains a regular whole number and a new fraction. In fact, some math teachers rewrite this expression without the addition sign altogether as

$1+\frac{1}{2}=1\frac{1}{2}$

While this notation for mixed numbers is common in early math courses, mathematicians and math students largely abandon it once they move on to other courses, and they do so for good reason. It is very easy to confuse "mixed number notation" for multiplication, reading the above incorrectly as "1 times $\frac{1}{2}$". It is useful to keep in mind that mixed number notation hides a + sign, one that connects a whole number and a fraction. Nevertheless, it can be useful to understand and apply such notation.

But it is more important for you to understand that every mixed number is simply another way of writing an improper fraction. In our example above, you can also think about the one whole candy bar as two halves, so that $1=\frac{1}{2}+\frac{1}{2}$. This allows you to count how much candy you have in a different way. You can combine your two halves with the half candy bar in your pocket, so you have a total of three halves.

In purely math terms:

$1+\frac{1}{2}=1\frac{1}{2}=\frac{3}{2}$ which is an improper fraction.