Proper and improper fractions

We call a proper fraction the one in which the numerator is less than the denominator.

If we want to share a pizza, the process to be followed is the same as previously seen with the blue square: the number of pieces in which we divide it gives us the denominator, and the number of pieces that we take/eat gives us the numerator.

In this way, if we have a pizza divided into $n$ parts, all of the same size, and we eat a quantity m that is less that n, it is not possible that we have eaten up the entire pizza: we will always have left some pieces, exactly we will have left $n-m>0.$ In other words, the pizza fraction that is eaten will never be the entire pizza.

For example, we can think that we are three people and we are eating a pizza, which is divided into four pieces. Undoubtedly, a slice of pizza is going to be left.

In general, the value of a proper fraction, or that of the implicit division, is always less than the unit.

On the other hand, an improper fraction is a fraction whose denominator is less than the numerator.

We have now divided the pizza into $8$ slices and it turns out that we are $10$ people. So, if we want everyone to eat one slice, we are going to need an entire pizza and a couple of slices from another one, and then there will be six slices left (or maybe five because I am sure that I will have two slices!)

Its value, or its implicit division, is always greater that the unit, that is to say, we are going to need more than one pizza to have the fraction.

Finally, if the numerator and the denominator are equal, it means that we are taking as many slices of pizza as we have, which could be four out of four, eight out of eight, or n out of n. So we are taking the entire pizza, that is to say, the fraction is exactly equal to $1$.

Now that we have eaten enough, let's see some examples:

In general:

• If $n<m$ then ${\displaystyle \frac{n}{m}}<1$, and it is called a proper fraction.
• If $n>m$ then ${\displaystyle \frac{n}{m}}>1$, and it is called an improper fraction.
• If $n=m$ then ${\displaystyle \frac{n}{m}}=1$, and the fraction is the unit.