What is a fraction and how to understand it?

Fraction of a Whole

Let's look at a real-life example. We bought a cake to eat over several days. Let's say we decided to divide it into four parts so that we could gradually eat one piece each day.

Imagine that this picture shows our cake divided into four pieces. Each piece of cake is a fraction, because each piece is individually part of the whole cake.

Let's say we ate one piece. How do we write this down? It's very simple. First, draw a small line that separates the numerator and denominator of the fraction:

Below this line we write down how many pieces the cake was divided into. The cake was divided into four pieces, so we write the number four at the bottom of the line:

And above this line we write the number of pieces eaten. One piece was eaten, so we write one 1 above. Thus, we get a fraction:

These are called fractions, and each fraction consists of a numerator and a denominator. The number written on top is called the numerator of the fraction. The number written underneath is called the denominator of the fraction.

In our example, the numerator of the fraction is one, and the denominator is four. This fraction can be read as "one fourth," "one piece of four," "one fourth share," or simply "a quarter" - all of these expressions mean the same thing.

Now imagine that we ate another piece of the same cake, divided into four parts. How to write such a fraction? Very simple! The numerator will now be two, because two pieces were eaten, and the denominator remains the same - four, since the cake is still divided into four parts. So, we get a fraction:

This fraction is read as follows: "two fourths", "two pieces of four" or "two fourths of shares". All of these options mean the same number of parts eaten.

Proper and improper fractions

Proper Fractions

A proper fraction is a fraction whose numerator is less than the denominator. For example, the following fractions are proper:

Such fractions are called proper because they express a part that is smaller than the whole. A fraction is a part of something, and naturally, this part should be smaller than the whole from which it is taken. For example, if you divide a cake into four parts and take one fourth (one piece), then this piece will be smaller than the whole cake. Thus, proper fractions reflect the logical connection between the part and the whole, which is why they got their name.

Improper Fractions

An improper fraction is a fraction in which the numerator is greater than or equal to the denominator. Unlike proper fractions, such fractions show that more than one whole is taken. For example, the following fractions are improper:

Such fractions are called improper because they express a quantity that exceeds the whole. After all, the denominator shows how many parts the whole is divided into, and the numerator shows how many of these parts were taken. If the numerator is greater than the denominator, this means that more was taken than the whole, which makes the fraction "improper" from the point of view of representing a share.

The fraction represents the division operation

The line in a fraction separating the numerator and denominator indicates a division operation. It means that the numerator can be divided by the denominator.

For example, let's take the fraction 6/3 (six thirds). The fraction bar indicates that six can be divided by three. We know that six divided by three equals two. We write this using the equal sign (=) and indicate the answer. Thus, any division of numbers can be expressed as fractions. For example:

Conclusion:

any division operation can be written as a fraction.