Fractions, decimals and percentages: Rational numbers. 3rd year of secondary education.

 Changing fractions to decimals A fraction can be changed into a decimal by dividing the numerator by the denominator. You can use a calculator, or see how to do it in the following window. The result could be: 1.- A WHOLE NUMBER.- For example: 72/9=8. This is the case when the numerator is a multiple of the denominator. 2.- A TERMINATING DECIMAL. - For example: 197/40=4.925. This is the case when the numerator is multiplied by a power of 10 as the result is an exact multiple of the denominator. 3.- A PURE RECURRING DECIMAL - For example: 4/11=0.36363636... The decimal numbers form a pattern that is repeated indefinitely after the decimal point.  4.- A MIXED RECURRING DECIMAL.- For example: 87/66=1.3181818... The first decimal(s) after the decimal point are not repeated, but the subsequent numbers form a pattern which is repeated indefinitely, as in the previous example. Look at these different examples in the window.

Rational numbers
We have just seen, in the section above, how we can get four different types of numbers when we divide a fraction (a whole number, a terminating decimal or a pure or mixed recurring decimal). All of these numbers could be said to fall into the category of recurring decimals, as a whole number, such as 4, could be written as 4.00000... and a terminating decimal, such as 0.25, could be written as 0.250000...

Therefore, we could say that any fraction can be expressed as a recurring decimal. We can also say that the opposite is true, i.e. that any recurring decimal can be expressed as a fraction. From now on we shall refer to these numbers as rational numbers.

To sum up:

 ANY NUMBER THAT CAN BE EXPRESSED AS A FRACTION IS KNOWN AS A RATIONAL NUMBER

We have just seen how we can get four different types of numbers when we divide a fraction. All of these numbers are RATIONAL. We are going to refer to this group of numbers with the letter Q.

We can classify the set Q of RATIONAL NUMBERS as follows:

 Ángela Núñez Castaín Spanish Ministry of Education. Year 2001