Decimals as we saw are numbers of the form 0.25, 0.3333333, 0.4545454……….
We saw that these numbers are obtained from fractions by actual division.
Now let us classify the decimal numbers as well.
Terminating Decimals
Look at the fractions: $$\frac{1}{2} , \frac{3}{100},\frac{1}{8} ,\frac{2}{25} $$etc….
In decimal form these will be written as:
$$\frac{1}{2} = 0.5 $$
$$\frac{3}{100} = 0.03 $$
$$\frac{1}{8} = 0.125 $$
$$\frac{2}{25} = 0.04 $$
As we see all the decimals above have only finite number of numbers after the decimal point. These are perfect examples of terminating decimal.
In short terminating decimals are defined as decimals that terminate, they just stop.
Non- Terminating Decimals
Some decimals don’t stop at all. They go on and on. Such decimals are called non- terminating decimals.
In other words, while converting a fraction into decimal by division method if the division process continues indefinitely i.e. the 0 remainder is never attained then it is called a non - terminating decimal.
Examples of non – terminating decimals are 0.22222222222222…, 0.2525252525……. etc.
Again there are two types of non-terminating decimals. The examples shown above are the type in which the numbers after the decimal point repeat in a certain pattern though they don’t end. Such decimals are called recurring decimals and are denoted by a bar on the repeating terms.
Examples of recurring decimals are:
The non-terminating decimals which don’t follow any pattern like the recurring but they just go on and on are actually perfect examples of irrational numbers.
They are the non-terminating non -repeating decimals
Examples: 0.4538294398402…, 0.00986735123…….. Etc
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