The Decimal representation of a Number

There are three major types of decimal representations of numbers in mathematics. Everyone interested in mathematics should be aware of these three types and the way of identifying them. They are named as,

1. Finite Decimal (Terminating Decimal)
2. Recurring Decimal
   
3. Infinite Decimal

Let’s go through them one by one and have a better understanding!

1. Finite Decimal

As its name sense, finite decimals are the values where there is an end for the division of two numbers. Since it terminates the division with a fully accurate answer, these are also named as terminating decimals. Actually, division terminates means that it would give 0 again and again forever after some a point in the division. Some examples for finite decimal representation for rational numbers are as follows (in Figure 1).

Figure 1

2. Recurring Decimal

There are some decimals where their decimal parts have a recurring portion which is not zero. Such decimal numbers are called as recurring decimals as its name senses. The important fact is that there might be a single number recurring, a pair of numbers recurring, or more than two numbers recurring. For example;

• 13.22222222 … is a situation where a single number recurs
• 13.23232323 … is a situation where a pair of numbers recurs
• 13.2345623456 … is a situation where more two numbers recur

There are specific ways to represent all these situations in mathematics. It makes the calculation easier for by reducing the length of the answer. The way of representing is nothing but putting a dot (.) on top of the single number recurring, or the pair of numbers recurring, or first and last numbers of the recurring portion if its more than two numbers. Figure 2 shows some examples and way to represent them in the proper way. 

Figure 2

Since there is no end of the division of a number for recurring decimal as in the finite decimal, it should be accurately decided whether it is a recurring decimal or not. It is good practice to divide a number until getting at least three recurring portions if the fraction is not familiar. Then, it is secure to show in short forms as in figure 2.

What about the decimal value of 22÷7? This is nothing but the π as we all know. 3.141 is taken for most calculation as an approximate value. Does it mean 22÷7 = 3.141? NO!!! π is neither finite decimal nor recurring decimal. As shown above, every rational number can be written as either finite decimal or recurring decimal (If you want to learn about rational numbers, please learn from the article on “Number Sets”). Therefore, if it is not a finite decimal or a recurring decimal, it cannot be a rational number. We already know from an example in number sets, π is an irrational number!!!

3. Infinite Decimal

Answer for the above argument is that 22÷7 gives an infinite decimal. If the decimal representation of a number is not finite, it is named as an infinite decimal. Further, it can be said that all the irrational numbers have infinite decimal representations. Figure 3 shows examples of infinite decimals below.

Figure 3

The next problem might be how to obtain infinite decimal values for irrational numbers like the square root of 2, square root of 3, so on and so forth. There will be another article on how to find/calculate decimal values for irrational numbers with square root in the author’s page. If you are interested, you may look into that. For now, all you must keep in mind is that such square roots are irrational numbers and therefore, their decimal values must be infinite. Hence, no need to find hundreds of digits for the decimal portion. It is enough to find up to some length as you require.