Non-Terminating Decimals

Non-terminating decimals are numbers with never-ending decimals; the digits after the decimal point continue indefinitely, either repeating or not, and that is why it is called non-terminating. There are two types of non-terminating decimals:

1. Non-terminating, recurring decimal
   
    
2. Non-terminating, non-recurring decimal
    

A non-terminating recurring decimal is called a non-terminating and repeating decimal. The numbers after the decimal point do not end, and they keep repeating in a pattern. For example, 0.252525…, where 25 is the pattern that repeats. These decimals can be written as fractions and are called rational numbers.
 

A non-terminating non-recurring decimal is also called a non-terminating non-repeating decimal. This means the digits after the decimal point are non-terminating and lack a repeating pattern. For example, 0.123456…… These cannot be expressed as fractions, as they do not follow any specific pattern. Since they cannot be written as fractions, we call them irrational numbers.

How to Convert Non-Terminating Decimal to Rational Number?

As discussed earlier, non-terminating repeating decimals are rational numbers. They can be converted into rational numbers using the steps below:

Steps to convert a non-terminating decimal to a rational number:

Step 1: Let us consider the recurring decimal as x.

Step 2: Write the number and place the repeating bar above it. The bar is used to show which digits are repeating.

Step 3: Count how many digits are repeating.

Step 4: If the repeating digit after the decimal point is 1 (0.11111111…), then multiply both sides of the equation by 10 to shift the repeating part left of the decimal. If the repeating digits are two (0.23232323…), then multiply them by 100, and so on.

Step 5: Subtract the two equations to make the repeating part disappear.

Step 6: Solve for x to get the final result as a fraction, and simplify the fraction if needed.

Let us take an example: 0.6666….

Let x = 0.6666…

Multiply both sides by 10 
10x = 6.666…

Subtract x from 10x to find the value of x 
10x - x = 6.666… – 0.666….

9x = 6 

x = 6/9 = 2/3

The final answer is 2/3.

Here are some real-life applications of non-terminating decimals to understand the concept more clearly.

Construction and Architecture
           
Architects often deal with measurements involving square roots or fractions, which may result in non-terminating decimals. Let us take an example where an architect is working with measurements involving square roots of non-perfect squares. When represented in their decimal form, these square roots are non-terminating and non-repeating decimals. As an architect, it is important to know how to work with these decimals to make accurate designs.

Chemistry and Formulas
       
In chemistry, calculations like molar mass and atomic mass involve irrational numbers and non-terminating decimals. Consider an example of Avogadro’s number(6.02214179 x 1023) it often leads to long decimal results in calculations. So, chemists need to work with accurate decimal approximations.

Sports and Statistics

Non-terminating decimals are also used in sports to calculate the average. In cricket, batting or bowling averages often result in repeating decimals. Let’s consider an example: a team has won 2 out of 3 matches and its winning rate is 0.666…, which is a recurring decimal. Understanding how to round and read these decimals helps in sports analysis.