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 a decimal whose digits repeat indefinitely. 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, therefor are called as rational numbers. We will see how a non-terminating recurring decimal can be expressed in the form of fraction in the following section clearly. 

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, π (3.14159265358979…) and √2 (1.41421356237…). These cannot be expressed as fractions; they do not follow any specific pattern. When a decimal neither terminates nor repeats, it cannot be written as a ratio of two integers. 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:

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 part has one digit (e.g., 0.111…), multiply both sides by 10 to shift it 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 \(x\) 

\(10x - x = 6.666… – 0.666….\)
 

\(9x = 6 \)
 

\(x = \frac{6}{9} = \frac{2}{3} \)
 

The final answer is \(\frac{2}{3} \).

Non-terminating decimals can seem tricky because they go on forever. But with simple ways, you can easily identify, understand, and work with them. Here are some tips and tricks to help you master them:

• Remember that non-terminating decimals are either recurring (repeating) or non-recurring (irrational). Identifying the type is the first step.
   
• For recurring decimals, try to spot the repeating block of digits. This makes it easier to write them as fractions.
   
• Practice converting fractions to decimals using long division to see which ones terminate and which ones repeat.
   
• For non-recurring decimals like 𝜋 or \(\sqrt{2} \), learn to round them to a few decimal places for calculations.
   
• Regularly work on examples from textbooks or worksheets. The more you practice, the easier it is to spot patterns and understand non-terminating decimals.

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.  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 × 10²³) which 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.
   
• Scientific constants and logarithms: The values of constants like 𝑒 (2.71828...) or ln2 are non-terminating, non-repeating decimals. They are used in growth or decay models, compound interest formulas, population growth and many natural phenomena. 
   
• Computer graphics and digital modeling: To draw curves, circles or smooth transitions, computers approximate irrational numbers like 𝜋 and square roots, with non-terminating decimals. These approximations are essential for rendering smooth graphics.