Non-Terminating Decimals

Non-terminating decimals are decimal numbers whose digits continue forever after the decimal point. It means the digits after the decimal point never come to an end, they keep on going without an end. Because of this characteristics, they are called non-terminating decimals. These decimals cannot be expressed as a fraction with finite digits, instead they continue without repeating or terminating. 

Non-terminating decimals are usually shown using an ellipsis ( … ) or a bar (  ̅  ) over the repeating decimals. 

Definition of Non-Terminating Decimals
 

A non-terminating decimal is a decimal number that goes on endlessly after the decimal point. The digits will not either repeat itself or terminate, and continues infinitely. 

Non-terminating decimals are of two types: 

• Non-terminating recurring decimals 
• Non-terminating non-recurring decimals. 

Examples of Non-Terminating Decimals
 

Some common examples of non-terminating decimals are: 

• π (pi), which is written as 3.14159….. . Its digits continue forever without repeating. 
• √2, whose value is 1.41421356….
• 0.333…, which comes from dividing 1 by 3. The digit 3 repeats endlessly. It is non-terminating and recurring.
• 0.121212…, where the digit 12 is repeating continuously. This is another recurring decimal.

A non-terminating decimal expansion is the decimal form of a number that continues forever without ending. It normally appears when a division does not end completely, leaving zero as the remainder. As a result, the decimal digits keep extending infinitely. 

The non-terminating decimal expansion is categorized into two: 

• Non-terminating repeating decimal expansion
• Non-terminating non-recurring decimal expansion

Non-Terminating 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. 

 Non-Terminating Non-Recurring Decimal

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.

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 \(\frac{2}{3} \).

In decimal expansion, some decimals stop after a certain number of digits, whereas others continues forever. It is important to know the difference between terminating and non-terminating decimals to identify how decimals behave when written in expanded form. Let us look into the major differences between terminating and non-terminating decimals from the table below: 

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.
   
• Parents and teachers can show how non-terminating decimals appear in daily lives, such as using π in measuring circles or irrational numbers. This will help them in learning meaningful. 
   
• Encourage the use of visual tools like number line, charts or color coded decimal patterns to help students differentiate between recurring and non-recurring decimals. 
   
• Provide students with practice problems involving long division, and help them observe when digits start repeating or when the remainder do not become zero. 
   
• Use decimal calculators, math apps or online decimal worksheets to help students get more experience with recurring and non-recurring decimals.
   
• Explain to students how decimals can be converted to fractions, and show them why irrational numbers cannot be converted. This helps to give a better understanding of the concepts to the students. 
   

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.