Last updated on July 5th, 2025
Non-terminating decimals are decimals that never end. For example, 0.33333… is a non-terminating decimal. In this article, we are going to learn more about 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:
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.
When students start learning about the non-terminating decimals, they can sometimes get confused between recurring and non-recurring decimals. Here are some common mistakes made by students and how to avoid those mistakes.
Convert 0.333… as a fraction?
0.333… = 1/3.
Let x = 0.333…..
Multiply both sides by 10, which gives us 10x = 3.333….
Now, subtract the equations to form another equation,
Which is 10x - x = 3.333… - 0.333…,
Which is equal to 9x = 3.
Now divide both sides by 9.
Simplify the fraction x = 1/3.
The answer is 1/3.
Sally ate 0.666… of a pizza. What fraction of the pizza did she eat?
She ate 2/3 of the pizza.
Sally ate 0.666… of a pizza, which means x = 0.666…
By following the steps of converting decimals into fractions,
we get 0.666… is equal to 2/3.
So she ate 2/3 of the pizza.
Convert 2.454545…. to a fraction?
x = 2.4545….
100x = 245.4545…
100x - x = 245.45454… - 2.454545….
99x = 243
x = 27/11
Let us consider x = 2.454545…. Next, we have to multiply x and the number by 100. Now subtract the two equations to get a new equation, which is 99x = 243. Solving for x, we get the answer as 27/11.
Identify the non-terminating recurring decimal in the following 1.23456… 1.675864…. 1.232323…
1.232323… is a non-terminating and recurring decimal.
1.232323… is a non-terminating and recurring decimal because it has repeating, never-ending digits after the decimal point.
Convert 1.656565… into a rational number?
x = 1.6565….
100x = 165.6565….
99x = 164
x = 164/99
Let us consider x = 1.6565… Now multiply both sides by 100. Now subtract both equations to get a new equation that is 99x = 164. Now, solving for x, we get 164/99 as the final result.
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.