BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon103 Learners

Last updated on July 4th, 2025

Math Whiteboard Illustration

Decimal Representation of Irrational Numbers

Professor Greenline Explaining Math Concepts

The decimal representation of an irrational number is a non-repeating, non-terminating decimal number. A decimal is a set of numbers with a decimal point; the numbers to the left of the point are integers, and the numbers to the right are decimal numbers.

Decimal Representation of Irrational Numbers for US Students
Professor Greenline from BrightChamps

What is the Decimal Representation of a Number?

The decimal representation simply expresses any given number using decimal digits. It depends on whether the digits repeat, terminate, or continue infinitely after the decimal point. Let’s see how decimal numbers are categorized. 

 

 

  • Terminating decimals: These decimals end after a finite number of digits. For example, 3.25, 0.5, 27.2, etc.

 

  • Non-terminating decimals: These decimals continue indefinitely without ending. They are further classified into two types. 

    Recurring decimals: In these decimals, a specific sequence of digits repeats at regular intervals. For example,0.333…, 94346.747474…, 573.636363…, etc.

    Non-Recurring Decimals: These decimals go on infinitely without any repeating pattern. For example, 743.872367346…, 7043927.78687564…, etc.
Professor Greenline from BrightChamps

What are Irrational Numbers?

Real numbers that cannot be simplified into fractions are called irrational numbers. Since irrational numbers are decimals with non-repeating, non-terminating decimal numbers, it is impossible to convert them into fractions. For example, π ≈ 3.14159265… is an irrational number because its decimal form never ends or follows a repeating pattern. However, for practical calculations, π is often approximated as 22/7 or 3.14, even though this is not its exact value.

Professor Greenline from BrightChamps

Real Life Applications of Decimal Representation of Irrational Numbers

Irrational numbers may seem abstract, but their decimal representations play a crucial role in many real-world applications. From engineering to finance, these numbers help ensure accuracy in various fields.

 

 

  • Science and Medicine: In the fields of chemistry or physics, there are mentions of values like  'π' 'e’, which belong to the set of irrational numbers. Also, these values are also used in MRI or CT scans for analyzing issues.

     
  • Engineering and Construction: In civil engineering, precise measurements also include irrational numbers. For decimal approximations, these values are applicable.

     
  • Finance and Economics: In the financial field, compounding interest, or continuous growth calculation require Euler’s number application. These kinds of irrational number applications give accurate financial predictions.

     
  • Technology and Computing: Computer algorithms, encryption methods, and simulations often involve irrational numbers for precision in calculations and security applications. 
     
Max Pointing Out Common Math Mistakes

Common Mistakes of Decimal Representation of Irrational Numbers and How to Avoid Them

While working with irrational numbers, it’s important to understand their unique decimal properties to avoid common mistakes. Here are five frequent errors people make when representing irrational numbers as decimals, along with tips to avoid them. 
 

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Shortening instead of rounding
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

While writing an irrational number, students might truncate the decimal numbers instead of rounding them off. For example, π can be wrongly written as 3.141 instead of 3.142 (if rounding to three decimal places). Always round correctly according to the given decimal place, rather than simply cutting off digits.
 

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Writing a repeating decimal for an irrational number
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Know that repeating decimals represent rational numbers, while irrational numbers never repeat in a pattern. For example, thinking π = 3.141414… (repeating) instead of its actual infinite, non-repeating decimal.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Rounding too early in calculations
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Rounding 2 to 1.41 early in a calculation, leading to less accuracy in the final result. Keep as many decimal places as possible during calculations and only round in the final step.
 

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Assuming irrational numbers will terminate
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

 Thinking 2 = 1.414 instead of understanding that it continues infinitely. Remember that irrational numbers have non-repeating, infinite decimal expansions. They never terminate

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Using an overly simplified fraction as the exact value.
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Use the proper symbol (e.g.,  or 2) when exact values are needed, and specify when using an approximation. 
 

arrow-right
Max from BrightChamps Saying "Hey"

Solved Examples of Decimal Representation of Irrational Numbers

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Is 0.101100110011000111…an irrational number?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

Yes
 

Explanation

This decimal does not terminate and does not have a repeating pattern, which means it is an irrational number.
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 2

If π is approximately 3.141592653, what is its value rounded to three decimal places?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

3.142
 

Explanation

To round  to three decimal places, look at the fourth digit (5). Since it is 5 or greater, we round up the third decimal place from 1 to 2, giving 3.142. 
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 3

How is the golden ratio ( 1.618033988) represented as a decimal rounded to two decimal places?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

1.62
 

Explanation

In the golden ratio, the third decimal digit is 8, which is greater than 5. So, we can round the second decimal digit from 1 to 2. Hence, the answer is 1.62. 
 

Max from BrightChamps Praising Clear Math Explanations
Math Teacher Background Image
Math Teacher Image

Hiralee Lalitkumar Makwana

About the Author

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.

Max, the Girl Character from BrightChamps

Fun Fact

: She loves to read number jokes and games.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
Dubai - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom