What is Recurring Decimal?

S3-SA4-0055

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A recurring decimal is a decimal number that has one or more digits that repeat infinitely after the decimal point. It happens when you divide two numbers and the division never ends, but the remainder starts repeating.

Simple Example

Quick Example

Imagine you want to share 1 gulab jamun equally among 3 friends. Each friend gets 1/3 of a gulab jamun. If you try to write 1/3 as a decimal, you get 0.3333... where the '3' keeps repeating forever. This '0.3333...' is a recurring decimal.

Worked Example

Step-by-Step

Let's convert the fraction 2/9 into a decimal to see if it's recurring.

Step 1: Set up the division. We need to divide 2 by 9.

Step 2: Since 2 is smaller than 9, add a decimal point and a zero to 2, making it 2.0. Write 0. in the quotient.

Step 3: Divide 20 by 9. 9 goes into 20 two times (9 x 2 = 18). Write '2' after the decimal point in the quotient (0.2).

Step 4: Subtract 18 from 20, which leaves a remainder of 2.

Step 5: Bring down another zero, making the new number 20. Divide 20 by 9 again. It goes in 2 times.

Step 6: You will notice the remainder is always 2, and the digit '2' keeps repeating in the quotient. So, 2/9 = 0.2222...

Answer: 2/9 is a recurring decimal, written as 0.2 with a bar over the 2 (0.2 bar).

Why It Matters

Understanding recurring decimals is important in fields like computer science and engineering, especially when dealing with precise calculations. In data science, knowing how to handle these numbers ensures accuracy in models. Even in finance, small recurring decimals can add up to significant amounts over time, making them crucial for accurate calculations.

Common Mistakes

MISTAKE: Thinking all decimals that don't end quickly are recurring. | CORRECTION: Only decimals where a specific digit or block of digits repeats infinitely are recurring. Decimals like 0.123456789... where no pattern repeats are not recurring.

MISTAKE: Not using the bar notation correctly, or forgetting to use it. | CORRECTION: The bar (vinculum) must be placed ONLY over the digit(s) that repeat. For 0.333..., write 0.3 bar. For 0.121212..., write 0.12 bar.

MISTAKE: Confusing recurring decimals with terminating decimals. | CORRECTION: Terminating decimals end after a finite number of digits (e.g., 0.5, 0.25). Recurring decimals never end and have repeating digits (e.g., 0.333..., 0.181818...).

Practice Questions

Try It Yourself

QUESTION: Is 0.7777... a recurring decimal? | ANSWER: Yes

QUESTION: Convert the fraction 1/3 into a decimal. Is it recurring? | ANSWER: 0.333... (0.3 bar). Yes, it is a recurring decimal.

QUESTION: Which digit(s) repeat in 0.454545...? Write it using the bar notation. | ANSWER: The digits '45' repeat. It is written as 0.45 bar.

MCQ

Quick Quiz

Which of the following is a recurring decimal?

0.5

0.125

0.6666...

0.75

The Correct Answer Is:

Option C (0.6666...) has the digit '6' repeating infinitely, which is the definition of a recurring decimal. Options A, B, and D are terminating decimals as they end after a certain number of digits.

Real World Connection

In the Real World

In cricket analytics, when calculating a batsman's strike rate (runs scored per 100 balls), sometimes the division results in a recurring decimal. For example, if a batsman scores 33 runs off 33 balls, their strike rate is 100.0. But if they score 33 runs off 30 balls, it's 110.0. If it's 33 runs off 27 balls, it's 122.222... which is a recurring decimal. Analysts need to understand these to report stats accurately.

Key Vocabulary

Key Terms

DECIMAL: A number that includes a whole part and a fractional part, separated by a decimal point (e.g., 3.5) | RECURRING: Repeating endlessly | TERMINATING DECIMAL: A decimal that ends after a finite number of digits (e.g., 0.25) | FRACTION: A way to represent a part of a whole (e.g., 1/2) | QUOTIENT: The result obtained when one number is divided by another.

What's Next

What to Learn Next

Great job learning about recurring decimals! Next, you can explore how to convert recurring decimals back into fractions. This will help you understand the relationship between fractions and decimals even better and build a stronger foundation for higher-level math concepts.