What is a Recurring Decimal?

S3-SA4-0248

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 endlessly after the decimal point. We show these repeating digits by placing a bar over them. It's also called a repeating decimal.

Simple Example

Quick Example

Imagine you want to divide 1 samosa equally among 3 friends. Each friend gets 1/3 of the samosa. If you try to write 1/3 as a decimal, you get 0.3333... where the '3' goes on forever. This is a recurring decimal, written as 0.bar(3).

Worked Example

Step-by-Step

Let's convert the fraction 2/3 into a recurring decimal.

1. Divide the numerator (2) by the denominator (3).
---2. Set up long division: 2 divided by 3.
---3. Since 2 is smaller than 3, put a '0' in the quotient and add a decimal point, then add a '0' to the 2, making it 20.
---4. Now divide 20 by 3. 3 times 6 is 18. Write '6' after the decimal point in the quotient.
---5. Subtract 18 from 20, which leaves 2.
---6. Add another '0' to the remainder 2, making it 20 again.
---7. Divide 20 by 3 again. You get 6, and a remainder of 2. This will keep repeating.
---8. So, 2/3 as a decimal is 0.6666... which is written as 0.bar(6).

Answer: 2/3 = 0.bar(6)

Why It Matters

Recurring decimals are important in fields like computer science and engineering, especially when dealing with precise calculations. They help AI and data science models handle numbers accurately. Understanding them can open doors to careers in data analysis or software development, where precision is key.

Common Mistakes

MISTAKE: Writing 1/3 as 0.3 or 0.33 | CORRECTION: The '3' repeats infinitely, so it must be written as 0.bar(3) to show it never ends.

MISTAKE: Only putting the bar over the last digit when multiple digits repeat, e.g., 0.121212... written as 0.12bar(2) | CORRECTION: The bar must cover ALL the repeating digits, e.g., 0.bar(12).

MISTAKE: Confusing a recurring decimal with a terminating decimal | CORRECTION: A terminating decimal ends (like 0.5), while a recurring decimal has digits that go on forever (like 0.bar(3)).

Practice Questions

Try It Yourself

QUESTION: Convert 1/9 into a recurring decimal. | ANSWER: 0.bar(1)

QUESTION: Which of these is a recurring decimal: 0.25, 0.12345..., 0.7777...? | ANSWER: 0.7777... (or 0.bar(7))

QUESTION: Write the fraction 5/6 as a recurring decimal. | ANSWER: 0.8bar(3)

MCQ

Quick Quiz

Which of the following represents a recurring decimal?

0.5

0.125

0.6666...

0.75

The Correct Answer Is:

Option C, 0.6666..., shows the digit '6' repeating infinitely, which is the definition of a recurring decimal. Options A, B, and D are terminating decimals because their digits end.

Real World Connection

In the Real World

When calculating the exact fuel efficiency of a car in kilometers per liter, sometimes the division results in a recurring decimal. For example, if a car travels 100 km on 7 liters of petrol, the efficiency is 100/7 km/L, which is approximately 14.285714... km/L. Understanding recurring decimals helps engineers and data scientists make accurate models for vehicles.

Key Vocabulary

Key Terms

DECIMAL POINT: A dot used to separate the whole number part from the fractional part of a number | NUMERATOR: The top number in a fraction | DENOMINATOR: The bottom number in a fraction | TERMINATING DECIMAL: A decimal that has a finite number of digits after the decimal point | BAR NOTATION: A line placed over repeating digits in a recurring decimal

What's Next

What to Learn Next

Great job understanding recurring decimals! Next, you can explore how to convert recurring decimals back into fractions. This will deepen your understanding of the relationship between fractions and decimals and help you solve more complex problems.