What is a Pure Recurring Decimal?

S3-SA4-0249

Grade Level:

Class 6

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

Definition

What is it?

A pure recurring decimal is a decimal number where ALL the digits after the decimal point repeat in a fixed pattern. This repeating pattern starts immediately after the decimal point. We often show this by putting a bar over the repeating digits.

Simple Example

Quick Example

Imagine you are trying to share 1 gulab jamun equally among 3 friends. If you try to divide 1 by 3, you get 0.3333... The digit '3' repeats forever right after the decimal point. So, 0.333... is a pure recurring decimal.

Worked Example

Step-by-Step

Let's convert the fraction 2/3 into a decimal to see if it's a pure recurring decimal.

STEP 1: Set up the division: 2 divided by 3.

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STEP 2: Since 2 is smaller than 3, we put a '0.' in the quotient and add a zero to 2, making it 20.

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STEP 3: Now, divide 20 by 3. 3 multiplied by 6 is 18. So, write '6' after the decimal point in the quotient. Subtract 18 from 20, which leaves 2.

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STEP 4: Bring down another zero, making it 20 again.

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STEP 5: Divide 20 by 3 again. We get 6, and a remainder of 2. You can see a pattern emerging.

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STEP 6: If you continue, you will keep getting '6' and a remainder of '2'.

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ANSWER: So, 2/3 as a decimal is 0.6666... This is a pure recurring decimal because the digit '6' repeats immediately after the decimal point. We can write it as 0.bar(6).

Why It Matters

Understanding recurring decimals helps in fields like computer science for precise calculations and in data science for analyzing patterns. Engineers use these concepts to design systems where exact values are crucial, like in building bridges or coding software for mobile apps.

Common Mistakes

MISTAKE: Thinking 0.1232323... is a pure recurring decimal. | CORRECTION: In 0.1232323..., the '1' does not repeat. Only '23' repeats, and it doesn't start immediately after the decimal point. This is a mixed recurring decimal, not pure.

MISTAKE: Forgetting the bar notation or putting it incorrectly, like writing 0.bar(3) for 0.3333... and 0.3bar(2) for 0.3222... | CORRECTION: The bar should be placed ONLY over the digits that are repeating. For 0.3333..., it's 0.bar(3). For 0.3222..., it's 0.3bar(2).

MISTAKE: Confusing recurring decimals with terminating decimals. | CORRECTION: Terminating decimals (like 0.5 or 0.25) have a finite number of digits after the decimal point. Recurring decimals have digits that go on forever in a repeating pattern.

Practice Questions

Try It Yourself

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

QUESTION: Which of these is a pure recurring decimal: 0.121212..., 0.123, 0.1222...? | ANSWER: 0.121212...

QUESTION: Convert the fraction 5/9 into a decimal and state if it is a pure recurring decimal. | ANSWER: 5/9 = 0.5555... Yes, it is a pure recurring decimal (0.bar(5)).

MCQ

Quick Quiz

Which of the following represents a pure recurring decimal?

0.45

0.454545...

0.45555...

0.456

The Correct Answer Is:

Option B (0.454545...) is a pure recurring decimal because the pattern '45' repeats immediately after the decimal point. Options A and D are terminating decimals, and Option C is a mixed recurring decimal.

Real World Connection

In the Real World

When you calculate your average marks for a subject, sometimes the division doesn't give a neat number. For example, if you get 10 marks in 3 tests and want the average, it's 10/3 = 3.333... This pure recurring decimal shows up in everyday calculations like this, even for things like calculating cricket run rates or sharing food bills equally.

Key Vocabulary

Key Terms

DECIMAL POINT: The dot that separates the whole number part from the fractional part of a number | RECURRING: Repeating in a continuous pattern | TERMINATING DECIMAL: A decimal that has a finite number of digits after the decimal point | FRACTION: A way to represent parts of a whole, written as numerator/denominator

What's Next

What to Learn Next

Great job understanding pure recurring decimals! Next, you can explore 'Mixed Recurring Decimals'. You'll see how they are similar but also different, and how to convert both types of recurring decimals back into fractions.