What is Pure Recurring Decimal?

S3-SA4-0056

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, without any non-repeating digits in between. It's like a never-ending pattern right after the point.

Simple Example

Quick Example

Imagine you are sharing 1 ladoo equally among 3 friends. Each friend gets 1/3 of the ladoo. When you convert 1/3 to a decimal, you get 0.3333... Here, only the digit '3' repeats forever right after the decimal point. This is a pure recurring decimal.

Worked Example

Step-by-Step

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

1. Set up the division: Divide 2 by 3.

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2. Start dividing: 2 cannot be divided by 3 directly, so we put 0. and add a zero to 2, making it 20.

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3. Divide 20 by 3: 3 goes into 20 six times (3 x 6 = 18). Write down 6 after the decimal point.

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4. Subtract: 20 - 18 = 2. We have a remainder of 2.

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5. Bring down another zero: The remainder is 2, so we add another zero, making it 20 again.

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6. Repeat division: 3 goes into 20 six times (3 x 6 = 18). Write down 6 again.

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7. Observe the pattern: You will keep getting a remainder of 2 and adding a zero, making the digit '6' repeat forever. So, 2/3 = 0.6666...

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Answer: Since only the digit '6' repeats right after the decimal point, 0.6666... is a Pure Recurring Decimal.

Why It Matters

Understanding recurring decimals helps in fields like Computer Science and Engineering, especially when dealing with precise calculations and data representation. Economists use these concepts when analyzing growth rates or financial models that involve repeating patterns. This foundational math skill is crucial for problem-solving in many advanced subjects.

Common Mistakes

MISTAKE: Thinking 0.12333... is a pure recurring decimal. | CORRECTION: In 0.12333..., the '1' and '2' do not repeat. Only '3' repeats. A pure recurring decimal has ALL digits after the decimal repeating.

MISTAKE: Confusing pure recurring decimals with terminating decimals like 0.5. | CORRECTION: Terminating decimals end, like 0.5 or 0.25. Pure recurring decimals go on forever with a repeating pattern, like 0.333...

MISTAKE: Not identifying the *entire* repeating block. For example, thinking 0.121212... is 0.1 repeating. | CORRECTION: The repeating block is '12'. The correct way to write it is 0.bar(12), showing that both 1 and 2 repeat together.

Practice Questions

Try It Yourself

QUESTION: Is 0.7777... a Pure Recurring Decimal? | ANSWER: Yes

QUESTION: Convert the fraction 5/9 to a decimal. Is it a pure recurring decimal? | ANSWER: 5/9 = 0.5555... Yes, it is a pure recurring decimal.

QUESTION: Which of these is a pure recurring decimal: A) 0.125 B) 0.232323... C) 0.1234 | ANSWER: B) 0.232323...

MCQ

Quick Quiz

Which of the following is a Pure Recurring Decimal?

0.5

0.121212...

0.123

0.12333...

The Correct Answer Is:

Option B, 0.121212..., has all digits after the decimal point ('12') repeating. Options A and C are terminating decimals. Option D has non-repeating digits ('12') before the repeating digit ('3'), making it a mixed recurring decimal, not a pure one.

Real World Connection

In the Real World

In computer programming, when you divide numbers, sometimes you get these repeating decimals. For example, if you're writing code for a billing system for a chai shop and need to calculate 1/3rd of a total amount, the computer has to approximate 0.333... This shows how understanding these decimals is important for accurate calculations in digital systems, from UPI transactions to weather simulations.

Key Vocabulary

Key Terms

DECIMAL: A number that includes a decimal point, like 3.5 | RECURRING: Repeating forever in a pattern | FRACTION: A part of a whole, like 1/2 | TERMINATING: Ending, not continuing forever

What's Next

What to Learn Next

Great job understanding pure recurring decimals! Next, you should explore 'Mixed Recurring Decimals'. These are a bit different because some digits after the decimal don't repeat, but then a pattern starts. It's an interesting step up from what you just learned!