What is a Non-Terminating Repeating Decimal?

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Grade Level:

Class 6

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

Definition

What is it?

A non-terminating repeating decimal is a decimal number that goes on forever without ending (non-terminating) and has a pattern of one or more digits that repeats endlessly (repeating). When you divide two numbers, sometimes the answer doesn't stop, but the digits start coming in a fixed cycle.

Simple Example

Quick Example

Imagine you have 10 rupees and want to share it equally among 3 friends. Each friend gets 3 rupees and 33 paise, but there's still a tiny bit left! If you keep dividing, you'll see 3.3333... where the '3' keeps repeating. This is a non-terminating repeating decimal.

Worked Example

Step-by-Step

Let's divide 2 by 3.

1. Set up the division: 2 ÷ 3.
---2. 3 cannot go into 2, so we put a 0 and a decimal point: 0.
---3. Add a zero to 2, making it 20. Now divide 20 by 3. 3 goes into 20 six times (3 x 6 = 18).
---4. Write down 6 after the decimal point: 0.6.
---5. Subtract 18 from 20, which leaves 2.
---6. Add another zero to the remainder 2, making it 20 again.
---7. Divide 20 by 3 again. It's 6, with a remainder of 2.
---8. You can see the '6' will keep repeating forever. So, 2 ÷ 3 = 0.666... (or 0.6 with a bar over the 6).

Why It Matters

Understanding these decimals is important in fields like computer science and data science, where precision matters. Engineers use them to calculate exact measurements, and economists might use them in financial models. Knowing this helps you handle numbers that don't have perfect, clean answers.

Common Mistakes

MISTAKE: Thinking all non-terminating decimals are repeating | CORRECTION: Some decimals, like pi (3.14159...), go on forever but never repeat a pattern. These are called non-terminating non-repeating decimals.

MISTAKE: Stopping the division too early and rounding off | CORRECTION: To identify a repeating decimal, you must continue the division until you see a clear pattern of digits starting to repeat. Don't round off unless asked.

MISTAKE: Writing 0.33 instead of 0.333... for 1/3 | CORRECTION: The three dots (...) or a bar over the repeating digit(s) are important to show that the decimal goes on forever and repeats. 0.33 is an approximation, not the exact value.

Practice Questions

Try It Yourself

QUESTION: Is 0.777... a non-terminating repeating decimal? | ANSWER: Yes

QUESTION: Convert the fraction 1/9 into a decimal. Is it non-terminating repeating? | ANSWER: 0.111... (Yes, it is)

QUESTION: Which of these fractions will result in a non-terminating repeating decimal: 1/2, 3/4, or 2/11? | ANSWER: 2/11 (It is 0.181818...)

MCQ

Quick Quiz

Which of the following decimals is a non-terminating repeating decimal?

0.5

0.125

0.333...

0.101101110...

The Correct Answer Is:

Option C (0.333...) clearly shows the digit '3' repeating endlessly, making it a non-terminating repeating decimal. Options A and B terminate (end), while Option D is non-terminating but non-repeating.

Real World Connection

In the Real World

When you use a calculator on your phone or computer to divide numbers like 10 by 3, it often shows you a few decimal places and then rounds off. However, internally, these systems understand that the actual value is a repeating decimal. This precision is crucial in banking software to ensure every paisa is accounted for, or in scientific simulations where even tiny errors can lead to big problems.

Key Vocabulary

Key Terms

TERMINATING: A decimal that ends after a finite number of digits. | NON-TERMINATING: A decimal that goes on forever. | REPEATING: A pattern of digits that occurs again and again. | DECIMAL POINT: The dot separating the whole number part from the fractional part. | FRACTION: A way to represent parts of a whole, like 1/2 or 3/4.

What's Next

What to Learn Next

Great job understanding repeating decimals! Next, you can explore how to convert these repeating decimals back into fractions. This will help you see the strong connection between fractions and decimals and build your number sense even further!