What is Non-terminating Repeating Decimals? | Simple Explanation for Class 7

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What is Non-terminating Repeating Decimals as Rational Numbers?

Grade Level:

Class 7

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, but its digits after the decimal point repeat in a fixed pattern. These decimals can always be written as a fraction (p/q, where q is not zero), which means they are rational numbers.

Simple Example

Quick Example

Imagine you're trying to share 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' keeps repeating forever. This is a non-terminating repeating decimal, and since it came from a fraction (1/3), it's a rational number.

Worked Example

Step-by-Step

Let's convert the non-terminating repeating decimal 0.777... into a fraction (p/q form).

Step 1: Let x = 0.777... (Equation 1)

Step 2: Since only one digit (7) is repeating, multiply both sides of Equation 1 by 10. So, 10x = 7.777... (Equation 2)

Step 3: Subtract Equation 1 from Equation 2. (10x - x) = (7.777... - 0.777...)

Step 4: This simplifies to 9x = 7

Step 5: Divide both sides by 9 to find x. x = 7/9

Answer: So, 0.777... is equal to the rational number 7/9.

Why It Matters

Understanding these decimals helps in fields like Data Science and Computer Science where precise calculations are crucial, even with repeating numbers. Engineers use this to design systems, and economists might use it when dealing with interest rates that repeat over time. It's foundational for advanced math and practical problem-solving.

Common Mistakes

MISTAKE: Thinking that 0.1234567891011... (where digits don't repeat in a pattern) is a repeating decimal. | CORRECTION: A non-terminating decimal must have a clear, repeating block of digits to be considered 'repeating'. If there's no pattern, it's an irrational number.

MISTAKE: Confusing terminating decimals (like 0.5) with non-terminating repeating decimals. | CORRECTION: Terminating decimals end after a finite number of digits. Non-terminating repeating decimals go on forever with a repeating pattern.

MISTAKE: Forgetting that *all* non-terminating repeating decimals are rational numbers. | CORRECTION: By definition, any number that can be expressed as a fraction p/q (where q is not 0) is rational. We can always convert repeating decimals to fractions.

Practice Questions

Try It Yourself

QUESTION: Is 0.121212... a non-terminating repeating decimal? | ANSWER: Yes

QUESTION: Convert 0.444... into a rational number (fraction). | ANSWER: 4/9

QUESTION: Convert 0.232323... into a rational number. (Hint: Multiply by 100 in the second step) | ANSWER: 23/99

MCQ

Quick Quiz

Which of the following is a non-terminating repeating decimal that is also a rational number?

0.1010010001...

0.75

0.6666...

pi (3.14159...)

The Correct Answer Is:

0.6666... has a repeating digit (6) and goes on forever, making it a non-terminating repeating decimal. It can be written as 2/3, so it's rational. Options A and D are irrational, and B is a terminating decimal.

Real World Connection

In the Real World

When you see cricket statistics like a player's strike rate (runs per balls faced) or average, sometimes these numbers are rounded. But the exact calculation might result in a non-terminating repeating decimal. For example, if a player scores 10 runs off 3 balls, their runs per ball is 10/3 = 3.333... This fundamental concept helps in understanding how computers handle such numbers in sports analytics platforms.

Key Vocabulary

Key Terms

NON-TERMINATING: Does not end, continues infinitely | REPEATING DECIMAL: A decimal where a sequence of digits repeats infinitely | RATIONAL NUMBER: Any number that can be expressed as a fraction p/q, where p and q are integers and q is not zero | FRACTION: A way to represent parts of a whole, written as a numerator over a denominator

What's Next

What to Learn Next

Great job understanding repeating decimals! Next, you can explore 'Irrational Numbers'. You'll learn about decimals that are non-terminating AND non-repeating, like pi, and see how they fit into the bigger picture of real numbers.