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What are Non-terminating Repeating Decimals as Rational Numbers?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Non-terminating repeating decimals are decimal numbers that go on forever (non-terminating) but have a pattern of digits that repeats endlessly (repeating). These special decimals can always be written as a fraction, which means they are rational numbers.
Simple Example
Quick Example
Imagine you're sharing 1 laddoo equally among 3 friends. Each friend gets 1/3 of the laddoo. 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.
Worked Example
Step-by-Step
Let's convert the non-terminating repeating decimal 0.666... into a fraction (a rational number).
Step 1: Let 'x' be equal to the decimal. So, x = 0.666... (Equation 1)
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Step 2: Since only one digit (6) is repeating, multiply both sides of Equation 1 by 10.
10x = 6.666... (Equation 2)
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Step 3: Subtract Equation 1 from Equation 2.
10x - x = 6.666... - 0.666...
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Step 4: Simplify both sides.
9x = 6
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Step 5: Solve for x by dividing both sides by 9.
x = 6/9
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Step 6: Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3.
x = 2/3
Answer: So, 0.666... is equal to the rational number 2/3.
Why It Matters
Understanding these decimals is crucial for computer science and engineering, especially when dealing with precise calculations. In data science, accurate representation of numbers helps in building better AI models. Even in economics, these concepts ensure financial calculations are correct.
Common Mistakes
MISTAKE: Thinking repeating decimals are not rational numbers. | CORRECTION: Repeating decimals can always be written as a fraction (p/q, where q is not zero), so they are indeed rational numbers.
MISTAKE: Confusing non-terminating repeating decimals with non-terminating non-repeating decimals. | CORRECTION: Repeating decimals have a clear pattern (e.g., 0.121212...), while non-repeating ones do not (like pi, 3.14159...). Only repeating ones are rational.
MISTAKE: Not multiplying by the correct power of 10 when converting to a fraction. | CORRECTION: Multiply by 10 if one digit repeats, by 100 if two digits repeat, by 1000 if three digits repeat, and so on.
Practice Questions
Try It Yourself
QUESTION: Is 0.777... a non-terminating repeating decimal? | ANSWER: Yes
QUESTION: Write 0.444... as a fraction. | ANSWER: 4/9
QUESTION: Convert 0.121212... into a rational number (fraction). | ANSWER: 12/99 or 4/33
MCQ
Quick Quiz
Which of these is a non-terminating repeating decimal?
0.5
0.12345...
0.333...
0.25
The Correct Answer Is:
C
Option C, 0.333..., has the digit '3' repeating infinitely, making it a non-terminating repeating decimal. Options A and D are terminating decimals, and Option B is non-terminating but non-repeating.
Real World Connection
In the Real World
When calculating percentages for interest rates in banks or sharing profits in a small business, sometimes you get repeating decimals like 33.33% (which is 1/3). Understanding these helps ensure everyone gets their fair share and calculations are precise, just like when UPI transactions show exact amounts.
Key Vocabulary
Key Terms
TERMINATING DECIMAL: A decimal that ends after a finite number of digits (e.g., 0.5) | NON-TERMINATING DECIMAL: A decimal that goes on forever (e.g., 0.333...) | REPEATING DECIMAL: A non-terminating decimal where a digit or block of digits repeats endlessly (e.g., 0.1212...) | RATIONAL NUMBER: Any number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.
What's Next
What to Learn Next
Great job understanding repeating decimals! Next, you can explore 'Irrational Numbers'. These are decimals that are non-terminating AND non-repeating, which means they cannot be written as a simple fraction. It's an exciting step towards understanding all types of numbers!
