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What is 0.666... (Two Thirds as Decimal)?
Grade Level:
Class 5
Maths, Computing, AI, Number Theory
Definition
What is it?
0.666... is a decimal number where the digit '6' repeats endlessly after the decimal point. It is the decimal form of the fraction two-thirds (2/3). This kind of decimal is called a repeating or recurring decimal.
Simple Example
Quick Example
Imagine you have 3 gulab jamuns and you want to share 2 of them equally among 3 friends. Each friend gets 2/3 of a gulab jamun. If you try to write 2/3 as a decimal, you get 0.666..., meaning each friend gets a little more than half, but not quite a full one.
Worked Example
Step-by-Step
Let's convert the fraction 2/3 to a decimal:
1. We need to divide the numerator (2) by the denominator (3).
2. Set up a long division: 2 ÷ 3.
3. Since 3 cannot go into 2, we add a decimal point and a zero to 2, making it 2.0. So, 2.0 ÷ 3.
4. 3 goes into 20 six times (3 x 6 = 18).
5. Write down '6' after the decimal point in the quotient. Subtract 18 from 20, which leaves 2.
6. Bring down another zero, making it 20 again.
7. 3 goes into 20 six times again, leaving 2.
8. This process will keep repeating, giving us '6' endlessly. So, 2/3 = 0.666...
ANSWER: 0.666...
Why It Matters
Understanding repeating decimals like 0.666... is crucial in Maths for accurate calculations. In computing and AI, these concepts help in representing and processing numbers efficiently. Engineers and scientists use such values in calculations for everything from satellite trajectories to designing new gadgets.
Common Mistakes
MISTAKE: Writing 0.66 or 0.67 as the exact value of 2/3 | CORRECTION: 0.66 or 0.67 are approximations. The exact decimal form of 2/3 is 0.666... with the 6 repeating infinitely, often written as 0.6 with a bar over the 6.
MISTAKE: Thinking that 0.666... is a slightly smaller number than 2/3 | CORRECTION: 0.666... is exactly equal to 2/3. It's not 'almost' 2/3; it IS 2/3, just in a different form.
MISTAKE: Confusing 0.6 with 0.666... | CORRECTION: 0.6 is six-tenths (6/10), which is 3/5. 0.666... is two-thirds (2/3). They are different fractions and different values.
Practice Questions
Try It Yourself
QUESTION: What fraction is exactly equal to 0.666...? | ANSWER: 2/3
QUESTION: If you divide a pizza into 3 equal slices and eat 2 of them, what decimal portion of the pizza did you eat? | ANSWER: 0.666...
QUESTION: A recipe calls for 2/3 cup of milk. If your measuring cup only has markings for decimals, what decimal value would you try to measure? | ANSWER: Approximately 0.67 cups, knowing that the exact value is 0.666...
MCQ
Quick Quiz
Which of these is the correct way to write 2/3 as a decimal?
0.6
0.66
0.666...
0.7
The Correct Answer Is:
C
When you divide 2 by 3, the digit 6 repeats infinitely. So, 0.666... with the '...' showing it continues is the correct representation. Options A, B, and D are approximations or incorrect values.
Real World Connection
In the Real World
Imagine you're checking cricket statistics for a player. If a player has scored 2 runs from 3 balls, their strike rate is often calculated as a decimal. While usually rounded, the exact ratio is 0.666... runs per ball. Similarly, in financial calculations or while programming, exact fractions are sometimes converted to their repeating decimal forms for precise data handling.
Key Vocabulary
Key Terms
REPEATING DECIMAL: A decimal number where one or more digits repeat infinitely after the decimal point. | NUMERATOR: The top number in a fraction, representing the number of parts being considered. | DENOMINATOR: The bottom number in a fraction, representing the total number of equal parts. | 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 how to convert other fractions into decimals, especially those that result in terminating decimals (like 1/2 = 0.5) and other types of repeating decimals, such as 0.333... or 0.142857142857.... This will strengthen your understanding of number systems.
