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What is the Decimal Representation of a Rational Number?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The decimal representation of a rational number is simply writing a rational number (a fraction) as a decimal. This means dividing the numerator by the denominator. It can result in either a decimal that ends (terminating) or a decimal that repeats a pattern (non-terminating repeating).
Simple Example
Quick Example
Imagine you scored 3 out of 5 marks in a quiz. To show this as a decimal, you divide 3 by 5. 3 ÷ 5 = 0.6. So, 0.6 is the decimal representation of the fraction 3/5.
Worked Example
Step-by-Step
Let's find the decimal representation of the rational number 7/4.
Step 1: Understand that 7/4 means 7 divided by 4.
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Step 2: Perform the division. Set up a long division problem.
4 ) 7
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Step 3: Divide 7 by 4. 4 goes into 7 one time (1 x 4 = 4).
1
4 ) 7
-4
---
3
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Step 4: Since there's a remainder (3), add a decimal point and a zero to the dividend (7 becomes 7.0), and a decimal point to the quotient.
1.
4 ) 7.0
-4
---
3 0
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Step 5: Bring down the zero. Now divide 30 by 4. 4 goes into 30 seven times (7 x 4 = 28).
1.7
4 ) 7.0
-4
---
3 0
-2 8
-----
2
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Step 6: Add another zero to the dividend (7.00) and bring it down. Divide 20 by 4. 4 goes into 20 five times (5 x 4 = 20).
1.75
4 ) 7.00
-4
---
3 0
-2 8
-----
2 0
-2 0
-----
0
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Step 7: The remainder is 0, so the division is complete.
Answer: The decimal representation of 7/4 is 1.75.
Why It Matters
Understanding decimal representations is crucial in fields like Data Science and Economics for analyzing trends and making predictions. Engineers use them to measure and design structures precisely, and even in Cryptography, decimals are foundational for complex calculations.
Common Mistakes
MISTAKE: Confusing terminating decimals with non-terminating repeating decimals. | CORRECTION: A terminating decimal ends (like 0.5), while a non-terminating repeating decimal has digits that repeat forever (like 0.333...).
MISTAKE: Incorrectly performing long division, especially when adding decimal points and zeros. | CORRECTION: Practice long division carefully. Remember to add a decimal point to the quotient as soon as you add one to the dividend.
MISTAKE: Not recognizing that all rational numbers have either a terminating or repeating decimal form. | CORRECTION: If a decimal doesn't terminate, keep dividing to see if a pattern of digits starts repeating. If it does, it's a rational number.
Practice Questions
Try It Yourself
QUESTION: What is the decimal representation of 1/2? | ANSWER: 0.5
QUESTION: Convert 5/8 into its decimal form. | ANSWER: 0.625
QUESTION: Find the decimal representation of 2/3. What kind of decimal is it? | ANSWER: 0.666... (or 0.6 with a bar over 6). It is a non-terminating repeating decimal.
MCQ
Quick Quiz
Which of the following is the decimal representation of 3/4?
0.34
0.75
0.43
0.25
The Correct Answer Is:
B
To convert 3/4 to a decimal, you divide 3 by 4. 3 ÷ 4 = 0.75. Options A, C, and D are incorrect divisions.
Real World Connection
In the Real World
When you buy vegetables at the market, the price per kilogram is often given in decimals, like ₹25.50 for potatoes. Similarly, when checking your mobile data usage, you might see 1.75 GB used. These are all decimal representations of quantities.
Key Vocabulary
Key Terms
RATIONAL NUMBER: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. | TERMINATING DECIMAL: A decimal that ends after a finite number of digits. | NON-TERMINATING REPEATING DECIMAL: A decimal that has one or more digits that repeat infinitely. | NUMERATOR: The top number in a fraction. | DENOMINATOR: The bottom number in a fraction.
What's Next
What to Learn Next
Now that you understand decimal representations, you can learn about comparing and ordering rational numbers, both in fraction and decimal form. This will help you understand how different values relate to each other.
