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What are Terminating Decimals as Rational Numbers?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Terminating decimals are decimal numbers that have a finite number of digits after the decimal point, meaning they stop. When these decimals can be written as a fraction p/q (where p and q are whole numbers and q is not zero), they are called rational numbers.
Simple Example
Quick Example
Imagine buying a packet of biscuits for Rs. 15.50. This is a terminating decimal because the digits after the decimal point (5 and 0) stop. We can write Rs. 15.50 as the fraction 1550/100, which is a rational number.
Worked Example
Step-by-Step
Let's convert the terminating decimal 0.75 into a rational number (a fraction).
Step 1: Write the decimal as a fraction by placing the decimal number over 1.
0.75 / 1
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Step 2: Count the number of digits after the decimal point. Here, there are two digits (7 and 5).
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Step 3: Multiply both the numerator and the denominator by 10 for each digit after the decimal point. Since there are two digits, we multiply by 10 x 10 = 100.
(0.75 x 100) / (1 x 100) = 75 / 100
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Step 4: Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor. Both 75 and 100 can be divided by 25.
75 / 25 = 3
100 / 25 = 4
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Answer: So, 0.75 as a rational number is 3/4.
Why It Matters
Understanding terminating decimals helps in fields like Computer Science for accurate calculations in programming, or in Economics for dealing with money and financial data. Engineers use them to measure and design precisely, ensuring structures are safe and efficient.
Common Mistakes
MISTAKE: Thinking that 1/3 (0.333...) is a terminating decimal. | CORRECTION: Terminating decimals must have a fixed, finite number of digits after the decimal point. 1/3 is a repeating decimal, not terminating.
MISTAKE: Forgetting to simplify the fraction after converting a decimal. | CORRECTION: Always simplify the fraction to its lowest terms. For example, 50/100 should be simplified to 1/2.
MISTAKE: Confusing the number of zeros needed in the denominator. | CORRECTION: The number of zeros in the denominator (10, 100, 1000, etc.) must match the number of digits after the decimal point. For 0.123, use 1000 (3 digits, 3 zeros).
Practice Questions
Try It Yourself
QUESTION: Convert 0.2 into a rational number (fraction). | ANSWER: 1/5
QUESTION: Is 1.25 a terminating decimal? If yes, write it as a rational number. | ANSWER: Yes, it is a terminating decimal. 5/4
QUESTION: A tailor used 3.75 meters of cloth for a kurta. Express this length as a rational number in its simplest form. | ANSWER: 15/4 meters
MCQ
Quick Quiz
Which of the following is a terminating decimal that can be written as a rational number?
1/3
sqrt(2)
0.125
pi
The Correct Answer Is:
C
0.125 is a terminating decimal because it stops after three digits, and it can be written as 125/1000, which simplifies to 1/8. The other options are either repeating decimals or irrational numbers.
Real World Connection
In the Real World
When you check your mobile data usage, it might show 1.5 GB remaining. This is a terminating decimal. In UPI transactions, amounts like Rs. 500.75 are terminating decimals, ensuring precise money transfers without any leftover bits.
Key Vocabulary
Key Terms
TERMINATING DECIMAL: A decimal number that ends or has a finite number of digits after the decimal point. | RATIONAL NUMBER: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. | NUMERATOR: The top number in a fraction, representing the part. | DENOMINATOR: The bottom number in a fraction, representing the total parts.
What's Next
What to Learn Next
Great job understanding terminating decimals! Next, you can explore 'Non-Terminating Repeating Decimals as Rational Numbers'. This will show you how decimals that go on forever but have a repeating pattern can also be written as fractions!
