NCERT Solutions for Class 9 Maths Exercise 1.3 Question 4
Understanding the Question 🧐
This question asks us to convert the repeating decimal &&0.99999…&& into the fractional form &&\frac{p}{q}&&. The second part of the question encourages us to think deeply about the answer we get, as it might seem surprising at first. Let’s tackle both parts.
Express &&0.99999…&& in the form &&\frac{p}{q}&&. Are you surprised by your answer? With your teacher and classmates, discuss why the answer makes sense.
Part 1: Converting &&0.999…&& to &&\frac{p}{q}&& Form 📝
We will use the same algebraic method as in the previous question.
- Let the number be &&x&&.
&&x = 0.999…&& —(1)
- Multiply to shift the decimal.
Since one digit (‘9’) is repeating, we multiply both sides by 10.
&&10x = 9.999…&& —(2)
- Subtract the equations.
Now, we subtract equation (1) from equation (2) to eliminate the infinite repeating part.
&&10x – x = (9.999…) – (0.999…)||
&&9x = 9&&
- Solve for &&x&&.
&&x = \frac{9}{9} = 1&&
Answer: The &&\frac{p}{q}&& form of &&0.999…&& is &&\frac{1}{1}&&, which is 1.
Part 2: Discussing the “Surprising” Answer 🤔
Yes, the answer is surprising! It feels strange that &&0.999…&&, a number that seems to be “just less than 1”, is actually exactly equal to 1. Let’s discuss why this makes perfect sense.
Why does &&0.999… = 1&&?
In mathematics, &&0.999…&& is not an approximation of 1; it is another way of writing 1. Here are a couple of ways to understand this:
1. The “Gap” is Zero
If two numbers are different, you should be able to find a number between them. Can you name any number that lies between &&0.999…&& and &&1&&? No, you can’t. The difference between them is &&1 – 0.999… = 0.000…&&, which is exactly zero. Since there is no gap between them, they must be the same number.
2. The Fraction Proof (&&1/3&&)
This is a very convincing proof. We all agree that:
&&\frac{1}{3} = 0.333…&&
Now, let’s multiply both sides of this equation by 3:
&&3 \times \frac{1}{3} = 3 \times 0.333…&&
Calculating both sides gives us:
&&1 = 0.999…&&
Since both sides were equal before we multiplied, they must be equal after. This confirms that &&1&& and &&0.999…&& are just two different ways of representing the same value.
Conclusion and Key Points ✅
The conversion of &&0.999…&& using the standard algebraic method gives the result &&1&&. While this may seem counter-intuitive, it is a correct and fundamental concept in mathematics. These ncert solutions show that some numbers can have more than one decimal representation.
- &&0.999…&& is a valid decimal representation of the integer &&1&&.
- The difference between &&1&& and &&0.999…&& is mathematically zero.
- This shows that some rational numbers can have two different decimal forms (one terminating and one repeating). For example, &&0.5 = 0.4999…&&.
FAQ
Q: What is the fractional form of &&0.999…&&?
A: The fractional form of &&0.999…&& is &&1&&, which can be written as &&\frac{1}{1}&&.
Q: Is &&0.999…&& actually equal to 1, or just very close?
A: Mathematically, &&0.999…&& is exactly equal to &&1&&. They are two different representations of the same number, and there is no gap or difference between them.
Q: How can you prove that &&0.999… = 1&& using fractions?
A: We know that the fraction &&\frac{1}{3} = 0.333…&&. If you multiply both sides of this equation by 3, you get &&3 \times \frac{1}{3} = 1&& on the left side, and &&3 \times 0.333… = 0.999…&& on the right side. Therefore, &&1 = 0.999…&&.
Q: Why does the result &&0.999… = 1&& seem surprising?
A: It seems surprising because our intuition tells us that numbers with different digits should have different values. However, in the case of infinite decimals, this intuition can be misleading. The infinite trail of 9s completely closes the gap to 1, leaving no space between them.
Q: Do other numbers have two different decimal representations?
A: Yes, any number that has a terminating decimal also has an infinite repeating decimal form. For example, &&0.5&& is also exactly equal to &&0.4999…&&.
Further Reading
The idea that a single number can have multiple representations is a fascinating topic. To explore more, you can refer to the official NCERT textbooks and discuss with your teachers.
