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Q7:
Write three numbers whose decimal expansions are non-terminating non-recurring.
Solution :
Theoretical Foundation: Decimal Expansions of Real Numbers
In the real number system, numbers are broadly classified into two categories based on their decimal expansions: rational numbers and irrational numbers. [Per the fundamental classification of Real Numbers], a rational number's decimal expansion either terminates (ends) or is non-terminating but recurring (repeats a specific block of digits infinitely). Conversely, an irrational number cannot be expressed as a simple fraction $\frac{p}{q}$ (where $p$ and $q$ are integers and $q \neq 0$). Consequently, the defining characteristic of an irrational number is that its decimal expansion is strictly non-terminating and non-recurring.
To satisfy the prompt, we must provide three distinct examples of irrational numbers. We will derive these using three different mathematical approaches: pattern construction, algebraic roots, and transcendental constants.
Step 1: Constructing a Pattern-Based Irrational Number
The most direct way to generate a non-terminating, non-recurring decimal is to construct a sequence of digits that follows a predictable, yet non-repeating pattern. We can achieve this by systematically increasing the number of zeros between a repeating non-zero digit.
- Let the first decimal place be $0$.
- Follow it with a $1$, then one $0$ ($0.01\dots$).
- Follow it with a $1$, then two $0$s ($0.01001\dots$).
- Follow it with a $1$, then three $0$s ($0.010010001\dots$).
Number 1: $0.01001000100001\dots$
[Justification: Because the block of zeros expands by one with each iteration, there is no fixed finite block of digits that will ever repeat infinitely. Thus, it is non-recurring. The ellipsis ($\dots$) indicates it is non-terminating.]
Step 2: Utilizing Square Roots of Non-Perfect Squares
In algebra, the square root of any positive integer that is not a perfect square (such as $2, 3, 5, 6, 7$) is inherently an irrational number. [Per the theorem of irrationality of roots: If $p$ is a prime number, then $\sqrt{p}$ is irrational].
Let us take the square root of the first prime number, $2$.
Number 2: $\sqrt{2} = 1.41421356237309504\dots$
[Justification: The decimal expansion of $\sqrt{2}$ has been mathematically proven to extend infinitely without any repeating sequence of digits.]
Step 3: Referencing Transcendental Mathematical Constants
Certain fundamental constants in mathematics cannot be expressed as the root of any non-zero polynomial equation with rational coefficients. These are known as transcendental numbers, which are a subset of irrational numbers. The most famous example is Pi ($\pi$), which represents the ratio of a circle's circumference to its diameter.
Number 3: $\pi = 3.141592653589793238\dots$
[Justification: Johann Lambert proved in 1761 that $\pi$ is irrational. Therefore, its decimal representation goes on forever without settling into a permanently repeating pattern.]
Final Solution: Three distinct numbers whose decimal expansions are non-terminating and non-recurring are:
1) $0.01001000100001\dots$ (A constructed non-repeating pattern)
2) $\sqrt{2} = 1.41421356\dots$ (The square root of a non-perfect square)
3) $\pi = 3.14159265\dots$ (A transcendental mathematical constant)
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.3
- Q1(i): Write the following in decimal form and say what kind of decimal expansion each has : (i) $\frac{36}{100}$
- Q1(ii): Write the following in decimal form and say what kind of decimal expansion each has : (ii) $\frac{1}{11}$
- Q1(iii): Write the following in decimal form and say what kind of decimal expansion each has : (iii) $4\frac{1}{8}$
- Q1(iv): Write the following in decimal form and say what kind of decimal expansion each has : (iv) $\frac{3}{13}$
- Q1(v): Write the following in decimal form and say what kind of decimal expansion each has : (v) $\frac{2}{11}$
- Q1(vi): Write the following in decimal form and say what kind of decimal expansion each has : (vi) $\frac{329}{400}$
- Q2: You know that $\frac{1}{7} = 0.\overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of $\frac{1}{7}$ carefully.]
- Q3(i): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (i) $0.\overline{6}$
- Q3(ii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (ii) $0.4\overline{7}$
- Q3(iii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (iii) $0.\overline{001}$
- Q4: 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.
- Q5: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$? Perform the division to check your answer.
- Q6: Look at several examples of rational numbers in the form $\frac{p}{q}$ ($q \neq 0$), where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy?
- Q8: Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
- Q9(i): Classify the following numbers as rational or irrational : (i) $\sqrt{23}$
- Q9(ii): Classify the following numbers as rational or irrational : (ii) $\sqrt{225}$
- Q9(iii): Classify the following numbers as rational or irrational : (iii) $0.3796$
- Q9(iv): Classify the following numbers as rational or irrational : (iv) $7.478478...$
- Q9(v): Classify the following numbers as rational or irrational : (v) $1.101001000100001...$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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