NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.3: Number Systems

Understanding Decimal Expansions of Real Numbers

To conquer Exercise 1.3, you must understand how real numbers behave when expressed as decimals.

1. Rational Numbers: A rational number of the form $p/q$ (where $p$ and $q$ are integers and $q \neq 0$) has a decimal expansion that is either:

• Terminating: The division ends, and the remainder becomes zero after some steps (e.g., $7/8 = 0.875$). This happens only when the prime factors of the denominator $q$ consist only of 2s and/or 5s.
• Non-terminating Recurring (Repeating): The division never ends, but a block of digits repeats periodically (e.g., $1/3 = 0.3333... = 0.\overline{3}$). The bar on top of the digits represents the repeating cycle (called the period).

1. Irrational Numbers: An irrational number has a decimal expansion that is Non-terminating Non-recurring. The digits go on forever without forming any repeating pattern (e.g., $\sqrt{2} = 1.414213...$ or $\pi = 3.141592...$).

Step-by-Step Process: Converting Repeating Decimals to p/q Form

1. Step 1: Assign a Variable — Equate the given recurring decimal to a variable, say $x$. Write down the expanded form showing the repeating digits clearly. For example, let $x = 0.\overline{6}$, which means $x = 0.6666...$ (Equation 1).
2. Step 2: Determine Periodicity — Count how many digits are repeating under the bar. If 1 digit repeats (e.g., $0.\overline{6}$), the periodicity is 1. If 2 digits repeat (e.g., $0.\overline{27}$), the periodicity is 2.
3. Step 3: Multiply by Powers of 10 — Multiply both sides of Equation 1 by $10^n$, where $n$ is the periodicity. If periodicity is 1, multiply by $10^1 = 10$. If periodicity is 2, multiply by $10^2 = 100$. This shifts the decimal point past the first repeating block. (e.g., $10x = 6.6666...$ (Equation 2)).
4. Step 4: Subtract to Eliminate Decimals — Subtract the original equation (Equation 1) from the new equation (Equation 2). This eliminates the infinite decimal part. (e.g., $10x - x = 6.6666... - 0.6666... \implies 9x = 6$).
5. Step 5: Solve for x — Isolate $x$ to get the fraction. Simplify the fraction to its lowest terms. (e.g., $x = 6/9$, which simplifies to $2/3$).

Worked Textbook Examples Explained

• Example 1: Show that $0.2353535... = 0.2\overline{35}$ can be expressed in the form $p/q$. Step 1: Let $x = 0.2353535...$ (Eq. 1) Step 2: Here, the repeating block is '35', which has 2 digits. So, we multiply both sides of Equation 1 by 100. Step 3: $100x = 23.535353...$ (Eq. 2) Step 4: Subtract Eq. 1 from Eq. 2: $100x - x = (23.535353...) - (0.235353...)$ $99x = 23.3$ Step 5: Convert the decimal on the RHS into a fraction: $99x = 233/10$ $x = 233/990$. Since 233 and 990 have no common factors, our final $p/q$ form is $233/990$.
• Example 2: Find three different irrational numbers between the rational numbers $5/7$ and $9/11$. Step 1: Convert the given rational numbers into decimal form. $5/7 = 0.\overline{714285}$ $9/11 = 0.\overline{81}$ Step 2: We need to find three numbers that lie between $0.714285...$ and $0.818181...$ which have non-terminating and non-repeating decimal expansions. Step 3: Write three such non-repeating, non-terminating patterns: Number 1: $0.73073007300073...$ Number 2: $0.75075007500075...$ * Number 3: $0.801001000100001...$

Common Exam Mistakes & Board Preparation Tips

• Mistake 1: Multiplying by the wrong power of 10. Always count only the number of repeating digits under the bar. For $0.4\overline{7}$, only one digit (7) repeats. Do not multiply by 100 first; multiply by 10.
• Mistake 2: Confusing Non-terminating with Irrational. Remind yourself that non-terminating but repeating decimals are strictly rational. Only non-terminating and non-repeating decimals are irrational.
• Board Tip: When writing non-terminating non-recurring numbers, use a clear increasing zero pattern (like $0.120120012000...$) to clearly prove to the examiner that the digits do not repeat in a fixed block cycle.

Practice Questions with Solutions

• Q: Express $0.\overline{001}$ in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$. A: Step 1: Let $x = 0.\overline{001}$, which means $x = 0.001001001...$ (Equation 1). Step 2: Since three digits (001) repeat, we multiply both sides of Equation 1 by $10^3 = 1000$. Step 3: $1000x = 1.001001...$ (Equation 2). Step 4: Subtract Equation 1 from Equation 2: $1000x - x = (1.001001...) - (0.001001...)$ $999x = 1$. Step 5: Solve for $x$: $x = 1/999$. Final answer: $1/999$
• Q: Classify the following numbers as rational or irrational: (i) $\sqrt{23}$, (ii) $\sqrt{225}$, (iii) $0.3796$, (iv) $7.478478...$. A: Step 1: Analyze (i) $\sqrt{23}$. Since 23 is not a perfect square, its square root yields a non-terminating, non-repeating decimal. Thus, it is irrational. Step 2: Analyze (ii) $\sqrt{225}$. Since $\sqrt{225} = 15$, which is a whole number, it is rational. Step 3: Analyze (iii) $0.3796$. The decimal ends after 4 decimal places (terminates). Thus, it is rational. Step 4: Analyze (iv) $7.478478...$. The block '478' repeats infinitely. Since it is non-terminating but recurring, it is rational. Final answer: (i) Irrational, (ii) Rational, (iii) Rational, (iv) Rational
• Q: Find an irrational number between $1/7$ and $2/7$. A: Step 1: Write down the decimal values of $1/7$ and $2/7$. $1/7 \approx 0.142857...$ $2/7 \approx 0.285714...$ Step 2: Select a value between $0.1428$ and $0.2857$ that has a non-terminating and non-recurring pattern. Step 3: Let's pick a pattern starting with $0.15$. We write: $0.15015001500015...$ Final answer: $0.15015001500015...$ (or any similar non-repeating pattern between $0.14$ and $0.28$)

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