Complete Solutions and Summary of Number Systems – NCERT Class 9, Mathematics, Chapter 1 – Summary, Questions, Answers, Extra Questions
Detailed summary and explanation of Chapter 1 ‘Number Systems’ with all question answers, extra questions, and solutions from NCERT Class IX, Mathematics.
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Tags: Number Systems, Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers, Decimal Expansions, Laws of Exponents, Rationalisation, Class 9, NCERT, Mathematics, Chapter 1, Answers, Extra Questions
Number Systems
Chapter 1: Mathematics - Complete Study Guide
Chapter Overview
N
Natural
Z
Integers
Q
Rational
R
Real
What You'll Learn
Number Types
Natural, whole, integers, rationals, irrationals.
Irrational Numbers
Non-rational like \(\sqrt{2}\), \(\pi\).
Decimal Expansions
Terminating vs recurring.
Operations
On real numbers, exponents.
Key Highlights
Number systems include natural, whole, integers, rationals (terminating/recurring decimals), irrationals (non-terminating non-recurring). Real numbers combine both. Operations, square roots, rationalization, exponents for rationals.
Comprehensive Chapter Summary
1. Introduction to Number Systems
- The number line represents various types of numbers, starting from zero and extending in the positive direction with natural numbers.
- Natural numbers (N): Infinite list beginning with 1, 2, 3, ..., denoted by N.
- Whole numbers (W): Include 0 along with natural numbers, denoted by W.
- Integers (Z): Include positive and negative numbers along with zero, denoted by Z from the German word "zahlen" meaning "to count".
- Rational numbers (Q): Numbers of the form \(\frac{p}{q}\) where p and q are integers, q ≠ 0, include naturals, wholes, integers.
- Equivalent rational numbers: Different fractions representing the same value, like \(\frac{1}{2} = \frac{2}{4}\).
- Co-prime p and q: No common factors other than 1 for unique representation on the number line.
- Infinite rational numbers between any two rationals, e.g., five between 1 and 2: \(\frac{7}{6}, \frac{4}{3}, \frac{3}{2}, \frac{5}{3}, \frac{11}{6}\).
- Statements on number types: Every whole not natural (false, zero), every integer rational (true), every rational integer (false).
- Finding rationals: Midpoint method or equivalent fractions with higher denominator.
- Gaps on number line: Infinitely many numbers left, leading to irrationals.
- Questions on remaining numbers: What are they called? How to distinguish from rationals?
- Number line visualization: Collecting numbers in a bag from natural to rational.
- Example problems: Verifying statements, finding rationals between numbers.
- Remark on infinite rationals: General property between any two rationals.
- Transition to irrationals: Numbers not picked are irrationals.
Example: Rational Numbers
Find rationals between 1 and 2: Methods include midpoint or equivalent fractions. Infinitely many exist.
2. Irrational Numbers
- Irrational definition: Cannot be written as \(\frac{p}{q}\), q ≠ 0, integers p,q.
- History: Pythagoreans discovered irrationals like \(\sqrt{2}\) around 400 BC, myths about Hippacus.
- Examples: \(\sqrt{2}, \sqrt{3}, \sqrt{15}, \pi, 0.101101...\)
- Positive square root: \(\sqrt{}\) denotes positive root, e.g., \(\sqrt{4}=2\).
- Proofs: \(\sqrt{2}\) by Pythagoreans, others by Theodorus in 425 BC.
- \(\pi\): Irrational proved in 1700s by Lambert, Legendre.
- Real numbers (R): Union of rationals and irrationals, every point on line is real.
- Cantor-Dedekind: Bijection between reals and number line points.
- Locating irrationals: Geometric construction using Pythagoras for \(\sqrt{2}, \sqrt{3}\).
- Extension: Locate \(\sqrt{n}\) for any positive integer n.
- Statements: Every irrational real (true), every point \(\sqrt{m}\) m natural (false), every real irrational (false).
- Square roots: Not all irrational, e.g., \(\sqrt{4}=2\) rational.
- Infinite irrationals: Like rationals, infinitely many.
- Examples of irrationals: Square roots, \(\pi\), non-repeating decimals.
- Square root spiral: Classroom activity to construct depicting \(\sqrt{2}, \sqrt{3}, ...\)
- Transition to decimal expansions: Study to distinguish rational/irrational.
Real Numbers Overview
All points on number line are reals: rationals + irrationals. Infinite in both.
Irrational Discovery
Pythagoreans, Theodorus proved many square roots irrational.
Locating Irrationals
Use compass, arcs for \(\sqrt{2}\), extend to higher.
3. Real Numbers and their Decimal Expansions
- Rational decimals: Terminating (remainder 0) or non-terminating recurring (remainders repeat).
- Examples: \(\frac{10}{3}=3.333...\), \(\frac{7}{8}=0.875\), \(\frac{1}{7}=0.142857...\)
- Patterns: Repeating length < divisor, bar notation for recurring blocks.
- Terminating: Finite steps, e.g., 0.5, 2.556.
- Recurring: Infinite repeat, e.g., 3.3, 0.142857.
- Converse: Terminating or recurring decimals are rational.
- Convert to fraction: For terminating, direct; recurring, algebraic method with x=decimal, multiply, subtract.
- Examples: 0.333...=\(\frac{1}{3}\), 1.27=\(\frac{14}{11}\), 0.235=\(\frac{233}{990}\).
- Irrational decimals: Non-terminating non-recurring.
- Examples: \(\sqrt{2}=1.414...\), \(\pi=3.141...\)
- History: Approximations in Sulbasutras, Archimedes, Aryabhatta; modern trillions digits.
- Generate irrationals: Between rationals, non-repeating like 0.1501500...
- Decimal types: Write and classify expansions.
- Predict expansions: For fractions like k/7, cyclic shifts.
- Max repeating digits: < divisor, e.g., 16 for 1/17.
- Terminating condition: Denominator 2^m 5^n after simplification.
- Non-terminating non-recurring: Three examples.
- Irrationals between rationals: Three different.
- Classify numbers: Rational or irrational based on form.
- Remark on \(\pi\): Approximation 22/7 rational, exact irrational.
Decimal Expansions
Rationals: Terminate or recur. Irrationals: Non-terminate non-recur.
Finding Irrationals
Between \(\frac{1}{7}\), \(\frac{2}{7}\): 0.1501500...
4. Operations on Real Numbers
- Rationals: Closed under +, -, ×, ÷ (≠0), commutative, associative, distributive.
- Irrationals: Satisfy commutative, associative, distributive for + and ×.
- Operations with irrationals: Sum/product with rational is irrational (non-zero rational).
- Two irrationals: Sum, difference, product, quotient may be rational or irrational.
- Examples: Rational + irrational = irrational, like 2 + \(\sqrt{3}\).
- Check irrationals: 7\(\sqrt{5}\), etc., by decimal non-terminating non-recurring.
- Simplify expressions: Add, multiply, divide irrationals, sometimes rational result.
- Square roots: \(\sqrt{a} = b > 0\), b² = a, extend to positives.
- Geometric location: For \(\sqrt{x}\), construct using semicircle, Pythagoras.
- Position on line: Arc intersection for \(\sqrt{x}\).
- nth roots: \(\sqrt[n]{a} = b > 0\), b^n = a.
- Identities: \(\sqrt{ab}=\sqrt{a}\sqrt{b}\), rationalize using conjugates.
- Simplify using identities: Expand, difference of squares.
- Rationalize denominator: Multiply conjugate, e.g., \(\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\).
- Examples: Rationalize with binomials like 2 + \(\sqrt{3}\).
- Remark: Simplify to rational + irrational form.
Rationalization
Denominators like \(\sqrt{2}\): Multiply conjugate.
5. Laws of Exponents for Real Numbers
- Basic laws: a^m a^n = a^{m+n}, (a^m)^n = a^{mn}, a^m / a^n = a^{m-n}, (ab)^m = a^m b^m.
- Zero exponent: a^0 = 1.
- Negative exponents: a^{-n} = 1/a^n.
- Examples: Simplify with negatives.
- Rational exponents: a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}.
- Definition: For a > 0, \sqrt[n]{a} = b > 0, b^n = a.
- Extended laws: For rational p,q, same as integer.
- Simplify: With fractional exponents.
- Equivalent forms: (a^{1/n})^m or (a^m)^{1/n}.
- Co-prime m,n: No common factors >1.
- Examples: 4^{3/2} = 8.
- Find roots: 64^{1/2}=8, 32^{1/5}=2, etc.
- Simplify expressions: Products, powers with fractions.
- Base positive real: For rational exponents.
- Extension: To real exponents later.
- Summary points: Laws hold for rationals.
Key Concepts and Definitions
Rational Number
\(\frac{p}{q}\), q ≠ 0, integers p,q.
Irrational
Not \(\frac{p}{q}\), e.g., \(\sqrt{2}\).
Real Number
Rational or irrational.
Terminating Decimal
Ends, e.g., 0.875.
Recurring
Repeats, e.g., 0.333...
Rationalize
Make denominator rational.
Exponent Laws
For rationals: a^{p+q}, etc.
Important Facts
\(\sqrt{2}\)
Irrational
\(\pi\)
3.14159...
a^{1/n}
nth Root
R
All Reals
Q
Rationals
Questions and Answers from Chapter
Short Questions (1 Mark)
Q1. Is zero a rational number? Can you write it in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0?
Answer: Yes, \(\frac{0}{1}\).
Q2. State whether the following statements are true or false. (i) Every natural number is a whole number.
Answer: True.
Q3. State whether the following statements are true or false. (ii) Every integer is a whole number.
Answer: False.
Q4. State whether the following statements are true or false. (iii) Every rational number is a whole number.
Answer: False.
Q5. State whether the following statements are true or false. (i) Every irrational number is a real number.
Answer: True.
Q6. State whether the following statements are true or false. (ii) Every point on the number line is of the form \(\sqrt{m}\), where m is a natural number.
Answer: False.
Q7. State whether the following statements are true or false. (iii) Every real number is an irrational number.
Answer: False.
Q8. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Answer: No, \(\sqrt{4}=2\).
Q9. Write the following in decimal form and say what kind of decimal expansion each has : (i) \(\frac{36}{100}\)
Answer: 0.36, terminating.
Q10. Write the following in decimal form and say what kind of decimal expansion each has : (ii) \(\frac{1}{11}\)
Answer: 0.0909..., recurring.
Q11. Write the following in decimal form and say what kind of decimal expansion each has : (iii) \(4 \frac{1}{8}\)
Answer: 4.125, terminating.
Q12. Write the following in decimal form and say what kind of decimal expansion each has : (iv) \(\frac{3}{13}\)
Answer: 0.230769..., recurring.
Q13. Write the following in decimal form and say what kind of decimal expansion each has : (v) \(\frac{2}{11}\)
Answer: 0.1818..., recurring.
Q14. Write the following in decimal form and say what kind of decimal expansion each has : (vi) \(\frac{329}{400}\)
Answer: 0.8225, terminating.
Q15. Classify the following numbers as rational or irrational : (i) \(\sqrt{23}\)
Answer: Irrational.
Q16. Classify the following numbers as rational or irrational : (ii) \(\sqrt{225}\)
Answer: Rational.
Q17. Classify the following numbers as rational or irrational : (iii) 0.3796
Answer: Rational.
Q18. Classify the following numbers as rational or irrational : (iv) 7.478478...
Answer: Rational.
Q19. Classify the following numbers as rational or irrational : (v) 1.101001000100001...
Answer: Irrational.
Q20. Classify the following numbers as rational or irrational : (i) \(2 - \sqrt{5}\)
Answer: Irrational.
Medium Questions (3 Marks)
Q1. Find six rational numbers between 3 and 4.
Answer: Use equivalent: 3 = \(\frac{21}{7}\), 4 = \(\frac{28}{7}\). Six: \(\frac{22}{7}\), \(\frac{23}{7}\), \(\frac{24}{7}\), \(\frac{25}{7}\), \(\frac{26}{7}\), \(\frac{27}{7}\).
Q2. Find five rational numbers between \(\frac{3}{5}\) and \(\frac{4}{5}\).
Answer: \(\frac{3}{5} = \frac{18}{30}\), \(\frac{4}{5} = \frac{24}{30}\). Five: \(\frac{19}{30}\), \(\frac{20}{30}\), \(\frac{21}{30}\), \(\frac{22}{30}\), \(\frac{23}{30}\).
Q3. Show how \(\sqrt{5}\) can be represented on the number line.
Answer: Construct square 2x2 +1, diagonal \(\sqrt{5}\). Compass arc.
Q4. You know that \(\frac{1}{7} = 0.142857\). 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?
Answer: Cyclic shifts: 0.285714, 0.428571, etc.
Q5. Express the following in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0. (i) 0.6
Answer: Let x=0.666..., 10x=6.666..., 9x=6, x=\(\frac{2}{3}\).
Q6. Express the following in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0. (ii) 0.47
Answer: Let x=0.4777..., 10x=4.777..., 90x=43, x=\(\frac{43}{90}\).
Q7. Express the following in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0. (iii) 0.001
Answer: Let x=0.001001..., 1000x=1.001..., 999x=1, x=\(\frac{1}{999}\).
Q8. Express 0.99999 .... in the form \(\frac{p}{q}\). Are you surprised by your answer?
Answer: x=0.999..., 10x=9.999..., 9x=9, x=1.
Q9. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \(\frac{1}{17}\)?
Answer: 16, as less than divisor.
Q10. Look at several examples of rational numbers in the form \(\frac{p}{q}\) (q ≠ 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?
Answer: q = 2^m 5^n.
Q11. Write three numbers whose decimal expansions are non-terminating non-recurring.
Answer: 0.101001..., 0.202002..., 0.303003...
Q12. Find three different irrational numbers between the rational numbers \(\frac{5}{7}\) and \(\frac{9}{11}\).
Answer: 0.720720072..., 0.730730073..., 0.740740074...
Q13. Classify the following numbers as rational or irrational : (i) \((\3 + \sqrt{23}) - \sqrt{23}\)
Answer: Rational (3).
Q14. Classify the following numbers as rational or irrational : (ii) \(\frac{2 \sqrt{7}}{7 \sqrt{7}}\)
Answer: Rational (2/7).
Q15. Classify the following numbers as rational or irrational : (iii) \(\frac{1}{\sqrt{2}}\)
Answer: Irrational.
Q16. Classify the following numbers as rational or irrational : (iv) 2\(\pi\)
Answer: Irrational.
Q17. Simplify each of the following expressions: (i) \((\sqrt{3} + \sqrt{3})(\sqrt{2} + \sqrt{2})\)
Answer: 2\(\sqrt{6}\) + 2\(\sqrt{6}\) = 4\(\sqrt{6}\).
Q18. Simplify each of the following expressions: (ii) \((\sqrt{3} + \sqrt{3})(\sqrt{3} - \sqrt{3})\)
Answer: 3 - 3 = 0.
Q19. Simplify each of the following expressions: (iii) \((\sqrt{5} + \sqrt{2})^2\)
Answer: 5 + 2\(\sqrt{10}\) + 2 = 7 + 2\(\sqrt{10}\).
Q20. Simplify each of the following expressions: (iv) \((\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})\)
Answer: 5 - 2 = 3.
Long Questions (6 Marks)
Q1. Recall, \(\pi\) is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, \(\pi = \frac{c}{d}\). This seems to contradict the fact that \(\pi\) is irrational. How will you resolve this contradiction?
Answer: Ratio of two irrationals can be irrational. c and d are both irrational but ratio \(\pi\) irrational. Approximation like 22/7 is rational but exact \(\pi\) irrational. No contradiction as exact c/d irrational.
Q2. Represent \(\sqrt{9.3}\) on the number line.
Answer: Mark 9.3 on line as AB. Add 1 to BC. Midpoint O of AC. Semicircle radius OC. Perpendicular from B intersects at D = \(\sqrt{9.3}\). Geometric proof using Pythagoras.
Q3. Rationalise the denominators of the following: (i) \(\frac{1}{\sqrt{7}}\)
Answer: \(\frac{\sqrt{7}}{7}\). Multiply numerator and denominator by \(\sqrt{7}\).
Q4. Rationalise the denominators of the following: (ii) \(\frac{1}{\sqrt{7} - \sqrt{6}}\)
Answer: \(\sqrt{7} + \sqrt{6}\). Conjugate multiply, denominator 1.
Q5. Rationalise the denominators of the following: (iii) \(\frac{1}{\sqrt{5} + \sqrt{2}}\)
Answer: \(\frac{\sqrt{5} - \sqrt{2}}{3}\). Denominator 5-2=3.
Q6. Rationalise the denominators of the following: (iv) \(\frac{1}{\sqrt{7} - 2}\)
Answer: \(\frac{\sqrt{7} + 2}{3}\). Denominator 7-4=3.
Q7. Find : (i) \(64^{\frac{1}{2}}\)
Answer: 8. Square root of 64.
Q8. Find : (ii) \(32^{\frac{1}{5}}\)
Answer: 2. Fifth root.
Q9. Find : (iii) \(125^{\frac{1}{3}}\)
Answer: 5. Cube root.
Q10. Find : (i) \(9^{\frac{3}{2}}\)
Answer: 27. (9^{1/2})^3 = 3^3.
Q11. Find : (ii) \(32^{\frac{2}{5}}\)
Answer: 4. (32^{1/5})^2 = 2^2.
Q12. Find : (iii) \(16^{\frac{3}{4}}\)
Answer: 8. (16^{1/4})^3 = 2^3.
Q13. Find : (iv) \(125^{-\frac{1}{3}}\)
Answer: 1/5. Reciprocal cube root.
Q14. Simplify : (i) \(2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}\)
Answer: \(2^{\frac{11}{15}}\). Add exponents.
Q15. Simplify : (ii) \(( \frac{1}{3^3} )^7\)
Answer: \(3^{-21}\). Power multiply.
Q16. Simplify : (iii) \(\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}\)
Answer: \(11^{\frac{1}{4}}\). Subtract exponents.
Q17. Simplify : (iv) \(7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}\)
Answer: \((56)^{\frac{1}{2}}\). Product under root.
Q18. State whether the following statements are true or false. Give reasons for your answers. (i) Every whole number is a natural number.
Answer: False, 0 not natural. Naturals start from 1, wholes include 0.
Q19. State whether the following statements are true or false. Give reasons for your answers. (ii) Every integer is a rational number.
Answer: True, m = \(\frac{m}{1}\).
Q20. State whether the following statements are true or false. Give reasons for your answers. (iii) Every rational number is an integer.
Answer: False, \(\frac{3}{5}\) not integer.
Interactive Knowledge Quiz
Test your understanding of Number Systems
Quick Revision Notes
Number Types
- N: 1,2,3...
- W: 0+N
- Z: ±N+0
- Q: p/q
- Irrational: Non p/q
- R: Q + Irrational
Decimals
- Terminating: Rational
- Recurring: Rational
- Non-term non-rec: Irrational
Operations
- Rat + Irr = Irr
- Rationalize denom
- Exponents: a^{m/n}
Exam Strategy Tips
- Prove irrationality
- Rationalize
- Simplify exponents
- Decimal to fraction
- Locate on line
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