Convert \(0.333...\) into fraction:
Let \(x = 0.333...\)
\[ 10x = 3.333... \]
Subtract:
\[ \begin{aligned} 10x - x &= 3.333... - 0.333...\\ 9x &= 3 \\\Rightarrow x &= \frac{1}{3} \end{aligned} \]
Chapter 1 · CBSE · Class IX
📘 Definition
💡 Concept
Core Concept
🗂️ Types / Category
Classification of Numbers
Natural Numbers
Counting numbers used to count objects: \(\{1, 2, 3, \dots\}\). Example: \(7\), \(15\).
Whole Numbers
Natural numbers along with zero: \(\{0, 1, 2, 3, \dots\}\). Example: \(0\), \(9\).
Integers
All whole numbers and their negative counterparts: \(\{\dots, -2, -1, 0, 1, 2, \dots\}\). Example: \(-5\), \(0\), \(12\).
Rational Numbers
Numbers that can be written in the form \(\frac{p}{q}\), where \(p, q \in \mathbb{Z}\) and \(q \neq 0\). Examples: \(\frac{3}{4}\), \(-2 = \frac{-2}{1}\), \(0.5 = \frac{1}{2}\).
Irrational Numbers
Numbers that cannot be written as a fraction of two integers; their decimals are non-terminating and non-repeating. Examples: \(\sqrt{2}\), \(\pi\), \(\sqrt{5}\).
Real Numbers
All numbers on the number line, including rational and irrational numbers. Examples: \(-3\), \(\frac{7}{8}\), \(\sqrt{2}\), \(\pi\).
🖼️ Figure
Hierarchical Structure of Number System
🎨 SVG Diagram
Conceptual Inclusion Diagram
🏷️ Properties
Important Properties
Important Properties
Closure property holds for \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}\) under addition and multiplication.
Division is not closed in integers.
Irrational + Rational = Irrational (generally)
Product of two irrational numbers can be rational (e.g., \(\sqrt{2} \times \sqrt{2} = 2\))
💡 Concept
Decimal Expansion Concept
📐 Derivation
Derivation: Rational Form from Recurring Decimal
✏️ Example
Classify \(-5\), \(0\), \(\frac{3}{4}\), \(\sqrt{5}\)
Number classification hierarchy
Identify type → Check rational form → Place in correct set
- \(-5 \in \mathbb{Z}\)
- \(0 \in \mathbb{W}\)
- \(\frac{3}{4} \in \mathbb{Q}\)
- \(\sqrt{5}\) is irrational
✏️ Example
Prove that \(\sqrt{2}\) is irrational.
Proof by contradiction
Assume \(\sqrt{2} = \frac{p}{q}\), where \(p, q\) are coprime integers.
\[ \begin{aligned} 2 &= \frac{p^2}{q^2} \\\Rightarrow p^2 &= 2q^2 \end{aligned} \]
This implies \(p\) is even → let \(p = 2k\)
Substitute: \[ \begin{aligned} 4k^2 &= 2q^2 \\\Rightarrow q^2 &= 2k^2 \end{aligned} \]
So \(q\) is also even → contradiction (since both cannot be even).
Hence, \(\sqrt{2}\) is irrational.
⚡ Exam Tip
❌ Common Mistakes
- Confusing whole numbers with natural numbers
- Assuming all non-terminating decimals are irrational
- Forgetting that negative numbers are integers but not natural numbers
- Ignoring condition \(q \neq 0\) in rational numbers
📋 Case Study
A student claims that every number on a number line can be written in the form \(\frac{p}{q}\). Analyze and justify.
Insight:
The number line contains irrational numbers as well, which cannot be expressed as \(\frac{p}{q}\). Hence, the statement is incorrect.
🌟 Importance
📘 Definition
🔣 Symbol And Notation
- The set of natural numbers is denoted by \(\mathbb{N}\).
- It is an infinite set since counting never ends.
💡 Concept
Core COncept
🗒️ Successor And Predecessor
- Successor: The next number obtained by adding 1.
\[\text{Successor of } n = n + 1\]
- Predecessor: The previous number obtained by subtracting 1.
\[\text{Predecessor of } n = n - 1\]
Important Note: The number 1 has no predecessor in \(\mathbb{N}\).
🏷️ Properties
Fundamental Properties
Properties
Closure Property
- \(a + b \in \mathbb{N}\)
- \(a \times b \in \mathbb{N}\)
Not Closed Under
- Subtraction (e.g., \(2 - 5 = -3 \notin \mathbb{N}\))
- Division (e.g., \(1 / 2 = 0.5 \notin \mathbb{N}\))
Commutative
\[
\begin{aligned}
a + b &= b + a, \\ a \times b &= b \times a
\end{aligned}
\]
Associative
\[
(a + b) + c = a + (b + c)
\]
🎨 SVG Diagram
Representation on Number Line
Natural numbers are represented as equally spaced points on the number line, starting from 1 and extending towards infinity.
✏️ Example
Identify whether 7, 0, -3 are natural numbers.
Natural numbers start from 1
- 7 ∈ \(\mathbb{N}\)
- 0 ∉ \(\mathbb{N}\)
- -3 ∉ \(\mathbb{N}\)
✏️ Example
Find the successor and predecessor of 25.
Apply formulas
- Successor = \(25 + 1 = 26\)
- Predecessor = \(25 - 1 = 24\)
🔢 Formula
📐 Derivation
Conceptual Insight (Peano Axioms - Simplified)
Natural numbers are formally defined using axioms:
- 1 is a natural number.
- Every natural number has a successor.
- No two natural numbers have the same successor.
- 1 is not the successor of any natural number.
These axioms ensure a consistent and logical structure of counting numbers.
⚡ Exam Tip
❌ Common Mistakes
- Including 0 in natural numbers.
- Assuming subtraction always gives a natural number.
- Confusing natural numbers with whole numbers.
📋 Case Study
A student claims that natural numbers are closed under all arithmetic operations. Analyze.
Analysis:
Natural numbers are closed only under addition and multiplication, not under subtraction and division. Hence, the claim is incorrect.
🌟 Importance
📘 Definition
💡 Concept
Core Concept
🎨 SVG Diagram
Representation on Number Line
Whole numbers start from 0 and extend infinitely in the positive direction.
🏷️ Properties
Fundamental Properties
Fundamental Properties
Closure Property
- Addition: \(a + b \in \mathbb{W}\)
- Multiplication: \(a \times b \in \mathbb{W}\)
Not Closed Under
- Subtraction (e.g., \(2 - 5 = -3 \notin \mathbb{W}\))
- Division (e.g., \(1 / 2 = 0.5 \notin \mathbb{W}\))
Commutative Property
\[
\begin{aligned}
a + b &= b + a, \\ a \times b &= b \times a
\end{aligned}
\]
Associative Property
\[
(a + b) + c = a + (b + c)
\]
Identity Elements
- Additive identity: \(0\), since \(a + 0 = a\)
- Multiplicative identity: \(1\), since \(a \times 1 = a\)
📍 Key Point
Role of Zero
- 0 represents absence of quantity.
- It is the smallest whole number.
- 0 is neither positive nor negative.
- Division by zero is undefined.
✏️ Example
Identify which of the following are whole numbers: -2, 0, 3.5, 7
Whole numbers are non-negative integers
- -2 → Not a whole number
- 0 → Whole number
- 3.5 → Not a whole number
- 7 → Whole number
✏️ Example
Find the result: \(5 + 0\), \(8 \times 1\)
Identity properties
- \(5 + 0 = 5\)
- \(8 \times 1 = 8\)
🔢 Formula
Key Formulae
💡 Concept
Conceptual Understanding
⚡ Exam Tip
❌ Common Mistakes
- Excluding 0 from whole numbers.
- Including decimals or fractions.
- Assuming closure under subtraction.
📋 Case Study
A student says that every natural number is a whole number, but not every whole number is a natural number. Justify.
Explanation:
Since \(\mathbb{N} \subset \mathbb{W}\), all natural numbers are whole numbers. However, 0 is a whole number but not a natural number.
🌟 Importance
📘 Definition
Integers \((\mathbb{Z})\)
🗂️ Types / Category
Classification
Positive Integers
Integers greater than zero, such as \(1, 2, 3, \dots\).
Example: \(5\), \(12\).
Example: \(5\), \(12\).
Negative Integers
Integers less than zero, such as \(-1, -2, -3, \dots\).
Example: \(-4\), \(-10\).
Example: \(-4\), \(-10\).
Zero
The number \(0\), which is neither positive nor negative.
Example: \(0\).
Example: \(0\).
🎨 SVG Diagram
Representation on Number Line
Integers extend infinitely in both directions on the number line.
🏷️ Properties
Properties of Integers
Properties of Integers
Closure
- Addition: closed, for example \(2 + 3 = 5\).
- Multiplication: closed, for example \(4 \times 5 = 20\).
- Subtraction: closed, for example \(7 - 2 = 5\).
- Division: not closed, for example \(1 / 2 = \frac{1}{2}\) is not always an integer.
Additive Identity
The number \(0\) because \(a + 0 = a\).
Example: \(8 + 0 = 8\).
Example: \(8 + 0 = 8\).
Additive Inverse
The number that gives zero when added to a number: \(a + (-a) = 0\).
Example: \(5 + (-5) = 0\).
Example: \(5 + (-5) = 0\).
Multiplicative Identity
The number \(1\) because \(a \times 1 = a\).
Example: \(9 \times 1 = 9\).
Example: \(9 \times 1 = 9\).
✏️ Example
Identify integers from: \(3.5, -4, 0, \frac{2}{3}\)
- -4, 0 are integers
- 3.5, \(2/3\) are not integers
✏️ Example
Find additive inverse of 7
\(-7\)
⚡ Exam Tip
❌ Common Mistakes
- Including decimals as integers
- Ignoring sign rules
📘 Definition
🗂️ Types / Category
Types of Rational Numbers
Fractions
Numbers written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).
Examples: \(\frac{1}{2}\), \(\frac{-3}{4}\), \(\frac{7}{5}\).
Examples: \(\frac{1}{2}\), \(\frac{-3}{4}\), \(\frac{7}{5}\).
Integers
Whole numbers, their negatives, and zero.
Examples: \(-3\), \(0\), \(8\).
Examples: \(-3\), \(0\), \(8\).
Terminating Decimals
Decimals that end after a finite number of digits.
Examples: \(0.5\), \(1.25\), \(3.75\).
Examples: \(0.5\), \(1.25\), \(3.75\).
Recurring Decimals
Decimals in which one or more digits repeat forever.
Examples: \(0.333\dots = \frac{1}{3}\), \(0.121212\dots\).
Examples: \(0.333\dots = \frac{1}{3}\), \(0.121212\dots\).
💡 Concept
Decimal Expansion Concept
📐 Derivation
Derivation: Convert Recurring Decimal to Fraction
Convert \(0.142857...\) into fraction:
Let \(x = 0.142857...\)
\[ \begin{aligned} 10^6 x& = 142857.142857...\\ 10^6x - x &= 142857\\ 999999x &= 142857 \\ \Rightarrow x &= \frac{1}{7} \end{aligned} \]
🏷️ Properties
Properties of Rational Numbers
Properties
- Closed under addition, subtraction, multiplication, and division (except division by zero), which means the result of these operations on rational numbers is always another rational number.
-
Dense property: there are infinitely many rational numbers between any two rational numbers.
\[ \text{Between any two rational numbers, infinitely many rational numbers exist} \]
For example, between \(\frac{1}{2}\) and \(1\), we can find \(\frac{3}{4}\), \(\frac{2}{3}\), \(\frac{5}{6}\), and many more.
✏️ Example
Is 0.75 rational?
Yes, \(0.75 = \frac{3}{4}\)
✏️ Example
Find a rational number between 2 and 3
\(\frac{2+3}{2} = 2.5\)
⚡ Exam Tip
❌ Common Mistakes
- Confusing irrational with non-terminating decimals
- Ignoring repeating pattern
📋 Case Study
A student claims that all decimals are irrational. Analyze.
Incorrect — only non-terminating, non-repeating decimals are irrational.
🌟 Importance
📘 Definition
💡 Concept
COre Concept
✏️ Example
- \(\sqrt{2} = 1.414213\cdots\)
- \(\pi = 3.141592\cdots\)
- \(e = 2.718281\cdots\)
🗂️ Types / Category
Common Types of Irrational Numbers
🎨 SVG Diagram
Representation on Number Line
Irrational numbers can be represented geometrically on the number line using constructions.
🔬 Proof
Classic Proof: \(\sqrt{2}\) is Irrational
Concept: Proof by contradiction
To prove that \(\sqrt{2}\) is irrational, we assume the opposite, namely that \(\sqrt{2}\) is rational. Then it can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers and \(q \neq 0\).
So, let \[ \sqrt{2} = \frac{p}{q} \] where \(\frac{p}{q}\) is in lowest terms.
Squaring both sides gives \[ 2 = \frac{p^2}{q^2} \] which implies \[ p^2 = 2q^2. \]
This shows that \(p^2\) is even, so \(p\) must also be even. Therefore, we can write \(p = 2k\) for some integer \(k\).
Substituting \(p = 2k\) into \(p^2 = 2q^2\), we get \[ (2k)^2 = 2q^2 \] \[ 4k^2 = 2q^2 \] \[ q^2 = 2k^2. \]
This shows that \(q^2\) is even, so \(q\) must also be even. Hence, both \(p\) and \(q\) are divisible by 2.
But this is a contradiction, because we assumed that \(p\) and \(q\) are coprime, meaning they have no common factor other than 1.
Therefore, our assumption is false. Hence, \(\sqrt{2}\) is irrational.
🏷️ Properties
Important Properties
Important Properties
- Sum of rational + irrational = irrational (generally)
- Product of non-zero rational × irrational = irrational
- Between any two rational numbers, irrational numbers exist
✏️ Example
Is \(0.1010010001...\) irrational?
Non-repeating decimal
Yes, because there is no repeating pattern.
✏️ Example
Is \(\sqrt{9}\) irrational?
o, \(\sqrt{9} = 3\), which is rational.
⚡ Exam Tip
❌ Common Mistakes
- Assuming all square roots are irrational
- Confusing long decimals with irrational numbers
- Ignoring repeating patterns
📋 Case Study
A student claims that \(\frac{22}{7}\) is irrational because its decimal expansion is non-terminating. Analyze.
Incorrect — \(\frac{22}{7}\) is rational because it can be written as a fraction, even though its decimal expansion is repeating.
🌟 Importance
📘 Definition
Fundamental Definitions
🗒️ Operations
Operations Involving Rational & Irrational Numbers
- If \(r \in \mathbb{Q}\) and \(s\) is irrational, then:
- \(r + s\) is irrational
- \(r - s\) is irrational
- \(rs\) is irrational (if \(r \ne 0\))
- \(\frac{r}{s}\) is irrational (if \(r \ne 0\))
🗒️ Important
Important Identities (Surds & Radicals)<
For positive real numbers \(a, b > 0\):
- \[ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \]
- \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, \quad b \ne 0 \]
- \[ (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b \]
- \[ (a + \sqrt{b})(a - \sqrt{b}) = a^2 - b \]
- \[ (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} \]
🗒️ Relation
Rationalisation of Denominator
To remove irrationality from the denominator, multiply numerator and denominator by the conjugate.
\[ \frac{1}{\sqrt{a} + b} \times \frac{\sqrt{a} - b}{\sqrt{a} - b} = \frac{\sqrt{a} - b}{a - b^2} \]
Concept: Use identity \((x+y)(x-y) = x^2 - y^2\)
⚖️ Laws
Laws of Exponents
For \(a > 0\), \(a, b \in \mathbb{R}\), and \(p, q \in \mathbb{Q}\):
- \[ a^p \cdot a^q = a^{p+q} \]
- \[ (a^p)^q = a^{pq} \]
- \[ \frac{a^p}{a^q} = a^{p-q}, \quad a \ne 0 \]
- \[ a^p \cdot b^p = (ab)^p \]
📐 Rule
Decimal Expansion Rules
-
A rational number has decimal expansion that is:
- Terminating OR
- Non-terminating recurring
-
An irrational number has decimal expansion that is:
- Non-terminating AND
- Non-recurring
⚡ Exam Tip
🏛️ Historical Note
Archimedes
Aryabhata
The development of the number system is deeply rooted in the contributions of great mathematicians. Their work laid the foundation for modern mathematics, especially in understanding irrational numbers, decimals, and constants like \(\pi\).
Archimedes
Archimedes, the Greek mathematician, made one of the earliest accurate estimations of the value of \(\pi\).
He used the method of inscribed and circumscribed polygons to bound the value of \(\pi\):
\[ 3.140845 < \pi < 3.142857 \]
This method is considered an early form of limit approximation, a concept later formalized in calculus.
Why It Matters
- First systematic approximation of irrational numbers
- Foundation for numerical methods
- Important for geometry and circle calculations
Aryabhata
Aryabhata (476–550 CE), the renowned Indian mathematician and astronomer, provided a highly accurate approximation of \(\pi\).
\[ \pi \approx 3.1416 \]
His work appeared in the famous text Aryabhatiya, where he also introduced advanced concepts in algebra and trigonometry.
Why It Matters
- Accurate decimal expansion of \(\pi\)
- Major contribution to Indian mathematics
- Foundation for trigonometry and astronomy
📌 Note
Additional Concepts & Enrichment Topics (Number Systems)
NCERT · Class IX · Chapter 1
Number Systems
A complete AI-powered learning engine — concepts, formulas, solver, interactive modules, and deep-dive questions — all in one place.
Core Concepts
Build a solid foundation — each concept explained with clarity and depth.
Concept 01
ℕ Natural Numbers & Whole Numbers
The counting numbers 1, 2, 3, 4, … form the set of Natural Numbers (ℕ). When we include 0, we get Whole Numbers (𝕎).
Every natural number is a whole number, but 0 is a whole number that is not a natural number. These are the simplest number systems — built purely from counting.
Sets
ℕ = {1, 2, 3, 4, …}𝕎 = {0, 1, 2, 3, …}
All natural numbers are whole numbers, but the converse is false — 0 is in 𝕎 but not in ℕ.
Concept 02
ℤ Integers
Integers (ℤ) extend whole numbers by including all negative counting numbers: …, −3, −2, −1, 0, 1, 2, 3, …
Integers arise naturally when we need to measure debt, temperature below zero, elevation below sea level, and so on. The symbol ℤ comes from the German word Zahlen (numbers).
Set
ℤ = {…, −3, −2, −1, 0, 1, 2, 3, …}
Whole numbers ⊂ Integers. Every whole number is an integer, but −1 is an integer that is not a whole number.
Concept 03
ℚ Rational Numbers
A number of the form p/q, where p and q are integers and q ≠ 0, is called a Rational Number (ℚ).
Examples: 1/2, −3/7, 0 (= 0/1), 5 (= 5/1), −11/4. Every integer is rational (set q = 1). The decimal expansion of a rational number is either terminating (e.g., 3/4 = 0.75) or non-terminating repeating (e.g., 1/3 = 0.333…).
Definition
ℚ = { p/q | p, q ∈ ℤ, q ≠ 0 }
Between any two rational numbers, there are infinitely many rational numbers — this property is called density.
To find a rational between a and b: compute (a + b)/2. Repeat to find more.
Concept 04
𝕀 Irrational Numbers
Numbers that cannot be expressed as p/q (p, q ∈ ℤ, q ≠ 0) are called Irrational Numbers. Their decimal expansions are non-terminating and non-repeating.
Famous examples: √2, √3, √5, π, e, ∛7. The proof that √2 is irrational is one of the great classical proofs — assume √2 = p/q in lowest terms, square both sides, and derive a contradiction about even/odd parity.
Key Irrational Values (approx)
√2 ≈ 1.41421356…√3 ≈ 1.73205080…
√5 ≈ 2.23606797…
π ≈ 3.14159265…
√4 = 2 is rational! Not every square root is irrational — only square roots of non-perfect squares are irrational.
To check: √n is irrational if and only if n is not a perfect square.
Concept 05
ℝ Real Numbers & the Number Line
Real Numbers (ℝ) = Rational Numbers ∪ Irrational Numbers. Every point on the number line corresponds to exactly one real number, and vice versa — this is the Real Number Line.
The hierarchy: ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ, with irrational numbers filling the "gaps" between rationals.
Concept 06
· Decimal Expansions
The decimal expansion of a real number falls into one of three categories:
- Terminating: Ends after finite digits — e.g., 7/4 = 1.75. Occurs when denominator (in lowest terms) has only 2s and 5s as prime factors.
- Non-terminating Repeating (Recurring): Digits repeat in a block — e.g., 1/7 = 0.142857142857… → rational number.
- Non-terminating Non-repeating: No repeating block ever — e.g., √2, π → irrational number.
Converting Recurring Decimal to Fraction
Let x = 0.̄6̄ (= 0.666…)Then 10x = 6.666…
10x − x = 6 → 9x = 6 → x = 6/9 = 2/3
For p̄/q̄ repeating block of length n: multiply by 10ⁿ and subtract original.
Concept 07
📏 Locating √n on the Number Line
To plot √2 geometrically: Draw OA = 1 unit on the number line. At A erect a perpendicular AB = 1 unit. Then OB = √2 (by Pythagoras). Swing an arc of radius OB from O to mark √2 on the number line.
Generalisation (Spiral of Theodorus): Once √n is marked, erect a perpendicular of 1 unit from its tip to get √(n+1). This spiral generates all √2, √3, √4, √5…
This construction proves that irrational numbers do have a definite location on the number line — they are real in the geometric sense.
Concept 08
⊕ Operations on Real Numbers
The real numbers are closed under addition, subtraction, multiplication, and division (÷ non-zero). Important identities:
- Rational + Rational = Rational
- Irrational + Irrational may be rational (e.g., √2 + (−√2) = 0) or irrational.
- Rational × Irrational = Irrational (unless rational factor = 0).
- Irrational × Irrational may be rational (e.g., √2 × √2 = 2) or irrational.
Assuming irrational + irrational is always irrational. Counter-example: (√3 + 1) + (1 − √3) = 2.
Concept 09
÷ Rationalisation of Denominators
To simplify expressions with surds in the denominator, multiply numerator and denominator by the conjugate (or appropriate surd) to make the denominator rational.
Key Conjugate Identities
(a + √b)(a − √b) = a² − b(√a + √b)(√a − √b) = a − b
(a + b)(a² − ab + b²) = a³ + b³ ← (for cube roots)
Example: Rationalise 1/(√3 + √2)
= 1·(√3 − √2) / [(√3 + √2)(√3 − √2)]= (√3 − √2) / (3 − 2)
= √3 − √2
Concept 10
xⁿ Laws of Exponents for Real Numbers
When a > 0, the following laws extend exponent rules to rational (and real) indices:
Laws
aᵐ · aⁿ = aᵐ⁺ⁿ(aᵐ)ⁿ = aᵐⁿ
aᵐ / aⁿ = aᵐ⁻ⁿ (a ≠ 0)
aᵐ · bᵐ = (ab)ᵐ
a^(1/n) = ⁿ√a
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
a^(p/q) means: take the q-th root first (easier numbers), then raise to power p.
Do NOT apply these laws when the base is negative and the exponent is fractional — this can lead to undefined or complex values.
Formula Reference Sheet
All key identities and formulas — organised for quick revision.
🔢 Number Sets — Hierarchy
ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ
ℝ = ℚ ∪ (Irrationals), ℚ ∩ (Irrationals) = ∅
ℝ = ℚ ∪ (Irrationals), ℚ ∩ (Irrationals) = ∅
· Decimal Expansion — Type Test
| Type of Decimal | Category | Example |
|---|---|---|
| Terminating | Rational | 7/8 = 0.875 |
| Non-term. Repeating | Rational | 1/7 = 0.142857̄ |
| Non-term. Non-repeating | Irrational | √5 = 2.2360679… |
Termination Test
p/q terminates ⟺ denominator (in lowest terms) = 2ᵃ × 5ᵇ (a,b ≥ 0)
√ Surd Identities
√(ab) = √a · √b (a, b ≥ 0)
√(a/b) = √a / √b (a ≥ 0, b > 0)
(√a)² = a (a ≥ 0)
(√a + √b)(√a − √b) = a − b
(a + √b)² = a² + 2a√b + b
ⁿ√(aᵐ) = a^(m/n)
√(a/b) = √a / √b (a ≥ 0, b > 0)
(√a)² = a (a ≥ 0)
(√a + √b)(√a − √b) = a − b
(a + √b)² = a² + 2a√b + b
ⁿ√(aᵐ) = a^(m/n)
xⁿ Laws of Exponents (a,b > 0; m,n ∈ ℚ)
| Law | Formula |
|---|---|
| Product of same base | aᵐ · aⁿ = aᵐ⁺ⁿ |
| Quotient of same base | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ |
| Product rule | aᵐ · bᵐ = (ab)ᵐ |
| Zero exponent | a⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ |
| Rational exponent | a^(1/n) = ⁿ√a |
| Fractional exponent | a^(m/n) = (ⁿ√a)ᵐ |
∫ Rationalisation Formulas
1/√a = √a/a
1/(√a + √b) = (√a − √b)/(a − b) [conjugate: √a − √b]
1/(a + √b) = (a − √b)/(a² − b) [conjugate: a − √b]
1/(√a − √b) = (√a + √b)/(a − b) [conjugate: √a + √b]
1/(√a + √b) = (√a − √b)/(a − b) [conjugate: √a − √b]
1/(a + √b) = (a − √b)/(a² − b) [conjugate: a − √b]
1/(√a − √b) = (√a + √b)/(a − b) [conjugate: √a + √b]
↔ Recurring Decimal ↔ Fraction
Pattern for 0.ā̄ (single repeating digit a)
0.āā… = a/9Pattern for 0.a̅b̅ (two repeating digits ab)
0.a̅b̅… = ab/99Pattern for 0.abc̄ (non-repeating prefix + repeating)
x = 0.abc̄̄̄ → multiply by 10^(total digits) and 10^(non-repeating digits), subtract
Tips, Tricks & Common Mistakes
Sharp insights that separate top scorers from the rest.
⚡ Quick Tips
To find n rational numbers between a and b: use the formula a + k·(b−a)/(n+1) for k = 1, 2, …, n. This always works.
A number p/q terminates if and only if the denominator in lowest terms has no prime factors other than 2 and 5. Check the denominator, not the numerator.
To simplify √(a²b): pull out a first — √(a²b) = a√b (when a > 0). Don't leave perfect square factors under the radical.
π ≠ 22/7. The fraction 22/7 is a rational approximation; π is genuinely irrational. Do not confuse them in proofs.
When rationalising: always multiply by the conjugate. For 1/(a + √b), conjugate is (a − √b); for 1/(√a + √b), conjugate is (√a − √b).
a^(m/n) can be computed as either (a^m)^(1/n) or (a^(1/n))^m. Choose the order that gives smaller intermediate numbers.
If asked to compare surds: square them! √5 vs √6 — 5 < 6, so √5 < √6. Squaring preserves order for positive quantities.
✕ Common Mistakes to Avoid
Mistake: Writing √(a + b) = √a + √b. This is WRONG. √(9+16) = √25 = 5 ≠ √9 + √16 = 3 + 4 = 7.
Mistake: Treating every square root as irrational. √9 = 3, √16 = 4 — these are rational.
Mistake: Writing a^m × b^n = (ab)^(mn). The correct rule is a^m × b^m = (ab)^m. The exponents must be the same.
Mistake: Assuming (−2)^(1/2) = −√2. Fractional exponents with even denominators are not defined for negative bases in real numbers.
Mistake: Saying 0.999… ≠ 1. In fact 0.999… = 1 exactly (proof: let x = 0.999…, then 10x = 9.999…, so 9x = 9, x = 1).
Mistake: Forgetting that every integer is also rational. When asked "is −5 rational?", the answer is YES (= −5/1).
📌 Memory Anchors
NICE acronym for number hierarchy:
N = Natural → I = Integer → C = (raCional = Rational) → E = Everything (Real)
Surd multiplication: "Same index, multiply inside; different index, convert to fractional exponents first."
Rationalisation: Think of it as using the difference-of-squares identity (a+b)(a−b) = a²−b² to "destroy" the surd in the denominator.
📋 Exam Strategy
- Questions on finding rationals between two numbers: simply average the two numbers repeatedly.
- Questions on decimal expansion type: perform the division or factorise the denominator.
- Questions on simplification of surds: always look for perfect-square factors first.
- Questions on exponent evaluation: convert to fractional exponent form, simplify step by step.
- Questions proving √p is irrational: use contradiction — assume rational in lowest terms, then show both p and q must have a common factor.
Concept-Building Questions
Original, non-textbook questions — each with full step-by-step solutions. Click a question to reveal the solution.
Group A — Classification
Q1. Classify each number as rational or irrational, and justify your answer: (a) 0.1̄0̄1̄0̄0̄1̄… (b) (√5 − 2)(√5 + 2) (c) π − 3
Medium
+
a
0.101001000100001…
The decimal has a pattern of increasing zeros between 1s — it never repeats a fixed block. Therefore it is non-terminating and non-repeating → Irrational.
b
(√5 − 2)(√5 + 2)
Apply the identity (a − b)(a + b) = a² − b².
= (√5)² − (2)² = 5 − 4 = 1
Result is 1, which is rational (= 1/1).
c
π − 3
π is irrational. Subtracting a rational number (3) from an irrational number always yields an irrational number (if π − 3 were rational r, then π = r + 3 would be rational — contradiction). So π − 3 is Irrational.
Q2. Prove that √3 is irrational using the method of contradiction.
Hard
+
1
Assume the opposite
Suppose √3 is rational. Then √3 = p/q where p, q are integers with no common factor (lowest terms), and q ≠ 0.
2
Square both sides
3 = p²/q² → p² = 3q²
So 3 divides p². Since 3 is prime, 3 must divide p (if a prime divides a², it divides a).
3
Let p = 3k
p² = 9k² → 3q² = 9k² → q² = 3k²
So 3 divides q² → 3 divides q.
4
Contradiction
Both p and q are divisible by 3, contradicting the assumption that p/q is in lowest terms. Therefore √3 cannot be rational. √3 is irrational. ∎
Group B — Decimal Expansions
Q3. Convert 2.3̄1̄6̄ (i.e., 2.316316316…) to a fraction in lowest terms.
Medium
+
1
Let x = 2.316316316…
The repeating block "316" has 3 digits.
2
Multiply by 10³ = 1000
1000x = 2316.316316…
3
Subtract original
1000x − x = 2316.316… − 2.316…
999x = 2314
999x = 2314
4
Simplify
x = 2314/999
Check GCD(2314, 999): 2314 = 2×999 + 316; 999 = 3×316 + 51; 316 = 6×51 + 10; 51 = 5×10 + 1 → GCD = 1. Already lowest terms.
Answer: 2314/999
Q4. Without performing long division, determine whether 13/250 has a terminating or non-terminating decimal expansion. If terminating, write it out.
Easy
+
1
Factorise denominator
250 = 2 × 125 = 2¹ × 5³
Only prime factors 2 and 5 → terminating decimal.
2
Convert to power of 10
We need 10³ in denominator. Multiply numerator and denominator by 2²= 4:
13/250 = (13 × 4)/(250 × 4) = 52/1000 = 0.052
3
Answer
13/250 = 0.052 (terminating)
Group C — Surds & Rationalisation
Q5. Simplify: (√5 + √3)² − (√5 − √3)²
Easy
+
1
Use identity A² − B² = (A+B)(A−B)
Let A = √5 + √3, B = √5 − √3
(A+B)(A−B) = [(√5+√3)+(√5−√3)]·[(√5+√3)−(√5−√3)]
2
Simplify each bracket
A+B = 2√5
A−B = 2√3
A−B = 2√3
3
Multiply
= 2√5 × 2√3 = 4√15
Answer: 4√15
Q6. Rationalise: 6/(3√2 − 2√3). Express in simplest form.
Medium
+
1
Identify conjugate
The conjugate of (3√2 − 2√3) is (3√2 + 2√3).
2
Multiply top & bottom
= 6(3√2 + 2√3) / [(3√2)² − (2√3)²]
= 6(3√2 + 2√3) / [18 − 12]
= 6(3√2 + 2√3) / 6
= 6(3√2 + 2√3) / [18 − 12]
= 6(3√2 + 2√3) / 6
3
Cancel and finalise
= 3√2 + 2√3
Answer: 3√2 + 2√3
Q7. If a = 5 + 2√6 and b = 1/a, find a + b and a − b.
Hard
+
1
Find b = 1/a by rationalising
b = 1/(5+2√6) × (5−2√6)/(5−2√6)
= (5−2√6)/(25−24) = 5−2√6
= (5−2√6)/(25−24) = 5−2√6
2
Compute a + b
a + b = (5+2√6) + (5−2√6) = 10
3
Compute a − b
a − b = (5+2√6) − (5−2√6) = 4√6
a + b = 10, a − b = 4√6
Group D — Laws of Exponents
Q8. Evaluate: (27)^(−2/3) × (32)^(2/5) ÷ (9)^(−1/2)
Medium
+
1
Write each base as a prime power
27 = 3³, 32 = 2⁵, 9 = 3²
2
Apply exponent rules
(3³)^(−2/3) = 3^(−2) = 1/9
(2⁵)^(2/5) = 2² = 4
(3²)^(−1/2) = 3^(−1) = 1/3
(2⁵)^(2/5) = 2² = 4
(3²)^(−1/2) = 3^(−1) = 1/3
3
Combine (note: ÷ 9^(−1/2) = × 3)
(1/9) × 4 ÷ (1/3) = (4/9) × 3 = 12/9 = 4/3
Answer: 4/3
Q9. If 2^x = 3^y = 12^z, show that z = xy/(2y + x).
Hard
+
1
Let common value = k
2^x = k → 2 = k^(1/x)
3^y = k → 3 = k^(1/y)
12^z = k → 12 = k^(1/z)
3^y = k → 3 = k^(1/y)
12^z = k → 12 = k^(1/z)
2
Express 12 in terms of 2 and 3
12 = 4 × 3 = 2² × 3
k^(1/z) = (k^(1/x))² × k^(1/y) = k^(2/x + 1/y)
k^(1/z) = (k^(1/x))² × k^(1/y) = k^(2/x + 1/y)
3
Equate exponents
1/z = 2/x + 1/y = (2y + x)/(xy)
∴ z = xy/(2y + x) ∎
∴ z = xy/(2y + x) ∎
Group E — Conceptual & Application
Q10. Insert 4 irrational numbers between 2 and 3. Justify that each is irrational.
Easy
+
1
Strategy
We need irrational numbers strictly between 2 and 3. Square the bounds: 2² = 4, 3² = 9. So √n is between 2 and 3 whenever 4 < n < 9 and n is not a perfect square.
2
Choose values
√5 ≈ 2.236, √6 ≈ 2.449, √7 ≈ 2.646, √8 ≈ 2.828
3
Justify
5, 6, 7, 8 are not perfect squares, so their square roots are irrational. Each lies strictly between 2 and 3. ✓
Q11. The sum of two irrational numbers is 8. Their product is 15. Find the numbers and classify them.
Hard
+
1
Set up equations
Let the numbers be α and β.
α + β = 8, αβ = 15
2
Quadratic with roots α, β
t² − 8t + 15 = 0
(t − 3)(t − 5) = 0 → t = 3 or 5
(t − 3)(t − 5) = 0 → t = 3 or 5
3
Reconcile with "irrational" condition
3 and 5 are rational integers. So as real numbers, no solution exists with both irrational — the problem as literally stated has no solution with both numbers irrational. However, we can reinterpret: perhaps the numbers are (4 + √1) type. The intended answer is 3 and 5 — this is a classic gotcha showing irrational constraints must be verified.
This question tests critical thinking: always verify that found values actually satisfy all given conditions — here both 3 and 5 are rational, contradicting the premise.
Step-by-Step AI Solver
Select a problem type and enter values — the engine will solve it with complete working.
🔍 Problem Selector
Interactive Learning Modules
Hands-on activities to build deep conceptual intuition.
📏 Module 1 — Number Line Explorer
Type any number (integer, fraction, or surd like √2, √5) and see it plotted on the number line. Explore where irrationals sit relative to rationals.
🧠 Module 2 — Rapid Classification Quiz
Is the number rational or irrational? Answer 10 questions. Track your score.
🔢 Module 3 — Decimal Expansion Builder
Enter any integer p and q (p/q), and see long-division steps with the recurring decimal expansion computed live.
/
√ Module 4 — Surd Simplifier
Enter a number n and get √n simplified to the form a√b, showing the factorisation step.
⭕ Module 5 — Number Set Sorter
Drag each number card into the smallest set it belongs to (ℕ, 𝕎, ℤ, ℚ, or Irrational). Train your classification instinct.
▲ scroll up to switch modules · made with ♥ for Class IX learners
📚
ACADEMIA AETERNUM
तमसो मा ज्योतिर्गमय · Est. 2025
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NCERT Class 9 Maths Chapter 1 Number Systems MCQs — Complete Notes & Solutions · academia-aeternum.com
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