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Chapter 3 · NCERT 2025-26

🔢 A Story of Numbers

Natural Numbers · Integers · Rationals · Irrationals · Real Numbers

ℕ
ℤ
ℚ
ℝ
π
√2
📐 Introduction: The Story Begins

Numbers are the oldest and most fundamental invention of mathematics. Long before humans built cities or invented writing, they needed to count — how many sheep in the herd, how many days till the next full moon, how many seeds to plant. This chapter tells the remarkable story of numbers — how humanity started with simple counting and gradually discovered an entire universe of numbers that stretches to infinity and beyond.

Each time humans encountered a problem that existing numbers could not solve, they invented a new type of number. Cannot subtract a larger number from a smaller one? Invent negative numbers. Cannot divide 3 by 7 exactly? Invent fractions (rational numbers). Cannot find the exact length of a diagonal? Discover irrational numbers. This chapter traces that journey from the simplest counting numbers all the way to the real number system.

🇮🇳 Indian Mathematical Heritage

India played a central role in the development of the number system that the entire world uses today.

🔢 The Decimal System

The Hindu-Arabic numeral system (0, 1, 2, ..., 9) with place values based on powers of 10 was developed in India. It reached the Arab world by the 8th century and then spread to Europe. Every calculator, computer, and phone in the world uses this system.

🟠 Zero — Brahmagupta (628 CE)

Brahmagupta was the first mathematician to formally define zero as a number and give rules for arithmetic involving zero and negative numbers. Without zero, our entire number system would collapse.

🔵 Negative Numbers

Indian mathematicians, especially Brahmagupta, were the first to treat negative numbers as legitimate quantities. He called positive numbers "fortunes" and negative numbers "debts" — a very practical way of thinking about them.

🌟 Baudhayana & √2

The Baudhayana Sulbasutras (~800 BCE) contain a remarkably accurate approximation of √2 as 1.4142156..., which is correct to five decimal places. This shows that ancient Indians were already grappling with irrational numbers.
💡 Did You Know? The word "zero" comes from the Sanskrit word shunya (meaning "void" or "empty"). The concept travelled from India to the Arab world (where it became sifr) and then to Europe (where it became zero). Every time you write 0, you are using an Indian invention!
📚 Chapter Overview
Topic Key Concepts
Natural Numbers Counting numbers: 1, 2, 3, ...
Whole Numbers Natural numbers + zero: 0, 1, 2, 3, ...
Integers ..., −3, −2, −1, 0, 1, 2, 3, ...
Rational Numbers Numbers of the form p/q (q ≠ 0); properties and operations
Number Line Representation of numbers; density of rationals
Irrational Numbers Numbers that cannot be expressed as p/q; examples like √2, π
Real Numbers Union of rational and irrational numbers
🔢 Natural Numbers

The natural numbers are the most basic numbers we learn as children: 1, 2, 3, 4, 5, ... They are the numbers we use for counting objects. The set of natural numbers is denoted by the symbol ℕ (or sometimes N).

ℕ = {1, 2, 3, 4, 5, 6, 7, ...} The set of Natural Numbers — starts from 1, goes on forever
💡 Key Properties of Natural Numbers

✅ Closure under Addition

Adding any two natural numbers always gives a natural number. Example: 3 + 5 = 8 (natural). Closed.

✅ Closure under Multiplication

Multiplying any two natural numbers gives a natural number. Example: 4 × 7 = 28 (natural). Closed.

❌ Not Closed under Subtraction

Subtracting may give a non-natural result. Example: 3 − 5 = −2 (not a natural number). Not closed.

❌ Not Closed under Division

Dividing may give a non-natural result. Example: 5 ÷ 2 = 2.5 (not a natural number). Not closed.
💡 Important: The smallest natural number is 1. There is no largest natural number — for any natural number n, n + 1 is also a natural number. The set is infinite.
💡 Memory Aid: Natural = Numbers you Naturally use for counting. Start from 1, and keep going. No zero, no negatives, no fractions.
🟡 Whole Numbers

When we include zero along with the natural numbers, we get the whole numbers. The set of whole numbers is denoted by W.

W = {0, 1, 2, 3, 4, 5, 6, ...} Whole Numbers = Natural Numbers + Zero

Why did we need zero? Imagine counting your apples. If you have some, natural numbers work fine. But what if you have none? You need a number to represent "nothing" — that number is zero. Zero is also essential for our place-value system: without it, we could not distinguish between 12 and 102 and 120.

💡 Relationship Between N and W

Every natural number is a whole number, but not every whole number is a natural number (because 0 is a whole number but not a natural number).

ℕ ⊂ W Natural numbers are a subset of whole numbers
💡 Key Difference: The smallest natural number is 1. The smallest whole number is 0.
🔵 Integers

Natural numbers cannot handle subtraction like 3 − 5. To solve this problem, we extend our number system to include negative numbers. The collection of all positive whole numbers, negative whole numbers, and zero is called the set of integers, denoted by ℤ (or Z, from the German word Zahlen meaning "numbers").

ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...} Integers extend infinitely in both directions
💡 Types of Integers

➕ Positive Integers

1, 2, 3, 4, 5, ... (same as natural numbers)

➖ Negative Integers

−1, −2, −3, −4, −5, ...

🟠 Zero

0 is an integer that is neither positive nor negative.
🛠 Closure Properties of Integers
Operation Closed? Example
Addition Yes −3 + 5 = 2 (integer)
Subtraction Yes 3 − 7 = −4 (integer)
Multiplication Yes −2 × 6 = −12 (integer)
Division No 7 ÷ 3 = 7/3 (not an integer)
💡 Why Not Closed under Division? Because dividing two integers does not always give an integer. For example, 1 ÷ 2 = 0.5, which is not an integer. This limitation led to the invention of the next type of number — rational numbers.
ℕ ⊂ W ⊂ ℤ Natural numbers ⊂ Whole numbers ⊂ Integers
🔶 Rational Numbers

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. The set of rational numbers is denoted by ℚ (from "quotient").

ℚ = { p/q : p, q ∈ ℤ, q ≠ 0 } Rational numbers — all numbers expressible as a fraction of two integers
📝 Examples of Rational Numbers

🔢 Fractions

1/2, 3/4, −5/7, 22/7 — these are obviously rational (already in p/q form).

🔢 Integers

Every integer is rational! For example, 5 = 5/1, −3 = −3/1, 0 = 0/1.

🔢 Terminating Decimals

0.25 = 1/4, 3.5 = 7/2, −0.8 = −4/5 — all terminating decimals are rational.

🔢 Repeating Decimals

0.333... = 1/3, 0.142857142857... = 1/7 — all repeating (recurring) decimals are rational.
💡 Properties of Rational Numbers
Property Addition Multiplication
Closure Yes: p/q + r/s is rational Yes: (p/q) × (r/s) is rational
Commutative Yes: a + b = b + a Yes: a × b = b × a
Associative Yes: (a + b) + c = a + (b + c) Yes: (a × b) × c = a × (b × c)
Identity 0 (additive identity): a + 0 = a 1 (multiplicative identity): a × 1 = a
Inverse Additive inverse of a is −a Multiplicative inverse of a/b is b/a (a ≠ 0)
Distributive a × (b + c) = a × b + a × c
✏️ Equivalent Rational Numbers

A rational number has infinitely many equivalent forms. For example:

1/2 = 2/4 = 3/6 = 4/8 = 5/10 = ... Multiply (or divide) both numerator and denominator by the same non-zero number

The standard form (or lowest terms) of a rational number is when p and q have no common factor other than 1, and q is positive. For example, the standard form of −6/8 is −3/4.

💡 Quick Test: Is a number rational? Ask: "Can I write it as a fraction of two integers?" If yes, it is rational. If the decimal terminates or repeats, it is rational. If the decimal goes on forever without repeating, it is not rational.
💡 Zero in Denominator: Division by zero is undefined. The expression 5/0 is meaningless. Always ensure q ≠ 0 in p/q. This is not just a rule — it reflects the fact that you cannot divide something into zero groups.
📏 Representation on the Number Line

Every rational number can be represented as a unique point on the number line. Conversely, every point on the number line whose position can be described by a terminating or repeating decimal corresponds to a rational number.

📍 Plotting Rational Numbers

To plot a fraction like 3/4 on the number line:

  1. Identify between which two consecutive integers the number lies. Since 0 < 3/4 < 1, it lies between 0 and 1.
  2. Divide the segment from 0 to 1 into 4 equal parts (because the denominator is 4).
  3. Count 3 parts from 0 to the right (because the numerator is 3).
  4. Mark and label the point as 3/4.

Example: Plot −5/3 on the number line.

Step 1: Since −5/3 = −1.666..., it lies between −2 and −1.
Step 2: Divide the segment from −2 to −1 into 3 equal parts.
Step 3: From −2, count 1 part to the right (towards −1). That point is −5/3.
Alternative: From −1, go 2/3 of a unit to the left. This also gives −5/3.
💡 Between Any Two Rational Numbers: Infinitely Many!

One of the most remarkable properties of rational numbers is their density: between any two distinct rational numbers, there exist infinitely many other rational numbers.

Between a/b and c/d, the number (a/b + c/d) / 2 always lies in between. The average of two rationals is always rational and lies between them

Example: Find 3 rational numbers between 1/3 and 1/2.

Method 1 (Averaging):
Between 1/3 and 1/2: average = (1/3 + 1/2)/2 = (2/6 + 3/6)/2 = (5/6)/2 = 5/12
Continue: Between 1/3 and 5/12: average = (4/12 + 5/12)/2 = (9/12)/2 = 9/24 = 3/8
Continue: Between 5/12 and 1/2: average = (5/12 + 6/12)/2 = (11/12)/2 = 11/24
Answer: Three rational numbers between 1/3 and 1/2 are 3/8, 5/12, and 11/24.

Example: Find 5 rational numbers between 1/4 and 1/3.

Method 2 (Equivalent fractions):
Write 1/4 and 1/3 with a common denominator having enough gap.
1/4 = 6/24 and 1/3 = 8/24. Only one integer (7) between 6 and 8 — not enough!
Use a larger denominator: 1/4 = 60/240 and 1/3 = 80/240.
Now we can pick: 61/240, 65/240 = 13/48, 70/240 = 7/24, 75/240 = 5/16, 79/240.
Answer: Five rational numbers between 1/4 and 1/3 are 61/240, 13/48, 7/24, 5/16, 79/240.
💡 Quick Method: To find n rational numbers between two fractions a/b and c/d, multiply both numerator and denominator by (n + 1) to create enough "room" between them. Then simply pick integers in between.
🛠 Operations on Rational Numbers
➕ Addition of Rational Numbers
a/b + c/d = (ad + bc) / bd Cross-multiply numerators, multiply denominators

Example 1: Add 2/3 + 4/5

Step 1: LCM of 3 and 5 = 15.
Step 2: 2/3 = 10/15, and 4/5 = 12/15.
Step 3: 10/15 + 12/15 = 22/15.
➖ Subtraction of Rational Numbers
a/b − c/d = (ad − bc) / bd Same as addition, but subtract the cross products

Example 2: Subtract −3/7 from 5/14

Step 1: 5/14 − (−3/7) = 5/14 + 3/7
Step 2: LCM of 14 and 7 = 14. So 3/7 = 6/14.
Step 3: 5/14 + 6/14 = 11/14.
✖ Multiplication of Rational Numbers
(a/b) × (c/d) = (a × c) / (b × d) Multiply numerators together and denominators together

Example 3: Multiply −2/5 × 3/4

Step 1: (−2 × 3) / (5 × 4) = −6/20
Step 2: Simplify: −6/20 = −3/10.
➗ Division of Rational Numbers
(a/b) ÷ (c/d) = (a/b) × (d/c) = (ad) / (bc) To divide by a fraction, multiply by its reciprocal (flip and multiply)

Example 4: Divide 7/9 by −2/3

Step 1: 7/9 ÷ (−2/3) = 7/9 × 3/(−2) = 7/9 × (−3/2)
Step 2: = (7 × −3) / (9 × 2) = −21/18
Step 3: Simplify: −21/18 = −7/6.
💡 "KFC" Rule for Division:
Keep the first fraction → Flip the second fraction → Change division to multiplication.
That is all division of fractions is!
🌟 Introduction to Irrational Numbers

Despite the richness of rational numbers, they do not fill up the entire number line. There are points on the number line that do not correspond to any rational number. The numbers that occupy these "gaps" are called irrational numbers.

An irrational number CANNOT be expressed as p/q for any integers p and q (q ≠ 0). Irrational = "not a ratio" of integers
💡 How to Recognise an Irrational Number

The decimal expansion of an irrational number is non-terminating and non-repeating. It goes on forever without ever settling into a repeating pattern.

√2 = 1.41421356...

The most famous irrational number. It is the length of the diagonal of a unit square. The ancient Greeks proved it cannot be expressed as a fraction.

√3 = 1.73205080...

The height of an equilateral triangle with side 2 units. Non-terminating, non-repeating.

π = 3.14159265...

The ratio of a circle's circumference to its diameter. Known to trillions of digits, never repeats.

√5 = 2.23606797...

Appears in the golden ratio: φ = (1 + √5)/2 ≈ 1.618. Found throughout nature and art.
📚 Proof that √2 is Irrational (Outline)

This is one of the most beautiful proofs in all of mathematics. It uses a method called proof by contradiction.

Proof: √2 is irrational

Assume the opposite: Suppose √2 is rational. Then we can write √2 = p/q where p and q are integers with no common factor (i.e., the fraction is in lowest terms) and q ≠ 0.
Square both sides: 2 = p²/q², which gives p² = 2q².
Conclusion 1: Since p² = 2q², p² is even. Therefore p must be even (because the square of an odd number is odd). Let p = 2k for some integer k.
Substitute: (2k)² = 2q² → 4k² = 2q² → q² = 2k².
Conclusion 2: Since q² = 2k², q² is even. Therefore q must be even.
Contradiction! Both p and q are even, so they have a common factor of 2. But we assumed p/q was in lowest terms (no common factors). This is a contradiction. Therefore, our assumption was wrong, and √2 is irrational.
🔢 Which Square Roots are Irrational?
💡 Key Rule: √n is irrational if and only if n is not a perfect square.
• √4 = 2 (rational, since 4 = 2²)
• √9 = 3 (rational, since 9 = 3²)
• √2, √3, √5, √6, √7, √8, √10, ... are all irrational.
💡 Quick Test for Irrationality: If the number under the square root sign is NOT a perfect square (1, 4, 9, 16, 25, 36, ...), then the square root is irrational. Works for cube roots too: √[3]{n} is irrational if n is not a perfect cube.
📈 The Real Number System

When we combine all rational numbers and all irrational numbers together, we get the complete set of real numbers, denoted by ℝ (or R).

ℝ = ℚ ∪ (Irrational Numbers) Real Numbers = Rational Numbers + Irrational Numbers

Every point on the number line corresponds to exactly one real number, and every real number corresponds to exactly one point on the number line. This is why the number line is sometimes called the real number line.

📊 The Number System Hierarchy
ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real

🔢 Natural Numbers ℕ

1, 2, 3, 4, 5, ...
Counting numbers. The innermost set.

🟡 Whole Numbers W

0, 1, 2, 3, 4, ...
Natural numbers + zero.

🔵 Integers ℤ

..., −2, −1, 0, 1, 2, ...
Whole numbers + negatives.

🔶 Rational Numbers ℚ

All p/q where q ≠ 0.
Includes all integers + fractions.

🌟 Irrational Numbers

√2, π, √3, e, ...
Non-terminating, non-repeating decimals.

📈 Real Numbers ℝ

Every number on the number line.
Rational + Irrational = Real.
⚠️ Rational vs Irrational — Key Differences
Property Rational Numbers Irrational Numbers
Form Can be written as p/q (q ≠ 0) Cannot be written as p/q
Decimal Terminating or repeating Non-terminating, non-repeating
Examples 1/2, 0.75, −3, 0.333... √2, π, √5, e
On number line Yes (dense, but with gaps) Yes (fills the gaps)
💡 The Big Picture: Think of the number line as a road. The rational numbers are like all the houses with proper addresses (fractions). The irrational numbers are like the spaces between houses — they are there, they have locations, but they cannot be described by a simple fraction. Together, they fill the entire road with no gaps at all.
✏️ NCERT-Style Worked Examples

Example 1: Classify the following numbers as rational or irrational: (a) √25, (b) √7, (c) 0.3333..., (d) 1.41421356..., (e) 22/7.

(a) √25 = 5 → 5 is an integer, hence rational.
(b) √7 → 7 is not a perfect square, so √7 is irrational.
(c) 0.3333... → This is a repeating decimal = 1/3, so it is rational.
(d) 1.41421356... → This is √2, a non-terminating non-repeating decimal, so it is irrational.
(e) 22/7 → This is a fraction of two integers (p = 22, q = 7), so it is rational. (Note: 22/7 is NOT equal to π; it is only an approximation.)

Example 2: Find the additive inverse and multiplicative inverse of −3/5.

Additive inverse: The number that gives 0 when added.
−3/5 + ? = 0 → ? = 3/5.
Multiplicative inverse (reciprocal): The number that gives 1 when multiplied.
(−3/5) × ? = 1 → ? = −5/3.

Example 3: Verify the distributive property: −3/4 × (2/3 + −5/6).

LHS: First compute the sum inside: 2/3 + (−5/6) = 4/6 − 5/6 = −1/6.
Then: −3/4 × (−1/6) = 3/24 = 1/8.
RHS: −3/4 × 2/3 + (−3/4) × (−5/6)
= −6/12 + 15/24 = −1/2 + 5/8 = −4/8 + 5/8 = 1/8.
Since LHS = RHS = 1/8, the distributive property is verified. ✅

Example 4: Represent 3/5 and −3/5 on the number line.

Step 1: 3/5 lies between 0 and 1. Divide the segment [0, 1] into 5 equal parts.
Step 2: Count 3 parts from 0 to the right. Mark this point as 3/5.
Step 3: −3/5 lies between −1 and 0. Divide the segment [−1, 0] into 5 equal parts.
Step 4: Count 3 parts from 0 to the left. Mark this point as −3/5.
Note: 3/5 and −3/5 are equidistant from 0, on opposite sides. They are additive inverses of each other.

Example 5: Find 5 rational numbers between −1 and 1.

Method: Write −1 and 1 with a denominator of 6: −1 = −6/6 and 1 = 6/6.
Pick integers between −6 and 6: −5/6, −2/6, 0/6, 1/6, 4/6
Simplify: −5/6, −1/3, 0, 1/6, 2/3.

Example 6: Express 0.overline{6} (0.666...) as a fraction.

Let x = 0.666...
Multiply by 10: 10x = 6.666...
Subtract: 10x − x = 6.666... − 0.666... → 9x = 6 → x = 6/9 = 2/3.

Example 7: Show that 0.101001000100001... is irrational.

Observe: The pattern is 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, ...
Key point: Although there is a pattern (increasing number of zeros between successive 1s), the decimal never repeats a fixed block.
Conclusion: Since the decimal is non-terminating and non-repeating, this number is irrational.

Example 8: Using the number line, show that every natural number is a rational number but not every rational number is a natural number.

Part 1: Any natural number n can be written as n/1, which is of the form p/q with q ≠ 0. So every natural number is rational.
Part 2: Consider the rational number 1/2. This is not a natural number because it is not a positive integer. On the number line, 1/2 lies between 0 and 1, not at any natural number position.
Similarly: −3 is rational (= −3/1) but not natural (negatives are not natural). 0 is rational (= 0/1) but not natural.
Conclusion: ℕ ⊂ ℚ but ℚ ⊄ ℕ. The set of rationals is strictly larger than the set of naturals.
📊 Chapter Summary
📋 Key Formulas & Facts
ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ The complete hierarchy of number systems
Rational: p/q (q ≠ 0) — terminates or repeats
Irrational: NOT p/q — non-terminating, non-repeating The two types of real numbers
📚 Quick Reference Table
Number Type Symbol Examples Closed Under
Natural ℕ 1, 2, 3, 4, ... + , ×
Whole W 0, 1, 2, 3, ... + , ×
Integer ℤ ..., −2, −1, 0, 1, 2, ... + , − , ×
Rational ℚ 1/2, −3/7, 0.75, 0.333... + , − , × , ÷ (except by 0)
Irrational — √2, π, √3, e Not closed under any operation
Real ℝ All of the above + , − , × , ÷ (except by 0)
⚠️ Common Mistakes to Avoid

❌ Thinking 22/7 = π

22/7 is only an approximation. 22/7 is rational (it is a fraction), but π is irrational. They are close but NOT equal.

❌ Saying √4 is irrational

√4 = 2, which is a natural number (and hence rational). Only square roots of non-perfect-squares are irrational.

❌ Dividing by zero

5/0 is undefined, not infinity or zero. Division by zero has no meaning in mathematics.

❌ Confusing "non-terminating" with "irrational"

0.333... is non-terminating but repeating, so it IS rational (= 1/3). A number is irrational only if it is non-terminating AND non-repeating.
💡 Revision Mantra:
• Natural → Counting from 1.
• Whole → Natural + 0.
• Integer → Whole + negatives.
• Rational → p/q, q ≠ 0 (terminates or repeats).
• Irrational → NOT p/q (never terminates, never repeats).
• Real → Rational + Irrational = everything on the number line.
🧠 Multiple Choice Questions (10 MCQs)

Click on an option to see if your answer is correct. The correct option will turn green.

  • Q1. Every natural number is a:
    • a) Whole number but not an integer
    • b) Integer but not a rational number
    • c) Rational number
    • d) Irrational number
    ✅ Answer: (c) Every natural number n can be written as n/1, making it rational.
  • Q2. Which of the following is irrational?
    • a) √9
    • b) √10
    • c) √16
    • d) √49
    ✅ Answer: (b) 10 is not a perfect square, so √10 is irrational. The others simplify to 3, 4, and 7.
  • Q3. The decimal expansion of a rational number is always:
    • a) Terminating only
    • b) Non-terminating and non-repeating
    • c) Either terminating or non-terminating repeating
    • d) Non-terminating only
    ✅ Answer: (c) Rational numbers have decimals that either terminate (like 0.25) or repeat (like 0.333...).
  • Q4. The additive identity for rational numbers is:
    • a) 0
    • b) 1
    • c) −1
    • d) 1/2
    ✅ Answer: (a) Adding 0 to any rational number gives the same number: a + 0 = a.
  • Q5. The multiplicative inverse of −2/7 is:
    • a) 2/7
    • b) −7/2
    • c) 7/2
    • d) −2/7
    ✅ Answer: (b) (−2/7) × (−7/2) = 14/14 = 1. The reciprocal keeps the sign.
  • Q6. Between any two rational numbers, there are:
    • a) Exactly one rational number
    • b) Exactly ten rational numbers
    • c) No rational numbers
    • d) Infinitely many rational numbers
    ✅ Answer: (d) This is the density property of rational numbers — between any two, infinitely many exist.
  • Q7. Which of the following is NOT a rational number?
    • a) 0
    • b) −17
    • c) 22/7
    • d) π
    ✅ Answer: (d) π is irrational. Note: 22/7 is rational (it is a fraction), but π ≠ 22/7.
  • Q8. The smallest whole number is:
    • a) 0
    • b) 1
    • c) −1
    • d) There is no smallest
    ✅ Answer: (a) The set of whole numbers W = {0, 1, 2, 3, ...} starts at 0.
  • Q9. The product of a non-zero rational number and an irrational number is always:
    • a) Rational
    • b) Irrational
    • c) Zero
    • d) An integer
    ✅ Answer: (b) For example, 2 × √3 = 2√3, which is irrational. A non-zero rational times an irrational is always irrational.
  • Q10. Which Indian mathematician first formally defined rules for zero and negative numbers?
    • a) Aryabhata
    • b) Brahmagupta
    • c) Baudhayana
    • d) Bhaskaracharya
    ✅ Answer: (b) Brahmagupta (628 CE) in his work Brahmasphutasiddhanta was the first to give formal rules for arithmetic with zero and negative numbers.
✍️ Fill in the Blanks (8 Questions)
  • 1. The collection of numbers 1, 2, 3, 4, ... is called __________ numbers.
    Natural numbers.
  • 2. A number that can be expressed in the form p/q, where p and q are integers and q ≠ 0, is called a __________ number.
    Rational number.
  • 3. The additive inverse of −7/3 is __________.
    7/3 (because −7/3 + 7/3 = 0).
  • 4. The multiplicative identity for rational numbers is __________.
    1 (because a × 1 = a for any rational a).
  • 5. √2 is an example of a(n) __________ number.
    Irrational number.
  • 6. Between any two rational numbers, there exist __________ many rational numbers.
    Infinitely many rational numbers.
  • 7. The decimal expansion of an irrational number is __________ and __________.
    Non-terminating and non-repeating.
  • 8. The set of all rational and irrational numbers together forms the __________ numbers.
    Real numbers.
✅ True or False (8 Questions)
  • 1. Every integer is a rational number.
    True. Any integer n can be written as n/1.
  • 2. Zero is a natural number.
    False. Zero is a whole number but not a natural number. Natural numbers start from 1.
  • 3. The sum of two irrational numbers is always irrational.
    False. Counter-example: (2 + √3) + (2 − √3) = 4, which is rational.
  • 4. √16 is an irrational number.
    False. √16 = 4, which is a natural number and hence rational.
  • 5. Every rational number has a multiplicative inverse.
    False. Zero is a rational number but has no multiplicative inverse (there is no number x such that 0 × x = 1).
  • 6. The number 0.10110111011110... is irrational.
    True. This decimal is non-terminating and non-repeating (the pattern of 1s changes), so it is irrational.
  • 7. Between 0 and 1, there are only finitely many rational numbers.
    False. Between any two rational numbers (including 0 and 1), there are infinitely many rational numbers.
  • 8. The product of two rational numbers is always rational.
    True. If a/b and c/d are rational (b, d ≠ 0), then (a/b) × (c/d) = ac/bd, which is also rational (bd ≠ 0).
📝 Short Answer Questions (8)
  • Q1. Define rational numbers. Give two examples.
    A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Examples: 3/4 and −5/2.
  • Q2. Is √3 rational or irrational? Justify.
    √3 is irrational. Since 3 is not a perfect square, √3 cannot be expressed as p/q for any integers p and q. Its decimal expansion (1.7320508...) is non-terminating and non-repeating.
  • Q3. Find the multiplicative inverse of −5/8.
    The multiplicative inverse of −5/8 is −8/5, because (−5/8) × (−8/5) = 40/40 = 1.
  • Q4. Find three rational numbers between 2/5 and 3/5.
    Write with a larger denominator: 2/5 = 8/20 and 3/5 = 12/20. Three rational numbers between them: 9/20, 10/20 (= 1/2), 11/20.
  • Q5. What is the difference between terminating and non-terminating repeating decimals?
    A terminating decimal has a finite number of digits after the decimal point (e.g., 0.75). A non-terminating repeating decimal goes on forever but has a block of digits that repeats (e.g., 0.333... where "3" repeats). Both represent rational numbers.
  • Q6. Express 0.overline{12} (0.121212...) as a fraction.
    Let x = 0.121212...
    100x = 12.121212...
    100x − x = 12 → 99x = 12 → x = 12/99 = 4/33.
  • Q7. State the closure property for integers under subtraction with an example.
    The set of integers is closed under subtraction: if a and b are integers, then a − b is also an integer. Example: 5 − 8 = −3 (an integer). This property does NOT hold for natural numbers (5 − 8 = −3, not natural).
  • Q8. Why is the set of natural numbers not closed under subtraction? Give an example.
    Because subtracting a larger natural number from a smaller one gives a negative number, which is not a natural number. Example: 3 − 7 = −4. Since −4 is not in the set {1, 2, 3, ...}, natural numbers are not closed under subtraction.
📖 Long Answer Questions (5)
Q1. Explain the hierarchy of number systems from natural numbers to real numbers. For each set, give its definition, symbol, examples, and state which operations it is closed under.
1. Natural Numbers (ℕ): {1, 2, 3, 4, ...} — counting numbers. Closed under addition and multiplication. Not closed under subtraction (3 − 5 = −2 ∉ ℕ) or division (5 ÷ 2 ∉ ℕ).

2. Whole Numbers (W): {0, 1, 2, 3, ...} — natural numbers plus zero. Closed under addition and multiplication. Not closed under subtraction or division.

3. Integers (ℤ): {..., −2, −1, 0, 1, 2, ...} — whole numbers plus negatives. Closed under addition, subtraction, and multiplication. Not closed under division (7 ÷ 3 ∉ ℤ).

4. Rational Numbers (ℚ): All p/q where p, q ∈ ℤ and q ≠ 0. Examples: 1/2, −3/7, 5 (= 5/1). Closed under all four operations (except division by zero). Decimal expansion is either terminating or repeating.

5. Irrational Numbers: Numbers that cannot be written as p/q. Examples: √2, π, √5. Decimal expansion is non-terminating and non-repeating. Not closed under any arithmetic operation (e.g., √2 × √2 = 2, which is rational).

6. Real Numbers (ℝ): ℚ ∪ Irrationals. Every point on the number line is a real number. Closed under addition, subtraction, multiplication, and division (except by 0).

Hierarchy: ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ. Each set is a proper subset of the next, meaning each extension adds new numbers to solve problems the previous set could not handle.
Q2. State and verify all the properties of rational numbers (closure, commutativity, associativity, identity, inverse, distributive) with respect to addition and multiplication. Give one example for each.
(i) Closure:
Addition: 2/3 + 4/5 = 22/15 (rational) ✅
Multiplication: 2/3 × 4/5 = 8/15 (rational) ✅

(ii) Commutativity:
Addition: 1/2 + 3/4 = 5/4 = 3/4 + 1/2 ✅
Multiplication: 2/3 × 5/7 = 10/21 = 5/7 × 2/3 ✅

(iii) Associativity:
Addition: (1/2 + 1/3) + 1/4 = 5/6 + 1/4 = 13/12 = 1/2 + (1/3 + 1/4) = 1/2 + 7/12 = 13/12 ✅
Multiplication: (1/2 × 2/3) × 3/4 = 1/3 × 3/4 = 1/4 = 1/2 × (2/3 × 3/4) = 1/2 × 1/2 = 1/4 ✅

(iv) Identity:
Additive identity: 3/7 + 0 = 3/7 ✅ (identity element is 0)
Multiplicative identity: 3/7 × 1 = 3/7 ✅ (identity element is 1)

(v) Inverse:
Additive inverse of 5/9 is −5/9: 5/9 + (−5/9) = 0 ✅
Multiplicative inverse of 5/9 is 9/5: 5/9 × 9/5 = 1 ✅

(vi) Distributive (multiplication over addition):
1/2 × (3/4 + 1/4) = 1/2 × 1 = 1/2
1/2 × 3/4 + 1/2 × 1/4 = 3/8 + 1/8 = 4/8 = 1/2 ✅
LHS = RHS, so the distributive property holds.
Q3. Prove that √2 is irrational. Also explain why √4 is NOT irrational.
Proof that √2 is irrational (by contradiction):

Assume √2 is rational. Then √2 = p/q, where p and q are co-prime integers (no common factor > 1) and q ≠ 0.

Squaring: 2 = p²/q² → p² = 2q².

Since p² is even (divisible by 2), p itself must be even (the square of an odd number is odd). Let p = 2m.

Substituting: (2m)² = 2q² → 4m² = 2q² → q² = 2m².

Since q² is even, q is also even. But now both p and q are even, meaning they share a common factor of 2. This contradicts our assumption that p/q is in lowest terms.

Therefore, √2 is irrational.

Why √4 is NOT irrational:
√4 = 2, because 2 × 2 = 4. The number 2 is an integer, which can be written as 2/1 (a fraction of two integers). Therefore, √4 is rational. In general, √n is rational if and only if n is a perfect square.
Q4. Find 6 rational numbers between 3/5 and 4/5. Explain the method you used and why infinitely many rational numbers exist between any two given rational numbers.
Method: To find 6 rational numbers between 3/5 and 4/5, we write equivalent fractions with a larger denominator.

Multiply numerator and denominator of both fractions by 7 (which is 6 + 1):
3/5 = 21/35 and 4/5 = 28/35.

Now we need 6 integers between 21 and 28: 22, 23, 24, 25, 26, 27.

The 6 rational numbers are: 22/35, 23/35, 24/35, 25/35 (= 5/7), 26/35, 27/35.

Why infinitely many exist: We could repeat this process between any two adjacent numbers we found. For example, between 22/35 and 23/35, we can find more by writing them as 220/350 and 230/350, then picking 221/350, 222/350, etc. This process can be repeated indefinitely. Mathematically, between any two rationals a and b (a < b), the number (a + b)/2 is also rational and lies strictly between them. Applying this repeatedly generates infinitely many distinct rational numbers.
Q5. Describe the contribution of Indian mathematicians to the development of the number system. Include at least three mathematicians and their specific contributions.
India has made foundational contributions to the number system that the entire world uses today:

1. Baudhayana (~800 BCE): In the Baudhayana Sulbasutras, he gave remarkably accurate approximations of √2 (correct to 5 decimal places: 1.4142156...). This shows awareness of irrational numbers nearly 2,300 years before European mathematicians formally recognised them. His work on the relationship between the diagonal and sides of a rectangle (Baudhayana-Pythagoras theorem) is foundational to the concept of irrational numbers arising from geometry.

2. Brahmagupta (598–668 CE): In his Brahmasphutasiddhanta (628 CE), he was the first mathematician in history to formally define zero as a number and give rules for arithmetic with zero. He also gave rules for negative numbers, calling positive numbers "fortunes" and negative numbers "debts." Without zero and negative numbers, the integer system (ℤ) and our place-value numeral system would not exist. He also stated rules like: "A debt subtracted from zero is a fortune" (−(−a) = a).

3. Aryabhata (476–550 CE): In his work Aryabhatiya, Aryabhata gave the approximate value of π as 3.1416, which is accurate to four decimal places. He introduced efficient algorithms for arithmetic using the decimal place-value system and contributed to algebra and astronomy. His work demonstrates early engagement with the concept of irrational numbers (π being irrational).

4. Bhaskaracharya (1114–1185 CE): Also known as Bhaskara II, he wrote Lilavati (on arithmetic) and Bijaganita (on algebra). He explored solutions to equations involving surds (square roots), showed properties of zero including that a/0 is infinite, and worked extensively with rational and irrational quantities. His work anticipated many concepts that were rediscovered in Europe centuries later.

The Hindu-Arabic Numeral System: The decimal positional system using digits 0–9 was developed in India, transmitted to the Arab world via traders and scholars (especially through the work of Al-Khwarizmi, who acknowledged the Indian origin), and then spread to Europe. This system is the most important contribution of India to world mathematics — it is the basis of all modern computing and science.
🃏 Quick Revision Flashcards

Use these flashcards for last-minute revision before your exam. Read the question, try to answer mentally, then check!

Q: What are natural numbers?

A: Counting numbers: 1, 2, 3, 4, ...

Q: Smallest whole number?

A: 0.

Q: Is 0 an integer?

A: Yes. 0 is an integer (neither positive nor negative).

Q: Is √5 rational?

A: No. 5 is not a perfect square, so √5 is irrational.

Q: Additive inverse of 3/7?

A: −3/7 (because 3/7 + (−3/7) = 0).

Q: Multiplicative inverse of −4?

A: −1/4 (because −4 × −1/4 = 1).

Q: Is 22/7 equal to π?

A: No! 22/7 is rational; π is irrational. 22/7 ≈ π but 22/7 ≠ π.

Q: How many rationals between 1/3 and 1/2?

A: Infinitely many.

Q: Who invented zero?

A: Brahmagupta (628 CE, India).

Q: ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ means?

A: Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real.

Q: 0.9999... = ?

A: 1. This is a repeating decimal equal to 1 exactly.

Q: Is √2 + √2 rational?

A: No. 2√2 is still irrational.
💡 Final Revision Mantra:
• Natural → Start from 1, count up.
• Whole → Add 0 to naturals.
• Integer → Add negatives to whole numbers.
• Rational → p/q; terminates or repeats.
• Irrational → Cannot be p/q; never terminates, never repeats.
• Real → All of the above combined.
• Density → Infinitely many rationals between any two rationals.

You have got this! Go ace that exam! 💪
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