Mathematics : Chapter 2: New Numbers

 📂 Topic 1: Understanding Irrational Numbers through Geometry
🔸 Concept:

Certain lengths like the diagonal of a square of side 1 cannot be expressed as fractions. Their square roots (like √2, √3) are irrational numbers.
🔹 Example 1: Diagonal of a Unit Square

Problem: A square has side length 1. Find the diagonal length.

Solution:

    A square’s diagonal splits it into two right triangles.

    Use Pythagoras Theorem:
    Diagonal² = side² + side²
    = 1² + 1² = 2
    ⇒ Diagonal = √2 ≈ 1.414 (irrational number)

🔹 Example 2: Height of an Equilateral Triangle of Side 2

Problem: Find the height of an equilateral triangle with side 2 units.

Solution:

    Split it into two right triangles:
    Base = 1, Hypotenuse = 2

    Use Pythagoras:
    Height² + 1² = 2²
    ⇒ Height² = 4 − 1 = 3
    ⇒ Height = √3 ≈ 1.732

🔹 Example 3: Side of a Cube of Volume 2

Problem: A cube has volume 2 cubic cm. Find the side length.

Solution:

    Volume = side³ ⇒ side = ∛2
    ∛2 ≈ 1.26 (irrational number)

📂 Topic 2: Approximation of Irrational Numbers (like √2)
🔸 Concept:

We can find decimal approximations of irrational numbers using trial and error or calculators.
🔹 Example 1: Approximate √2

Try decimals:

    1.4² = 1.96

    1.41² = 1.9881

    1.414² = 1.999396
    So, √2 ≈ 1.414 (correct to 3 decimal places)

🔹 Example 2: Approximate √3

Try:

    1.7² = 2.89

    1.73² = 2.9929

    1.732² = 2.999824
    So, √3 ≈ 1.732

🔹 Example 3: Approximate ∛2

Try:

    1.25³ = 1.953125

    1.26³ = 2.000376
    So, ∛2 ≈ 1.26

📂 Topic 3: Proof that √2 is Irrational
🔸 Concept:

We can prove √2 is irrational using contradiction.
🔹 Example 1: Proof Using Lowest Terms

Assume √2 = a/b (in lowest terms)
⇒ a² = 2b² ⇒ a² is even ⇒ a is even ⇒ a = 2k
⇒ (2k)² = 2b² ⇒ 4k² = 2b² ⇒ b² = 2k² ⇒ b is even
⇒ Both a and b are even — contradiction
∴ √2 is irrational
🔹 Example 2: Decimal Check

Observe:

    √2 = 1.4142135... (non-repeating, non-terminating)
    So, it is not a rational number

🔹 Example 3: Calculator Method

Using calculator:

    √2 = 1.414213562...
    No pattern emerges ⇒ irrational

📂 Topic 4: Decimal Expansion: Rational vs Irrational
🔸 Concept:

Rational numbers have terminating or repeating decimals. Irrational numbers do not.
🔹 Example 1: Rational – Terminating

1/4 = 0.25 (terminates) ⇒ rational
🔹 Example 2: Rational – Repeating

1/3 = 0.333... (repeats) ⇒ rational
🔹 Example 3: Irrational

√5 ≈ 2.2360679... (non-repeating) ⇒ irrational
📂 Topic 5: Use of Roots in Real Problems
🔸 Concept:

We often encounter roots in real-life geometry and measurement.
🔹 Example 1: Perimeter of a Right Triangle

Sides = 1, 1, and hypotenuse = √2
Perimeter = 1 + 1 + √2 ≈ 3.414
🔹 Example 2: Adding Roots

Find 2 + √3
≈ 2 + 1.732 = 3.732
🔹 Example 3: Difference of Roots

Find √3 − √2
≈ 1.732 − 1.414 = 0.318
📂 Topic 6: Symbolic Representation of Roots
🔸 Concept:

We use √x to represent square roots and ∛x for cube roots.
🔹 Example 1: √9 = 3

Because 3² = 9
🔹 Example 2: √(1/4) = 1/2

Because (1/2)² = 1/4
🔹 Example 3: ∛8 = 2

Because 2³ = 8

🔍 Summary

This section introduces a critical mathematical concept—New Numbers, particularly irrational numbers—through geometrical and algebraic reasoning. Starting with the diagonal of a unit square and extending to equilateral triangles and cube roots, students discover that not all lengths can be expressed as fractions or rational numbers. The lesson motivates the need for irrational numbers using real-life geometry-based scenarios (like side lengths, diagonals, and cube volumes), eventually leading to the symbolic representation of square roots and cube roots.

Key topics include:

    Understanding that √2, √3, etc., cannot be expressed as fractions.

    Introducing irrational numbers and decimal approximations.

    Applying Pythagoras’ Theorem for area and length calculation.

    The historical development of number systems, from natural to irrational numbers.

    Decimal approximations and comparing roots with decimals.

The section emphasizes conceptual clarity by using logical contradiction, geometrical construction, and real-life implications, preparing students to embrace irrational numbers with confidence.
📌 Capsule Notes

    Diagonal of a Square with side 1 = √2; cannot be a rational number.

    No fraction exists whose square equals 2 or 3 (proved via contradiction).

    Cube root of 2 (∛2) also not a rational number.

    Irrational numbers: Numbers that cannot be expressed as p/q.

    Examples of irrational numbers: √2, √3, √5, ∛2, π.

    Approximation of √2:

        √2 ≈ 1.4 (1 decimal),

        ≈ 1.41 (2 decimals),

        ≈ 1.4142 (4 decimals).

    √x means the side of a square with area x.

    Decimal representation of √2 never terminates or repeats.

    Rational numbers like 1/3 = 0.333... (repeating), but √2 = 1.414213... (non-repeating).

    Addition/Subtraction of roots possible (e.g., √2 + 2 ≈ 3.414).

    Historical note: Pythagoreans believed all measurements are rational—Hippasus disproved this.

❓ Questions with Answers (Q&A)

    Q: What is the length of the diagonal of a square with side 1 unit?
    A: √2 units.

    Q: Is √2 a rational number?
    A: No, it is an irrational number.

    Q: Approximate value of √2 up to 2 decimal places?
    A: 1.41

    Q: Why can't √2 be a fraction?
    A: Because no fraction (p/q) squared equals 2; it leads to contradiction.

    Q: What is the height of an equilateral triangle of side 2 units?
    A: √3 units.

    Q: True or False: All decimals are rational.
    A: False. Non-repeating, non-terminating decimals are irrational.

    Q: Write √2 approximately using "≈" symbol.
    A: √2 ≈ 1.414

    Q: What is the cube root of 2 written symbolically?
    A: ∛2

    Q: What is meant by irrational number?
    A: A number that cannot be written as a fraction (p/q) and has a non-repeating, non-terminating decimal.

    Q: Is 0.25 a rational number? Why?
    A: Yes, because it can be written as 1/4 (a fraction).

    Q: What is √4? Is it rational?
    A: √4 = 2; Yes, it is rational.

    Q: What is the perimeter of a triangle with sides 1, √2, and 2 units?
    A: 1 + √2 + 2 ≈ 4.414 units

    Q: Can repeating decimals be irrational?
    A: No, repeating decimals are rational.

    Q: Define "≈" symbol.
    A: It means "approximately equal to".

    Q: What is the approximate value of √3 up to 3 decimal places?
    A: 1.732

📚 Definitions & Key Terms

    Irrational Number: A number that cannot be expressed as a fraction (p/q). Eg: √2, π.

    Rational Number: A number that can be written as a fraction. Eg: 1/2, 0.75

    Square Root (√x): A number which, when squared, gives x.

    Cube Root (∛x): A number which, when cubed, gives x.

    Approximation (≈): A value that is close to, but not exactly equal.

    Non-Terminating Decimal: A decimal that goes on forever.

    Repeating Decimal: A decimal with a pattern of digits that repeats.

    Perfect Square: A number that is a square of a whole number (e.g., 1, 4, 9).

    Pythagoras Theorem: In a right triangle, hypotenuse² = base² + height².

🧠 Main Points to Remember (For Revision)

    √2 ≈ 1.414 (most important irrational number in geometry).

    No fraction squared = 2 ⇒ √2 is irrational.

    All square roots of non-perfect squares are irrational.

    Decimal of √2 = 1.4142135… (non-repeating, non-terminating).

    ∛2 is also irrational—no fraction cubed equals 2.

    Symbol “≈” is used for approximate equality.

    Irrational numbers help describe lengths like diagonals, cube sides, etc.

    Rational decimals: 0.5, 0.25; Irrational decimals: √2, √3.

    Use calculator to approximate roots for perimeter or area calculations.

📂 Topic-Wise Breakdown
1. Discovery of √2 and Irrationality

    Diagonal of square (side 1) = √2

    √2 cannot be expressed as a fraction

    Contradiction proof using even/odd properties

2. Other Irrational Lengths

    Height of equilateral triangle with side 2 = √3

    Side of cube with volume 2 = ∛2

3. Introduction to New Numbers

    Rational vs. Irrational numbers

    Decimal approximations

    Decimal expansion of irrational numbers ≠ repeating

4. Using Approximations

    √2 ≈ 1.41, √3 ≈ 1.732

    Calculating perimeter using approximate root values

5. Historical Note

    Pythagorean belief in rational measurements

    Hippasus discovered irrationality

6. Decimal Patterns

    Rational: Repeating or terminating

    Irrational: Non-repeating, non-terminating