🔢 Rational & Irrational Numbers

Class 8
Chapter 3
Score: 0
✍️ R.P. Pandey
🖥️ PROJECTOR MODE — High Contrast · Large Text · Bright Background
SECTION 3.0

🔁 पुनरवलोकन — Review

Let's warm up! Work through these review questions to prepare for the lesson.

What is a Rational Number? (आनुपातिक सङ्ख्या)

If a and b are integers and b ≠ 0, then any number that can be expressed in the form a/b is called a Rational Number.
Examples: 2, 5, −7, 5/8, 2.13, 1.6̄

Review Question (क) — Identify Rational Numbers

Which of these represent rational numbers? Click each to classify:

21/4
Avg. Nepali height (ft)
0°C
Water freezes
200 m/min
Runner's speed
−4/7
Fraction
−1
Integer
4/5
Fraction
👆 Click each number to confirm it's rational — all 6 are!

Review Question (आ) — Decimal Types

Convert these fractions to decimals, then identify the type:

FractionDecimalTypeRational?
4/50.8Terminating✅ Yes
5/31.6̄6̄6̄…Non-terminating recurring✅ Yes
4/70.571428…̄Non-terminating recurring✅ Yes
7/23.5Terminating✅ Yes
5/90.5̄5̄5̄…Non-terminating recurring✅ Yes
💡 Key Insight
Both terminating decimals and non-terminating recurring decimals can be converted to fractions → they are Rational. Numbers that are non-terminating AND non-recurring cannot → they are Irrational.
SECTION 3.1

📖 Rational & Irrational Numbers — Concepts

Understand definitions, examples, and the difference between rational and irrational numbers.

👩‍🏫
Teacher (गुरुआमा): Let's investigate whether √2, √(4/9), 0.25, 4.66…, and π are rational or irrational. First, let's explore each case.

🔬 क्रियाकलाप 1 — Activity 1: Investigate Each Case

√4
√(4/9)
0.25
4.6666…
√2
π
👩‍🏫
√4 = 2
2 can be written as 2/1, 4/2, 6/3, … — all in the form a/b.
∴ √4 = 2 is a Rational Number ✅

📜 Formal Definitions

Rational Number (आनुपातिक सङ्ख्या):
A number that can be expressed as a/b where a, b are integers and b ≠ 0.
Examples: 2, 5, −7, 5/8, 2.13, 1.6̄, 0.75, 0.3̄
Irrational Number (अनानुपातिक सङ्ख्या):
A number that CANNOT be expressed as a/b. These are non-terminating, non-recurring decimals.
Examples: √2, √5, ∛10, √(1/3), √7, 2.134…, π

📊 Decimal Classification Table

Decimal TypeExample→ Fraction?Classification
Terminating0.75, 0.25, 3.5✅ YesRational
Non-terminating Recurring0.3̄, 4.66…, 0.41̄✅ YesRational
Non-terminating Non-recurring√2=1.4142…, π=3.1415…❌ NoIrrational
SECTION 3.1 (continued)

🌳 Real Number System — The Number Tree

See how all number types relate to each other from क्रियाकलाप 2.

Real Numbers (वास्तविक सङ्ख्या) Hierarchy

The Real Numbers (R) are made up of Rational (Q) and Irrational (Ir) numbers. The relationship is: N ⊆ W ⊆ Z ⊆ Q ⊆ R, Ir ⊆ R

वास्तविक सङ्ख्या (R) आनुपातिक सङ्ख्या (Q) अनानुपातिक सङ्ख्या (Ir) पूर्णाङकहरू (Z) भिन्न सङ्ख्याहरू (F) पूर्ण सङ्ख्या (W) ऋणात्मक सङ्ख्या प्राकृतिक (N) शून्य (0) √2, √5, ∛10, π, 2.134… N ⊆ W ⊆ Z ⊆ Q ⊆ R Ir ⊆ R

🔵 Venn Diagram — Set Relationships

R Q Z W N 1,2,3… Ir √2,π,√5…
Real Numbers (R) = Rational Numbers (Q) ∪ Irrational Numbers (Ir)
where Q and Ir are disjoint sets (no overlap).
ACTIVITY

🎯 Drag & Drop Classifier

Classify each number as Rational (Q) or Irrational (Ir). Drag numbers into the correct box!

🃏 Drag Pool — Drag to classify

−5/2
√7
√5
2/5
10/20
3.57
3.5982…
−15
0.735…
−√169
√3
√26
2.5̄
35/9
∛9=2.08…→Q?

✅ RATIONAL (Q)

❌ IRRATIONAL (Ir)

⚡ Quick Fire — Rational or Irrational?

For each number, click the correct answer. Get a streak going!

0 STREAK 🔥
PRACTICE

🔄 Decimal ↔ Fraction Converter

Convert decimals to fractions using the algebraic method from the textbook (उदाहरण 2).

📐 Method: Converting Recurring Decimals to Fractions

One repeating digit (e.g. 0.3̄): Multiply by 10, subtract, solve for x.
Example: x = 0.3̄ → 10x = 3.3̄ → 9x = 3 → x = 1/3

Two repeating digits (e.g. 0.41̄): Multiply by 100, subtract, solve for x.
Example: x = 0.41̄ → 100x = 41.41̄ → 99x = 41 → x = 41/99

🖩 Interactive Converter

Enter a decimal below. Use a bar notation like 0.3r for 0.3̄ (one repeating digit) or 0.41r for 0.41̄
DECIMAL INPUT
→
FRACTION
—

📝 Practice — Exercise 3.1 Q3 (Textbook)

Convert these decimals to fractions (answers shown after 5 seconds):

ACTIVITY — क्रियाकलाप 3

📏 Plotting Irrationals on the Number Line

Using the Pythagorean theorem to locate √2, √3, √5 on a number line — exactly as taught in class!

👩‍🏫
Teacher: We can't mark 1.41421… directly on the number line. But using Pythagoras (p² + b² = h²), if p=1 and b=1, then h = √2. We draw a right triangle and use the hypotenuse as radius in a compass to mark √2 on the number line!

🖱 Interactive Number Line — Click to plot!

Click a button above to see the construction!

📐 How it works: Spiral of Theodorus

To plotRight triangle neededHypotenuse = ?
√2legs: 1, 1√(1²+1²) = √2 ✓
√3legs: √2, 1√(√2²+1²) = √3 ✓
√4 = 2legs: √3, 1√(√3²+1²) = √4 = 2 ✓
√5legs: 2, 1√(2²+1²) = √5 ✓
√nlegs: √(n−1), 1√(n−1+1) = √n ✓
QUIZ — अभ्यास 3.1

❓ Test Your Knowledge

Based on Exercise 3.1 from the textbook. Answer all questions, then check your score!

0
ANSWERED
0
CORRECT
0
WRONG
SUMMARY

🏆 Lesson Summary & Checklist

Review everything you've learned in Chapter 3 — Rational and Irrational Numbers.

📋 Learning Checklist

  • I can define Rational Numbers with examples
  • I can define Irrational Numbers with examples
  • I can identify terminating decimals
  • I can identify non-terminating recurring decimals
  • I can identify non-terminating non-recurring decimals
  • I can convert decimals to fractions using the algebraic method
  • I can plot √2 on the number line using a right triangle
  • I can draw the Venn diagram of real numbers
  • I understand why π ≈ 22/7 is used even though π is irrational
  • I can write numbers in scientific notation

⭐ Quick Reference Card

Rational (Q): Can be written as a/b (b≠0)
→ Integers, fractions, terminating decimals, recurring decimals

Irrational (Ir): Cannot be written as a/b
→ Non-terminating, non-recurring decimals
→ Examples: √2, √3, √5, π, ∛10

Real Numbers (R): Q ∪ Ir
Subset chain: N ⊆ W ⊆ Z ⊆ Q ⊆ R, Ir ⊆ R

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📚 Project Work (परियोजना कार्य)

1. On chart paper with 2cm = 1 unit, draw the number line and plot √2 and √3 using a compass. Present in class.

2. Take any 5 rational numbers. Convert to decimals. Identify whether they are terminating or non-terminating recurring decimals.