Real Numbers for Class 8 — Rational, Irrational Numbers & Absolute Value | Math with Raabi

 

🌸 Math with Raabi • Class 8

Real Numbers for Class 8 — Rational, Irrational & Absolute Value

📖 Table of Contents

• 🌟 What Are Real Numbers?
• 🔍 Rational & Irrational Numbers
• 💖 Understanding Absolute Value
• 📝 Worked Example
• 🎯 Practice Exercise
• 🎮 Mini Interactive Quiz
• ❓ FAQ
• 🔗 Related Topics

In this lesson, we explore the complete structure of Real Numbers (ℝ) — how rational and irrational numbers combine to form the real number system, and how absolute values help us measure distance on the number line. This is a Class‑8 friendly explanation with solved examples, practice, and a mini interactive quiz. 🌼

🌸 Fig: Real Numbers — Rational, Irrational, Natural, Whole & Integers

🌟 What Are Real Numbers?

Real numbers include all numbers that can be placed on the number line — whether they are whole, fractional, positive, negative, rational, or irrational.

We represent the set of real numbers as: ℝ = Q ∪ Q'
Here:

• Q = Set of Rational Numbers
• Q' = Set of Irrational Numbers

Together, they form the entire real number system.

🔍 1. Rational & Irrational Numbers

🌼 Rational numbers can be written in the form p/q, where p and q are integers and q ≠ 0.

💗 Irrational numbers cannot be written in fraction form and have non-terminating, non-repeating decimal expansions. (e.g., √2, π)

💖 2. Understanding Absolute Value

Absolute value shows the distance of a number from 0 on the number line.

It is written as |x|.
For example:

• |12| = 12
• |-12| = 12

Absolute value is always a non-negative real number.

💡 Tip: Distance is always positive! That’s why absolute value never gives a negative answer.

💡 Did You Know?

The absolute value symbol | | is called a “modulus”. It tells us the distance of a number from zero — no matter which direction it is on the number line! That’s why the answer is always positive. 📏✨

📝 Worked Example: Solving |x - 2| + |x + 1| = 5

To solve expressions with two absolute values, we check all possible sign combinations.

Case 1: (x - 2) ≥ 0 and (x + 1) ≥ 0

(x - 2) + (x + 1) = 5 → x = 3

Case 2: (x - 2) ≥ 0 and (x + 1) < 0

Equation becomes impossible — results in a contradiction.

Case 3: (x - 2) < 0 and (x + 1) ≥ 0

Equation becomes impossible — results in a contradiction.

Case 4: (x - 2) < 0 and (x + 1) < 0

-(x - 2) - (x + 1) = 5 → x = -8

✅ Therefore, the solutions are: x = 3, x = -8

🌍 Real-Life Connection

Absolute value is used to find distance. Example: If Ali is at position 0 and his home is at -5, the distance is |-5| = 5 units on the number line. The number is negative only because of direction (left side of zero). Distance is always positive! 😊📏

📏 Why Ali’s Distance Can’t Be Negative

Distance tells us “how far” someone is from a point — not the direction. So even if Ali walks left or right on the number line, the distance is always a positive value. That is why absolute value never gives negative answers. ✔✨

⚠️ Common Mistakes to Avoid

• Absolute value is never negative.
• For |x| = 5, there are two possible values: x = 5 and x = -5.
• Students often write only +5 and forget -5.
• |-a| = a — the negative sign only shows direction, not distance.

Concept	Short Notes
Absolute Value	Distance from zero — always positive
Equation Rule	|A| = B → A = B or A = -B
Real-Life Use	Used in temperature, elevation & number line distance

🎯 Practice Exercise

1. Find the absolute values of the following numbers:
   • |-5|
   • |7|
   • |-2|
   • |-18|
   • |13|
   • |-0.5|
   • |-36|
   • |-4.8|
   • |6.18|
   • |14.18|
   • |-16.34|
   • |-42|
   • |-418.3|
   • |-35.49|
   • |-147|
   • |156|
   • |-5/7|
   • |-3/4|
2. Solve the equation: |x + 1| = 3

🌟 Tip for Students

When solving absolute value equations like |A| = B, always remember the two cases: A = B and A = -B. This helps you find both possible solutions. 🧠📘

📘 Worked Example

Solve the equation |x - 4| = 6.

1. Use the formula: |A| = B → A = B or A = -B.
2. So, x - 4 = 6 → x = 10
3. And, x - 4 = -6 → x = -2

✔ Final Answer: x = 10 or x = -2

🎮 Mini Interactive Quiz

Try solving it on your own before clicking "Show Answer"! ✏️

Click "Show Answer" to reveal the step-by-step solution. 🌸

Q1. What is |-5|?

Step 1: |-5| means distance from 0.
Step 2: Distance is always positive.
✅ Answer: 5

Q2. √2 is rational or irrational?

Step 1: A rational number can be written as p/q where q ≠ 0.
Step 2: √2 = 1.41421… (non-terminating, non-repeating).
✅ Therefore, √2 is irrational.

Q3. Find |7|.

Step 1: |7| means distance from 0.
Step 2: Distance is always positive.
✅ Answer: 7

Q4. Solve |x + 1| = 3.

Step 1: |x + 1| = 3 → x + 1 = 3 or x + 1 = -3
Step 2: x + 1 = 3 → x = 2
Step 3: x + 1 = -3 → x = -4
✅ Therefore, x = 2 or x = -4

Q5. Which sets include 0?

Step 1: Natural numbers do NOT include 0.
Step 2: Whole numbers start from 0.
Step 3: Integers include negative numbers, 0, and positive numbers.
Step 4: Real numbers include all numbers on the number line.
✅ Therefore, sets that include 0: Whole Numbers, Integers, Real Numbers

🧑‍🏫 Teacher’s Note

Encourage students to sketch a simple number line when learning absolute value. Visual understanding greatly improves accuracy and confidence in solving equations. ✏️📏

🔍 Quick Summary

• Real Numbers include both Rational and Irrational numbers.
• Absolute value measures the distance from 0.
• Distance can never be negative.
• |A| = B → A = B or A = -B.

❓ Frequently Asked Questions (FAQ)

Q1: What are Real Numbers in Class 8?

Real numbers include all rational and irrational numbers shown on the number line.

Q2: What is the absolute value of a number?

Absolute value is the distance of a number from zero on the number line.

Q3: Is √2 rational or irrational?

√2 is an irrational number because it cannot be expressed as a fraction.

Q4: Which number sets include 0?

Whole numbers, integers, and real numbers include 0. (Natural numbers do NOT include 0.)

Q5: Are absolute values always positive?

Yes, because distance on the number line can never be negative.

🔗 Related Topics

Rational & Irrational Numbers — Simple Real Numbers — Decimal Fractions Venn Diagrams — Types of Sets De Morgan's Laws — Complement & Sets

🌸 By practicing these exercises and understanding the concepts above, you will master Class 8 Real Numbers with confidence. Bookmark this page and revisit the quizzes to strengthen your skills!

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