Math Notes For 9th Class Chapter 1: Real Numbers Easy Notes Solved Examples & Exam Tricks (Punjab Board 2025 2026)

Salaam! If you’re in 9th class right now and just opened Chapter 1 let me guess you’re thinking: “Yeh rational irrational wala chapter kitna boring lag raha hai…” 😅

But trust me bhai/behen this is actually one of the most scoring chapters if you understand it properly. Once you get the hang of it proofs become easy MCQs fly by and you start feeling confident for the whole book.

In this post I’m going to explain everything step-by-step like I explain to my own students no heavy language lots of examples common mistakes and exactly what comes in board exams every year.

Quick reality check for 9th graders:

• This chapter usually gives 8 12 marks (MCQs + short + long)
• √2 or √3 irrational proof almost 100% aata hai
• Number line wala diagram draw karna seekh lo 2 marks free
• Most students lose marks in proofs because they skip steps

Real Numbers Ko Samajhna Kitna Simple Hai?

Basically number line pe jitne bhi points hain woh sab real numbers hain.

Ab real numbers do types mein divide hote hain:

Rational Numbers

Jo p/q ke form mein likh sakein (p aur q integers q ≠ 0).

Examples jo har saal exam mein confuse karte hain:

• 3/4 → rational
• 0.75 → rational (terminating)
• 0.333… (1/3) → rational (repeating)
• 7 → rational (7/1)

Irrational Numbers

Jo kabhi bhi p/q mein nahi likh sakte decimal infinite aur non-repeating hota hai.

Famous examples jo board mein bohot aate hain:

• √2 ≈ 1.414213562…
• √3 ≈ 1.7320508…
• π ≈ 3.1415926535…
• √5 √7 etc.

Sabse Important Proof: √2 Irrational Hai (Yaad Rakhna!)

Yeh question har saal aata hai 5 ya 7 marks ka. Thoda dhyan se samjho ek baar samajh aa gaya toh life set!

Assume karo √2 rational hai → √2 = p/q (p aur q co-prime hain q ≠ 0)

⇒ 2 = p² / q²

⇒ p² = 2q² (p² even hai → p bhi even hoga)

Let p = 2k → (2k)² = 2q² → 4k² = 2q² → q² = 2k²

⇒ q² even → q bhi even

Ab dono p aur q even hain… lekin humne assume kiya tha woh co-prime hain. Contradiction!

Therefore √2 is irrational. Q.E.D. ✅

Tip: Har step likho “even → divisible by 2” wali baat highlight karo examiner impress ho jata hai.

Number Line Pe Irrational Numbers Kaise Dikhayein?

√2 draw karna bohot easy hai compass use karo:

1. Ek line banao AB = 1 unit
2. B pe perpendicular BC = 1 unit khicho
3. AC join karo → AC = √2 (Pythagoras se)
4. Compass se AC ki length ko number line pe mark kar do

Same method se √3 √5 sab bana sakte ho. Exam mein diagram saaf-saaf banao 2 marks pakke!

Exam Mein Aane Wale Top 8 MCQs (2024 2025 Papers Se Inspired)

1. Which one is irrational?

a) 0.25 b) 1/3 c) √4 d) √5

Answer: d) √5

2. Decimal expansion of rational number is always:

a) Terminating or repeating b) Non-terminating non-repeating c) Only terminating

Answer: a)

3. Between two rational numbers how many rationals are there?

Answer: Infinite

Short & Long Questions Jo Har Saal Repeat Hote Hain

• Differentiate rational and irrational numbers with two examples each. (3 marks)
• Prove that √3 is irrational. (Same method as √2 bas 3 ki jagah 2 likh dena 😄) (5 marks)
• Describe the method to represent √5 on the number line with steps. (Diagram zaroor banana) (4 marks)
• Give examples of terminating and non-terminating repeating decimals. (2 marks)

Stuck? Tension Na Lo!

Agar yeh chapter ab bhi mushkil lag raha hai ya live samajhna chahte ho hamare expert tutors se free demo le lo. WhatsApp pe message karo 5 minute mein reply aa jayega. Free Demo Book Karo

Roz 15 20 minutes practice karo yeh chapter clear ho gaya toh pura saal confidence rahega.
Allah hafiz aur best of luck for 9th board exams 2026! 🌟