hsanotes.com notes – short, clear, exam-focused – designed to help score 90+ marks in Punjab board exams.
Features of hsanotes.com notes
- Very short definitions
- Step-by-step solved answers
- Quick tricks & memory aid
- Exact PCTB / PECTAA book alignment
- Number line constructions simplified
- Past paper pattern questions included
- Fast revision format
- Trusted by thousands of Punjab students
For all Punjab boards (Lahore, Gujranwala, Rawalpindi, Faisalabad, Bahawalpur, Multan, Sargodha, Sahiwal, DG Khan) and top schools: The City School, Beaconhouse School System, Roots International Schools, Lahore Grammar School (LGS), Karachi Grammar School, Froebel’s International School, The Educators, Army Public School (APS), Divisional Public School, Government Model School.
Definitions
Rational number → p/q (p, q integers, q ≠ 0)
Irrational number → cannot be written as p/q; non-terminating & non-repeating decimal
Exercise 1.1 – Identify Rational or Irrational
(i) 2.353535… → Rational (terminating decimal)
(ii) 0.6̅ → Rational (repeating decimal)
(iii) 2.236067… → Irrational (non-repeating, non-terminating ≈ √5)
(iv) √7 → Irrational (7 not perfect square)
(v) e → Irrational (known constant)
(vi) π → Irrational (known constant)
(vii) 5 + √11 → Irrational (rational + irrational = irrational)
(viii) √3 + √13 → Irrational (sum of irrationals)
(ix) 15/4 → Rational (fraction)
Exercise 1.1 – Represent on Number Line
Numbers to mark:
(i) √2
(ii) √3
(iii) 4 1/3
(iv) –2 1/7
(v) 5 1/8
(vi) 2 3/4
Quick method
- Irrational (√2, √3): Draw right triangle with legs 1 unit → hypotenuse = √2 or √3 → use compass from 0 to mark.
- Rational/mixed numbers: Convert to improper fraction or decimal → divide unit segment into equal parts.
Full step-by-step constructions and diagrams are on original PDF pages 3–4.
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Keywords
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