Irrational Numbers, √2, π, and the Number Line

• Define irrational numbers and distinguish them from rationals (NCERT Ch. 1).
• Recognise familiar irrationals: √2, √3, √5, and π.
• Understand that decimal expansions are non-terminating and non-repeating.
• Simplify surds like √50 = 5√2 using prime factorisation.
• Locate √2 and √3 on the number line using geometric constructions.
• Apply rules: rational × non-zero irrational → irrational; sum/product properties.

1. Definition — A number is irrational if it cannot be written as p/q where p, q are integers and q ≠ 0.
2. Decimal form — Irrationals have decimals that never terminate and never show a repeating block.
3. Square roots — √n is rational only when n is a perfect square (1, 4, 9, 16, …). Otherwise √n is irrational.
4. Famous constants — π (circumference ÷ diameter) and √2 (diagonal of unit square) are irrational.
5. Simplifying surds — √72 = √(36×2) = 6√2; factor out largest perfect square.
6. Operations — √2 × √2 = 2 (rational). 3√5 remains irrational. √2 + (−√2) = 0 shows sums need care.
7. Number line — NCERT: construct unit square, diagonal length √2; extend to locate √3.
8. Real numbers — Every point on the number line is either rational or irrational (together: reals).
9. Applications — Diagonal of 3 m × 3 m room = 3√2 m; circle circumference C = 2πr.
10. NCERT Ex. 1.2 — Identify rational/irrational; represent √9.3 on number line approximately.

NCERT: Show √50 is irrational and simplify it

Step 1 — Factorise: √50 = √(25 × 2) = √25 × √2 = 5√2
Step 2 — 5 is rational; √2 is irrational (2 is not a perfect square)
Step 3 — Non-zero rational × irrational = irrational
Conclusion: √50 = 5√2 is irrational
Check: (5√2)² = 25 × 2 = 50 ✓
Application: Square tile side 5 cm → diagonal = 5√2 ≈ 7.07 cm

• Treating 22/7 as exactly equal to π (it is only an approximation).
• Assuming every root is irrational (√16 = 4 is rational).
• Claiming sum of two irrationals is always irrational (√2 + (−√2) = 0).
• Rounding √2 to 1.4 too early in multi-step surd simplification.
• Forgetting principal square root is non-negative: √(x²) = |x|.

• Is √45 rational or irrational? Simplify it.
• Between which consecutive integers does √20 lie?
• Simplify √8 + √2.
• Why is π not expressible as p/q?
• Locate √2 on the number line — what triangle is used?

Open the Practice tab for graded questions on Irrational Numbers, √2, π, and the Number Line.