Vardaan
Class 9 Maths • Chapter 1 • NCERT + RS Aggarwal + RD Sharma

Number Systems

Vardaan Learning Institute  |  School-Exam Focused Notes

?? 1. The Number System — Hierarchy

Real Numbers (R) Rational Numbers (Q) — p/q, q?0 Integers (Z) Whole Numbers (W) Natural Numbers (N) 1, 2, 3, 4, 5 ... + 0 + Negatives + Fractions & Decimals e.g. ½, -¾, 0.25, 3.333... Irrational Numbers v2, v3, v5, p, e Non-terminating, non-repeating decimals p = 3.14159265... v2 = 1.41421356...
Type Definition Examples Symbol
Natural Numbers Counting numbers starting from 1 1, 2, 3, 4, 5... N
Whole Numbers Natural numbers + zero 0, 1, 2, 3... W
Integers Whole numbers + negative numbers ...-2, -1, 0, 1, 2... Z
Rational Numbers Numbers of the form p/q where p,q?Z and q?0 ½, -¾, 0.25, 3, -7 Q
Irrational Numbers Non-terminating, non-repeating decimals. Cannot be written as p/q. v2, v3, p, e, v5 —
Real Numbers All rational + irrational numbers Every number on the number line R

?? 2. Rational Numbers — Key Properties

Identifying Rational from its Decimal Expansion:
• Terminating decimal ? Rational. Example: 0.25 = 25/100 = 1/4
• Non-terminating but repeating (recurring) ? Rational. Example: 0.3¯ = 1/3, 0.142857142857... = 1/7
• Non-terminating, non-repeating ? Irrational. Example: p = 3.14159..., v2 = 1.41421...

Converting Recurring Decimals to p/q Form (Very Important for Exams!)

Example 1: Convert 0.3¯ = 0.333... to p/q
Let x = 0.333...  ?  10x = 3.333...
10x - x = 3.333... - 0.333...  ?  9x = 3  ?  x = 1/3

Example 2: Convert 0.47¯ = 0.4777... to p/q
Let x = 0.4777...  ?  10x = 4.777...  ?  100x = 47.777...
100x - 10x = 47.777... - 4.777... = 43  ?  90x = 43  ?  x = 43/90

Example 3: Convert 0.6¯7¯ = 0.676767... to p/q
Let x = 0.676767...  ?  100x = 67.6767...
100x - x = 67  ?  99x = 67  ?  x = 67/99
?? Exam Shortcut For 0.a¯b¯... (k digits repeating from start): p/q = (repeating block) / (k nines)
e.g. 0.12¯3¯ ? digits before decimal don't repeat ? use subtraction method.

?? 3. Irrational Numbers — Key Concepts

Important Results on Irrationals (RD Sharma / Board exams):
?? Theorem (NCERT) — Prove v2 is irrational
Proof by Contradiction:
Assume v2 is rational. Then v2 = p/q where p, q are integers, q ? 0, and p/q is in lowest terms (HCF(p,q) = 1).
Squaring: 2 = p²/q²  ?  p² = 2q²
So p² is even  ?  p is even  ?  p = 2m for some integer m
Substituting: (2m)² = 2q²  ?  4m² = 2q²  ?  q² = 2m² ? q is even
But then both p and q are even, contradicting HCF(p,q) = 1.
? v2 is irrational.

Similarly v3, v5, v7 can be proved irrational. (Same method — important for board exams!)

?? 4. Representing Irrationals on Number Line

Method: Geometric construction (Spiral of Theodorus)

To represent v2 on a number line:
1. Draw OA = 1 unit on number line.
2. At A, draw AB ? OA such that AB = 1 unit.
3. OB = v(OA² + AB²) = v(1+1) = v2 (by Pythagorean theorem).
4. With O as centre and OB as radius, draw arc to cut number line at C. OC = v2.

For v3: From v2 point, draw perpendicular of 1 unit. Hypotenuse = v3.
For vn: Continue the spiral — from v(n-1) point, draw perpendicular of 1 unit; hypotenuse = vn.
0 1 1 1 v2 v2 v3 v3 Blue ? ? v2 Orange ? ? v3

?? 5. Operations on Real Numbers

Laws of Radicals (Very Frequently Tested)

Law Rule Example
Product Rule va × vb = v(ab) v3 × v5 = v15
Quotient Rule va ÷ vb = v(a/b) v18 / v2 = v9 = 3
Power Rule (va)² = a (v7)² = 7
a? × an = a??n 3² × 3³ = 35 = 243
a? ÷ an = a??n 56 ÷ 5² = 54 = 625
(a?)n = a?n (2³)² = 26 = 64
a° = 1 (a ? 0) 7° = 1
a?n = 1/an 2?³ = 1/8
a^(1/n) = nva 8^(1/3) = ?8 = 2
a^(m/n) = (nva)? = nv(a?) 8^(2/3) = (?8)² = 4

Rationalisation of Denominators (Very Important!)

The process of making the denominator free of radicals is called rationalisation.

Type 1 — Single radical in denominator:
1/v2 = 1/v2 × v2/v2 = v2/2

Type 2 — (a + vb) in denominator:
Multiply by conjugate: (a - vb)  ?  uses identity (a+b)(a-b) = a²-b²
1/(3+v2) = (3-v2)/[(3+v2)(3-v2)] = (3-v2)/(9-2) = (3-v2)/7

Type 3 — (va + vb) in denominator:
1/(v5+v3) = (v5-v3)/[(v5+v3)(v5-v3)] = (v5-v3)/(5-3) = (v5-v3)/2

Comparing Irrational Numbers

To compare va and vb: compare a and b. v5 > v3 since 5 > 3.
To compare va and n (a whole number): square both. Which is bigger, v7 or 2.5? ? 7 vs 6.25 ? v7 > 2.5
To compare nva and mvb where n ? m: make indices same using LCM.
Example: 2^(1/2) vs 3^(1/3). LCM of 2,3 = 6. ? 2^(3/6) vs 3^(2/6) ? 8^(1/6) vs 9^(1/6) ? 8 < 9 ? 2^(1/2) < 3^(1/3)

?? 6. Finding Rational / Irrational Numbers Between Two Numbers

Finding rationals between a and b:
Method 1: Take average: (a+b)/2
Method 2: Multiply both by 10/100: find numbers of the form m/n
Example: Rationals between 1/3 and 1/2 = 5/12 (average), 2/5, 3/7, etc.

Finding irrationals between a and b:
Simply pick any vn where n is not a perfect square and a < vn < b
Example: Irrationals between 2 and 3: v5, v6, v7, v8 (since 4 < 5,6,7,8 < 9)
?? Practice Questions — Number Systems (35 Questions)

Section A — 1 Mark Questions Easy

Q1. Is 0 a natural number, whole number, or integer?
Q2. Identify: is 0.101001000... rational or irrational?
Q3. Express 0.6¯ as p/q.
Q4. Is v4 rational or irrational?
Q5. Simplify: (v3)²
Q6. True/False: p = 22/7. Justify.
Q7. Find a rational number between 3 and 4.
Q8. Evaluate: 125^(1/3)
Q9. Are all integers rational numbers?
Q10. Simplify: v50 (in simplest radical form)

Section B — 2/3 Mark Questions Medium

Q11. Convert 0.47¯ to p/q form.
Q12. Convert 1.2¯7¯ to p/q form.
Q13. Find 3 irrational numbers between 2 and 3.
Q14. Find 3 rational numbers between 2/3 and 3/4.
Q15. Rationalise: 1/(2+v3)
Q16. Rationalise: 5/(v7-v2)
Q17. Simplify: (2+v3)(2-v3)
Q18. Represent v5 on the number line.
Q19. Simplify: v72 + v50 - v32
Q20. Evaluate: (64)^(-1/3) × (64)^(1/3 + 2)

Section C — 3/4 Mark Questions (NCERT/Board Style) Medium

  1. Q21. Simplify: (5 + 2v3)/(7 + 4v3). Rationalise and simplify completely.
  2. Q22. If x = 2 + v3, find: (i) x + 1/x   (ii) x² + 1/x²
  3. Q23. Simplify: (v2 + v3)² + (v5 - v2)²
  4. Q24. If a = 3 + 2v2, find va - 1/va. Hence find a + 1/a.
  5. Q25. Simplify: [5^(1/2) × 5^(1/3)] ÷ 5^(1/6)
  6. Q26. Show that 5 - v3 is irrational (using proof by contradiction).
  7. Q27. Show that 3v2 is irrational.

Section D — 5 Mark / Long Answer Hard

  1. Q28. Prove that v2 is irrational. (Full proof — NCERT Appendix / Board question)
  2. Q29. If x = (v3+v2)/(v3-v2) and y = (v3-v2)/(v3+v2), find x² + y² + xy.
  3. Q30. Simplify: [(v3 + v2)/(v3 - v2)] - [(v3 - v2)/(v3 + v2)]
  4. Q31. Simplify: [3^(n+1) + 3^n] / [3^(n+2) - 3^(n+1)]
  5. Q32. Arrange in ascending order: 2^(1/2), 3^(1/3), 4^(1/4)
  6. Q33. Simplify: [(x^a / x^b)^(a+b)] × [(x^b / x^c)^(b+c)] × [(x^c / x^a)^(c+a)]
  7. Q34. If 2^x = 3^y = 12^z, prove that 1/z = 1/y + 2/x.
  8. Q35. Convert 2.3¯5¯ (2.353535...) to p/q form and verify by long division.
? Key Answers: Q3: 2/3 | Q8: 5 | Q10: 5v2 | Q11: 43/90 | Q12: 126/99=14/11 | Q15: 2-v3 | Q16: 5(v7+v2)/5 | Q17: 1 | Q19: 7v2 | Q22(i): 4 | Q25: 5^(7/6) | Q30: 4v6 | Q31: 1/2 | Q35: 233/99

?? Quick Revision

  1. N ? W ? Z ? Q ? R. Every natural number is an integer, every integer is rational, every rational is real.
  2. Terminating or recurring decimal ? Rational | Non-terminating, non-recurring ? Irrational
  3. vp is irrational for any prime p. v4=2 is rational.
  4. Rationalise by multiplying by conjugate: 1/(a+vb) ? multiply by (a-vb)/(a-vb)
  5. Key laws: a?×an=a??n | a?÷an=a??n | (a?)n=a?n | a^(1/n)=nva | a^(m/n)=(nva)?
  6. Infinite rationals and irrationals exist between any two real numbers