Number Systems — Class 9 (Complete & Solved)

Mobile-first, exam-oriented notes with detailed solved examples (English)

Overview — What this file covers

Complete coverage of Chapter 1: Number Systems — definitions, classification, decimal behaviour, proofs of irrationality, surds, rationalization, Euclid's lemma, many solved examples with step-by-step solutions and practice problems arranged by difficulty (Easy → Challenging).

1. Classification of Numbers (Quick)

Natural (N)
1,2,3,...
Whole (W)
0,1,2,3,...
Integers (Z)
..., -2,-1,0,1,2,...
Rational (Q)
p/q where p,q ∈ Z, q≠0 (decimal terminates or repeats)
Irrational
non-terminating, non-repeating decimals (e.g., √2, π)
Real (R)
All rational + irrational numbers

2. Decimal Representation

  • Rational → decimal either terminates or repeats.
  • If q (in lowest p/q) has prime factors only 2 and/or 5 ⇒ terminating decimal.
  • Otherwise decimal repeats (periodic).
Example 1 — Convert decimal to fraction (simple repeating)
Convert 0.454545... to fraction.
Let x = 0.454545... (repeating block 45 of length 2). Then 100x = 45.454545... Subtract: 100x − x = 45 ⇒ 99x = 45 ⇒ x = 45/99 = 5/11.
Example 2 — Mixed repeating: 0.58(23)
x = 0.58232323... Repeating part = 23, non-repeating part = 58 before the repeat? Actually 0.58 23 repeating means 0.582323... Let r-length=2, non-repeat length=2.
10000x = 5823.2323..., 100x = 58.2323... Subtract: 9900x = 5765 ⇒ x = 5765/9900 = simplify divide by 5 → 1153/1980.

3. Rational vs Irrational — How to tell

  • Terminating or repeating decimal ⇒ rational.
  • Non-terminating & non-repeating ⇒ irrational.
  • Between any two real numbers, infinite rationals and irrationals exist (density).
Example 3 — Identify: 0.101001000100001...
Pattern has increasing groups of zeros; it does not repeat with a fixed period → non-terminating, non-repeating → irrational.

4. Standard Proofs of Irrationality (Worked)

Proof: √2 is irrational (classic)
Assume √2 = p/q in lowest terms. Then 2 = p^2/q^2 ⇒ p^2 = 2q^2. So p^2 even ⇒ p even (p=2k). Substitute: 4k^2 = 2q^2 ⇒ q^2 = 2k^2 ⇒ q even. Both even ⇒ contradicts lowest terms. Hence irrational.
Proof idea: √3 is irrational
Same parity argument fails directly because 3 is odd; instead use modulo 3 or prime factor reasoning. Assume √3 = p/q in lowest terms → p^2 = 3q^2 ⇒ p divisible by 3 ⇒ write p=3k ⇒ leads to q divisible by 3 ⇒ contradiction.

5. Surds & Rationalizing Denominators

Surd: irrational root expression that cannot be simplified to remove radical. Rationalization removes radicals from denominators by multiplying with conjugates.

Example 4 — Rationalize 1/(√3 + 1)
Multiply numerator and denominator by (√3 − 1): (√3 − 1)/(3 − 1) = (√3 − 1)/2.
Example 5 — Rationalize 2/(√7 − √5)
Multiply by conjugate (√7 + √5): numerator: 2(√7 + √5); denominator: (7 − 5) = 2 ⇒ result = √7 + √5.

6. Euclid's Division Lemma & GCD (short)

Given integers a and b>0, ∃ unique q, r with a = bq + r, 0 ≤ r < b. Repeated application finds gcd. Useful in proofs and simplifying fractions.

Example 6 — GCD of 252 and 105 (Euclidean algorithm)
252 = 105×2 + 42
105 = 42×2 + 21
42 = 21×2 + 0 ⇒ gcd = 21.

7. More Solved Problems (Step-by-step)

Problem A (Easy)
Convert 0.230230230... to fraction.
Let x = 0.230230... repeating block = 230 (length 3). 1000x = 230.230230... Subtract: 1000x − x = 230 ⇒ 999x = 230 ⇒ x = 230/999 (simplify? 230 and 999 have gcd 1) → final 230/999.
Problem B (Medium)
Express 0.058(3) where only 3 repeats after initial zeros: 0.058333...
Let x = 0.058333... repeat length 1, non-repeat length 3. 1000x = 58.333... and 10000x = 583.333... Subtract: 9000x = 525 ⇒ x = 525/9000 = divide by 75 → 7/120.
Problem C (Challenging)
Show that the decimal with pattern 0.123456789101112... (concatenation of integers) is irrational.
This decimal is non-periodic—no finite repeating block exists because blocks keep changing length—hence irrational. Formal proof uses contradiction: if periodic, there exists period T covering arbitrarily large appended integers which is impossible.

8. Practice Set (Solved + Unsolved)

Q1. Express 0.727272... as a fraction. (Solve)
Solution
x = 0.727272... ⇒ 100x = 72.7272... ⇒ 99x = 72 ⇒ x = 72/99 = 8/11.
Q2. Rationalize: 5/(√2 + √3). (Solve)
Solution
Multiply by (√3 − √2): numerator 5(√3 − √2), denominator 3 − 2 = 1 ⇒ 5(√3 − √2).
Q3. If p/q in lowest terms has q = 40, will decimal terminate? Explain. (Answer)
Solution
40 = 2^3 × 5, only primes 2 and 5 → decimal terminates.
Q4. Convert 0.581818... (where 18 repeats after 58) to fraction. (Try yourself)
Q5. Prove √6 is irrational (outline).
Q6 (Challenge). Show that between √2 and √3 there exists a rational whose denominator is ≤ 10. (Hint: check rationals with small denominators)

9. Quick Tips & Cheat-sheet

  • To convert repeating decimals: shift by powers of 10 to align repeating blocks, subtract, solve for x.
  • Terminating ⇔ denominator (lowest terms) factors only 2 and 5.
  • Rationalize by multiplying conjugate when two-term surds appear in denominator.
  • Use Euclidean algorithm for gcd quickly.

10. Chapter Summary

Focus on: (1) recognising rational and irrational numbers by decimal behaviour, (2) converting repeating decimals to fractions quickly, (3) rationalizing surds, and (4) Euclid's lemma and gcd algorithm. Practice many conversions and proofs — most exam questions are routine once steps are memorised.

Solved Answers Quick Table

  1. 0.454545... = 5/11
  2. 0.727272... = 8/11
  3. 0.230230... = 230/999
  4. 2/(√7 − √5) = √7 + √5