Cheat Sheet — Real Numbers

🔢 Number System Hierarchy

Symbol	Set	Examples
ℕ	Natural Numbers	1, 2, 3, …
𝕎	Whole Numbers	0, 1, 2, 3, …
ℤ	Integers	…−2, −1, 0, 1, 2…
ℚ	Rational (p/q form)	3/4, −7, 0.5
ℚ′	Irrational	√2, √3, π, e
ℝ	Real = ℚ ∪ ℚ′	All of the above

Subset Chain ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ ℚ ∩ ℚ′ = ∅ (disjoint)

22/7 ≈ 3.14285… is rational; π ≈ 3.14159… is irrational — they are NOT equal.

📐 Euclid's Division Lemma

STATEMENT

a = bq + r (0 ≤ r < b)

a = dividend, b = divisor, q = quotient, r = remainder.
q and r are unique for any given a and b.

ALGORITHM TO FIND HCF(a, b)

Step: a = bq+r → set a←b, b←r → repeat

When r = 0, the divisor at that step = HCF.

Example: HCF(225,135): 225=135×1+90 → 135=90×1+45 → 90=45×2+0 → HCF = 45

For 3 numbers: HCF(a,b,c) = HCF(HCF(a,b), c)

Use when: finding max tile size, equal-length cuts, grouping without remainder.

🌳 Fundamental Theorem of Arithmetic

Statement Every composite number can be expressed as a product of primes in exactly one way (ignoring order). Factorisation is unique.

360 = 2³ × 3² × 5 (unique — try any factor tree!)

420 = 2² × 3 × 5 × 7

HCF → LOWEST powers of COMMON primes

HCF(12,18): 12=2²×3, 18=2×3² → HCF = 2¹×3¹ = 6

LCM → HIGHEST powers of ALL primes

LCM(12,18): primes 2,3 → LCM = 2²×3² = 36

Verify: HCF × LCM = 6 × 36 = 216 = 12 × 18 ✓

⚖️ HCF & LCM — Key Formulas

GOLDEN FORMULA (two numbers only!)

HCF(a,b) × LCM(a,b) = a × b

⚠ Does NOT work for three or more numbers.

When to use HCF	When to use LCM
Dividing into equal groups	Events repeating together
Max size of equal pieces	Bells ringing simultaneously
"Largest number that divides…"	"Smallest number divisible by…"
No remainder left over	Synchronising cycles

Co-prime: HCF(a,b) = 1. Then LCM(a,b) = a×b. E.g.: HCF(8,15)=1 → LCM=120.

Remainder trick: Largest divisor of (626−1, 3127−2) = HCF(625, 3125) = 5⁴ = 625

∞ Irrational Numbers & Proofs

Definition A number that CANNOT be written as p/q (p,q integers, q≠0). Has a non-terminating, non-recurring decimal expansion.

Proof: √2 is Irrational

Assume √2 = p/q where HCF(p,q) = 1

2q² = p² ⟹ p² is even ⟹ p is even ⟹ p = 2m

q² = 2m² ⟹ q² even ⟹ q is even

Both p, q even ⟹ HCF ≥ 2 — Contradiction! ∴ √2 is irrational ■

Same proof works for √3, √5, √7 — any √prime.

If r is rational and x is irrational: r+x, r−x, r×x, r/x are all irrational (r≠0).

Sum of two irrationals can be rational: √2 + (−√2) = 0.

🔣 Decimal Expansions

RULE — When does p/q TERMINATE?

q = 2ⁿ × 5ᵐ (only) ⟺ terminating decimal

Check denominator after fully simplifying. Any other prime factor → non-terminating recurring.

Type	Condition	Example
Terminating	q = 2ⁿ×5ᵐ only	3/8=0.375
Non-term. Recurring	q has other primes	1/3=0.3̄
Non-term. Non-recurring	Irrational number	√2=1.414…

7/20: 20=2²×5 ✓ → terminates → 0.35

7/12: 12=2²×3 → has 3 → non-terminating recurring → 0.583̄

Recurring decimal → always rational. Non-term. non-recurring → always irrational.

🌳 Factor Tree Examples

360

360
╱ ╲
2 180
   ╱ ╲
  2 90
     ╱ ╲
    2 45
       ╱ ╲
      3 15
         ╱ ╲
        3 5

2³ × 3² × 5

420

420
╱ ╲
2 210
   ╱ ╲
  2 105
     ╱ ╲
    3 35
       ╱ ╲
      5 7

2² × 3 × 5 × 7

⚖️ HCF & LCM — Worked Examples

Euclid's: HCF(870,225)
870=225×3+195 → 225=195×1+30 → 195=30×6+15 → 30=15×2+0 → HCF=15

FTA: HCF & LCM(360,420)
360=2³×3²×5, 420=2²×3×5×7
HCF=2²×3×5=60 LCM=2³×3²×5×7=2520
Verify: 60×2520=151200=360×420 ✓

Bells: LCM(6,8,12)
6=2×3, 8=2³, 12=2²×3 → LCM=24 min

Ropes: HCF(840,560)
840=2³×3×5×7, 560=2⁴×5×7 → HCF=2³×5×7=280 cm

🔄 Methods Comparison — Euclid vs FTA

Feature	Euclid's Division Algorithm	Prime Factorisation (FTA)
Finds	HCF only	HCF and LCM both
Best for	Large numbers (fewer errors)	Smaller numbers; when LCM is needed
Process	Repeated division until remainder = 0	Factor tree → compare prime powers
Three numbers	Apply algorithm twice	All at once: lowest/highest powers
Error risk	Low — only division needed	Higher for large numbers
Product formula	HCF × LCM = a×b (only 2 numbers)	HCF × LCM = a×b (only 2 numbers)

🔣 Decimal Expansion — Quick Classification Table

Fraction	Simplified?	Denominator factors	Type	Decimal
3/8	Yes	2³ only ✓	Terminating	0.375
7/20	Yes	2²×5 only ✓	Terminating	0.35
77/210	77/210 = 11/30	2×3×5 — has 3 ✗	Non-term. Recurring	0.3666… = 0.36̄
1/6	Yes	2×3 — has 3 ✗	Non-term. Recurring	0.1666… = 0.16̄
2/11	Yes	11 — prime ✗	Non-term. Recurring	0.1818… = 0.1̄8̄
√2	—	Irrational	Non-term. Non-recurring	1.41421356…
π	—	Irrational	Non-term. Non-recurring	3.14159265…

🧠 Memory Tricks & Mnemonics

Number Chain: "Nice Wild Zebras Quickly Roam"

N ⊂ W ⊂ Z ⊂ Q ⊂ R — every subset contains the previous one.

HCF = Humble, LCM = Large

HCF takes the LOWEST (humble) powers of common primes. LCM takes the HIGHEST (large) powers of all primes.

Terminating Test: "Only 2s and 5s"

Simplify the fraction fully. If denominator = 2ⁿ × 5ᵐ and nothing else → terminating decimal.

Irrational Proof: "Both Even → Contradiction"

√2 proof key step: assume p/q in lowest terms → p even → q even → HCF ≥ 2. Contradicts lowest terms!

HCF vs LCM word problems

"Dividing / max size / equal groups" → HCF. "Together again / cycles / simultaneous" → LCM.

22/7 ≠ π — Remember!

22/7 ≈ 3.14285 (rational). π ≈ 3.14159 (irrational). 22/7 is only an approximation — both differ after 2 decimal places.

Euclid: "Keep Dividing, Last Divisor Wins"

Apply a=bq+r, replace a←b, b←r. When r=0, the last non-zero divisor is the HCF.

r + x = irrational (if r rational, x irrational)

Proof: if it were rational, then x = (r+x)−r = rational−rational = rational. Contradiction!