πŸ“‹ Cheat Sheet

Real Numbers Β· Mathematics Ch.1 Β· Class X CBSE Β· Quick Revision

πŸ”’ Number System Hierarchy
SymbolSetExamples
β„•Natural Numbers1, 2, 3, …
π•ŽWhole Numbers0, 1, 2, 3, …
β„€Integersβ€¦βˆ’2, βˆ’1, 0, 1, 2…
β„šRational (p/q form)3/4, βˆ’7, 0.5
β„šβ€²Irrational√2, √3, Ο€, e
ℝReal = β„š βˆͺ β„šβ€²All of the above
Subset Chain β„• βŠ‚ π•Ž βŠ‚ β„€ βŠ‚ β„š βŠ‚ ℝ    β„š ∩ β„šβ€² = βˆ… (disjoint)
22/7 β‰ˆ 3.14285… is rational; Ο€ β‰ˆ 3.14159… is irrational β€” they are NOT equal.
πŸ“ 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 β†’ set a←b, b←r β†’ repeat
When r = 0, the divisor at that step = HCF.
Example: HCF(225,135): 225=135Γ—1+90 β†’ 135=90Γ—1+45 β†’ 90=45Γ—2+0 β†’ 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 = 2Β³ Γ— 3Β² Γ— 5   (unique β€” try any factor tree!)
420 = 2Β² Γ— 3 Γ— 5 Γ— 7
HCF β†’ LOWEST powers of COMMON primes
HCF(12,18): 12=2Β²Γ—3, 18=2Γ—3Β² β†’ HCF = 2ΒΉΓ—3ΒΉ = 6
LCM β†’ HIGHEST powers of ALL primes
LCM(12,18): primes 2,3 β†’ LCM = 2Β²Γ—3Β² = 36
Verify: HCF Γ— LCM = 6 Γ— 36 = 216 = 12 Γ— 18 βœ“
βš–οΈ HCF & LCM β€” Key Formulas
GOLDEN FORMULA (two numbers only!)
HCF(a,b) Γ— LCM(a,b) = a Γ— b
⚠ Does NOT work for three or more numbers.
When to use HCFWhen to use LCM
Dividing into equal groupsEvents repeating together
Max size of equal piecesBells ringing simultaneously
"Largest number that divides…""Smallest number divisible by…"
No remainder left overSynchronising cycles
Co-prime: HCF(a,b) = 1. Then LCM(a,b) = aΓ—b. E.g.: HCF(8,15)=1 β†’ LCM=120.
Remainder trick: Largest divisor of (626βˆ’1, 3127βˆ’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
1.
Assume √2 = p/q where HCF(p,q) = 1
2.
2q² = p² ⟹ p² is even ⟹ p is even ⟹ p = 2m
3.
q² = 2m² ⟹ q² even ⟹ q is even
4.
Both p, q even ⟹ HCF β‰₯ 2 β€” Contradiction! ∴ √2 is irrational β– 
Same proof works for √3, √5, √7 β€” any √prime.
If r is rational and x is irrational: r+x, rβˆ’x, rΓ—x, r/x are all irrational (rβ‰ 0).
Sum of two irrationals can be rational: √2 + (βˆ’βˆš2) = 0.
πŸ”£ Decimal Expansions
RULE β€” When does p/q TERMINATE?
q = 2ⁿ Γ— 5ᡐ (only) ⟺ terminating decimal
Check denominator after fully simplifying. Any other prime factor β†’ non-terminating recurring.
TypeConditionExample
Terminatingq = 2ⁿ×5ᡐ only3/8=0.375
Non-term. Recurringq has other primes1/3=0.3Μ„
Non-term. Non-recurringIrrational number√2=1.414…
7/20: 20=2Β²Γ—5 βœ“ β†’ terminates β†’ 0.35
7/12: 12=2Β²Γ—3 β†’ has 3 β†’ non-terminating recurring β†’ 0.583Μ„
Recurring decimal β†’ always rational. Non-term. non-recurring β†’ always irrational.
🌳 Factor Tree Examples
360
360
β•± β•²
2 180
   β•± β•²
  2 90
     β•± β•²
    2 45
       β•± β•²
      3 15
         β•± β•²
        3 5
2Β³ Γ— 3Β² Γ— 5
420
420
β•± β•²
2 210
   β•± β•²
  2 105
     β•± β•²
    3 35
       β•± β•²
      5 7
2Β² Γ— 3 Γ— 5 Γ— 7
βš–οΈ HCF & LCM β€” Worked Examples
Euclid's: HCF(870,225)
870=225Γ—3+195 β†’ 225=195Γ—1+30 β†’ 195=30Γ—6+15 β†’ 30=15Γ—2+0 β†’ HCF=15
FTA: HCF & LCM(360,420)
360=2Β³Γ—3Β²Γ—5, 420=2Β²Γ—3Γ—5Γ—7
HCF=2Β²Γ—3Γ—5=60  LCM=2Β³Γ—3Β²Γ—5Γ—7=2520
Verify: 60Γ—2520=151200=360Γ—420 βœ“
Bells: LCM(6,8,12)
6=2Γ—3, 8=2Β³, 12=2Β²Γ—3 β†’ LCM=24 min
Ropes: HCF(840,560)
840=2Β³Γ—3Γ—5Γ—7, 560=2⁴×5Γ—7 β†’ HCF=2Β³Γ—5Γ—7=280 cm
πŸ”„ Methods Comparison β€” Euclid vs FTA
FeatureEuclid's Division AlgorithmPrime Factorisation (FTA)
FindsHCF onlyHCF and LCM both
Best forLarge numbers (fewer errors)Smaller numbers; when LCM is needed
ProcessRepeated division until remainder = 0Factor tree β†’ compare prime powers
Three numbersApply algorithm twiceAll at once: lowest/highest powers
Error riskLow β€” only division neededHigher for large numbers
Product formulaHCF Γ— LCM = aΓ—b (only 2 numbers)HCF Γ— LCM = aΓ—b (only 2 numbers)
πŸ”£ Decimal Expansion β€” Quick Classification Table
FractionSimplified?Denominator factorsTypeDecimal
3/8Yes2Β³ only βœ“Terminating0.375
7/20Yes2Β²Γ—5 only βœ“Terminating0.35
77/21077/210 = 11/302Γ—3Γ—5 β€” has 3 βœ—Non-term. Recurring0.3666… = 0.36Μ„
1/6Yes2Γ—3 β€” has 3 βœ—Non-term. Recurring0.1666… = 0.16Μ„
2/11Yes11 β€” prime βœ—Non-term. Recurring0.1818… = 0.1Μ„8Μ„
√2β€”IrrationalNon-term. Non-recurring1.41421356…
Ο€β€”IrrationalNon-term. Non-recurring3.14159265…
🧠 Memory Tricks & Mnemonics
Number Chain: "Nice Wild Zebras Quickly Roam"

N βŠ‚ W βŠ‚ Z βŠ‚ Q βŠ‚ R β€” 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 Γ— 5ᡐ and nothing else β†’ terminating decimal.

Irrational Proof: "Both Even β†’ Contradiction"

√2 proof key step: assume p/q in lowest terms β†’ p even β†’ q even β†’ HCF β‰₯ 2. Contradicts lowest terms!

HCF vs LCM word problems

"Dividing / max size / equal groups" β†’ HCF. "Together again / cycles / simultaneous" β†’ LCM.

22/7 β‰  Ο€ β€” Remember!

22/7 β‰ˆ 3.14285 (rational). Ο€ β‰ˆ 3.14159 (irrational). 22/7 is only an approximation β€” 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)βˆ’r = rationalβˆ’rational = rational. Contradiction!