Numbers
๐งฎ NUMBERS โ AP TET MASTER GUIDE
Comprehensive coverage of Number System, Operations, Factors, Multiples, Fractions, Rational Numbers with examples, shortcuts, and memory aids for competitive exams.
I. Number System and Numeration
Knowing about Numbers: Numbers help us count, compare, and measure. They can be natural, whole, integers, or rational.
| Type | Definition | Examples |
|---|---|---|
| Natural (โ) | Counting numbers | 1, 2, 3,โฆ |
| Whole (W) | Natural + 0 | 0, 1, 2,โฆ |
| Integers (โค) | Positive, negative & 0 | โ3, โ2, โ1, 0, 1,โฆ |
| Rational (โ) | p/q form (qโ 0) | ยฝ, โยพ, 5 |
| Irrational | Cannot be written as p/q | โ2, ฯ |
| Real (โ) | Rational + Irrational | All above |
๐ง Hierarchy: โ โ W โ โค โ โ โ โ
All real numbers can be placed on a number line.
All real numbers can be placed on a number line.
๐น Indian & International Numeration System
Indian: Uses lakhs and crores
International: Uses millions and billions
| International Unit | Equivalent in Indian System | Value |
|---|---|---|
| 100 Thousand | 1 Lakh | 1,00,000 |
| 1 Million | 10 Lakhs | 10,00,000 |
| 10 Million | 1 Crore | 1,00,00,000 |
| 1 Billion | 100 Crores (1 Arab) | 1,00,00,00,000 |
| 1 Trillion | 1 Lakh Crore (100 Arabs) | 1,00,00,00,00,00,000 |
๐ Example:
Indian โ 12,34,567 = 12 lakh 34 thousand 567
International โ 1,234,567 = 1 million 234 thousand 567
Indian โ 12,34,567 = 12 lakh 34 thousand 567
International โ 1,234,567 = 1 million 234 thousand 567
๐ก Quick Relation: Million โ 10 Lakhs | Crore โ 10 Million | Trillion โ 1 Lakh Crores
๐น Place & Face Value
In 6523 โ face value of 5 = 5; place value = 5ร10 = 50.
๐ก Commas: Indian (1,23,45,678), International (12,345,678)
II. Factors, Multiples, Prime & Composite
- Factors: Divide exactly โ 12 โ 1, 2, 3, 4, 6, 12
- Multiples: 7 โ 7, 14, 21, 28,โฆ
- Prime: Only 1 and itself (2, 3, 5, 7, 11,โฆ)
- Composite: More than two factors (12 โ 1, 2, 3, 4, 6, 12)
๐น Divisibility Rules
| Divisible By | Rule | Example |
|---|---|---|
| 2 | Last digit even | 824 โ |
| 3 | Sum of digits รท 3 | 5412 โ 5+4+1+2=12 โ |
| 4 | Last two digits รท 4 | 524 โ 24 รท 4 โ |
| 5 | Ends in 0 or 5 | 975 โ |
| 6 | Divisible by 2 & 3 | 432 โ |
| 8 | Last 3 digits รท 8 | 16248 โ 248 รท 8 โ |
| 9 | Sum of digits รท 9 | 729 โ 7+2+9=18 โ |
| 10 | Ends with 0 | 340 โ |
โก Quick Check: If divisible by both 2 and 3 โ divisible by 6.
๐น HCF & LCM
Formula: HCF ร LCM = Product of the numbers
Example: 18 ร 24 = 432 โ HCF = 6 โ LCM = 432 รท 6 = 72 โ
III. Fundamental Operations
- Addition: 245 + 378 = 623
- Subtraction: 785 โ 429 = 356
- Multiplication: 23 ร 15 = 345
- Division: 540 รท 9 = 60
๐ก BODMAS: Brackets โ Orders โ Division โ Multiplication โ Addition โ Subtraction
Example: 8 + 4ร(6โ2) = 8 + 4ร4 = 24
Example: 8 + 4ร(6โ2) = 8 + 4ร4 = 24
IV. Fractions and Decimals
| Type | Example |
|---|---|
| Proper | 3/5 |
| Improper | 7/4 |
| Mixed | 2 โ |
Conversion: 3/4 = 0.75, 0.6 = 3/5
| Fraction | Decimal | Type |
|---|---|---|
| ยฝ | 0.5 | Terminating |
| โ | 0.333โฆ | Repeating |
| ยพ | 0.75 | Terminating |
โก Rule: Terminating โ denominator only 2 or 5; Repeating โ other primes (1/8=0.125, 1/3=0.333โฆ)
V. Rational Numbers
- Form p/q, qโ 0 โ e.g., 3/4, โ5/2
- Closed under +, โ, ร, รท (except รท0)
- Between any two: (a+b)/2 โ e.g., (1/3, 1/2) = 5/12
- Reciprocal (inverse): 3/4 โ 4/3 โ product = 1
VI. Powers, Roots & Patterns
| Concept | Example |
|---|---|
| Square | 7ยฒ = 49 |
| Square Root | โ49 = 7 |
| Cube | 3ยณ = 27 |
| Cube Root | โ27 = 3 |
๐ก Squares (1โ10): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Memorize for fast mental math.
Memorize for fast mental math.
VII. Playing with Numbers & Tricks
- Shifting digits: Move decimal point to multiply/divide by powers of 10.
Example: 456 โ 45.6 (รท10), 4.56 (รท100). - Digital Sum: Sum digits to test divisibility by 3 or 9.
Example: 5724 โ 5+7+2+4=18 โ divisible by 9. - Unit-digit patterns: For last digits of powers (e.g., endings repeat every 4 for many bases).
- Reverse and add: Useful trick problems: reverse digits and perform operations to test patterns.
๐งฉ Quick Puzzles:
1) Which of these is divisible by 3? 293, 587, 672 โ 293(โ), 587(โ), 672(โ ).
2) Shift 378 two places left โ 37800 (ร100).
1) Which of these is divisible by 3? 293, 587, 672 โ 293(โ), 587(โ), 672(โ ).
2) Shift 378 two places left โ 37800 (ร100).
VIII. Exam-Oriented Shortcuts
- HCF ร LCM = Product โ use to find missing LCM/HCF quickly.
- For fraction comparison, cross-multiply instead of finding common denominators.
- Memorize squares & cubes up to at least 15 for speed.
- Use estimation to eliminate options in MCQs (approximate magnitudes).
โก Key formulas: (a+b)ยฒ = aยฒ + 2ab + bยฒ | (aโb)ยฒ = aยฒ โ 2ab + bยฒ
IX. Playing with Numbers & Exam Tips (Consolidated)
A. Practice Routine
- Daily 20-minute mental math: 10 mixed problems (HCF-LCM, fractions, squares).
- Alternate days: speed drills (tables, squares, simple factorization).
- Weekly: 1 full timed quiz (30 minutes) on Numbers topics.
B. Memory Hacks
- Memorize squares (1โ30) and cubes (1โ15) progressively.
- Remember divisibility patterns: 3 & 9 โ digit sum; 4 โ last two digits; 8 โ last three digits.
- For large numbers, convert to Indian terms mentally: Million โ ร10 = Lakhs, Billion โ ร100 = Crores.
C. Speed Techniques
- Cross-multiplication for comparing fractions quickly: a/b ? c/d โ compare ad ? bc.
- Use complements for subtraction close to base: 1000 โ 237 = 763 (complement method).
- Factorization to simplify HCF/LCM and fraction reductions.
D. Typical Exam Tricks & Sample Problems
- Q: If a number is divisible by 8, is it always divisible by 4? A: Yes โ 8 is multiple of 4.
- Q: Find a rational number between 2/5 and 3/5. A: (2/5 + 3/5)/2 = 5/10 = 1/2.
- Q: Which is larger: 999ยฒ or 1000ร998? A: 999ยฒ = 998001, 1000ร998 = 998000 โ 999ยฒ is larger by 1.
๐ Final Exam Tip: Accuracy > Speed. Build accuracy first with daily practice, then add speed drills. Use elimination and estimation for MCQs.