Uttar Pradesh Police Constable – Quantitative Aptitude
Topic: Number System
The “Number System” section tests your knowledge of different kinds of numbers, their properties, and the basic rules that govern operations on them. Mastery of these concepts gives you an edge in solving number‑related questions quickly and accurately.
1. Simple Explanation
| Type | Notation | Range | Key Property | Typical Use in Exams |
|---|---|---|---|---|
| Natural Numbers | ( \mathbb{N} ) | (1, 2, 3, \dots) | Positive integers only | Counting, series |
| Whole Numbers | ( \mathbb{W} ) | (0, 1, 2, \dots) | Includes 0 | Counting with zero |
| Integers | ( \mathbb{Z} ) | (\dots,-3,-2,-1,0,1,2,3,\dots) | Positive & negative | Divisibility, sign manipulation |
| Rational Numbers | ( \mathbb{Q} ) | (\frac{p}{q}) where (p,q\in\mathbb{Z}), (q\neq0) | Can be expressed as a fraction | Approximation, decimals |
| Irrational Numbers | – | Numbers that cannot be written as a fraction (e.g., (\sqrt{2}), (\pi)) | Non‑terminating, non‑repeating decimal | Approximation, limits |
| Real Numbers | ( \mathbb{R} ) | All rational + irrational | Complete number line | All arithmetic operations |
| Complex Numbers | ( \mathbb{C} ) | (a+bi) where (i^2=-1) | Includes imaginary unit | Advanced topics (rare in UPSC) |
Bottom line: Start with the natural numbers, then add 0, then negatives, then fractions, then irrationals. All other systems extend this hierarchy.
2. Key Points (Quick‑Reference)
A. Basic Operations & Properties
| Property | Rule | Example |
|---|---|---|
| Addition/Subtraction | Closest to “carry” or “borrow” rules | (73+28=101) |
| Multiplication | Distributive law: (a(b+c)=ab+ac) | (5(3+4)=5·7=35) |
| Division | Quotient (integer part) + remainder | (17÷4=4) rem (1) |
| Exponentiation | (a^b) – repeated multiplication | (2^3=8) |
| Order of Operations | PEMDAS/BODMAS | (2+3×4=14) |
B. Number Properties
| Property | Definition | Quick Check |
|---|---|---|
| Prime | Only divisible by 1 & itself | 13, 29 |
| Composite | More than two divisors | 12, 18 |
| Perfect Square | (n^2) | (4^2=16), (9^2=81) |
| Perfect Cube | (n^3) | (2^3=8) |
| Even/Odd | Divisible by 2? | 8 (even), 7 (odd) |
| Divisibility Rules | Quick test for divisibility | 9: sum of digits divisible by 9 |
C. Divisibility Rules (Shortcuts)
| Number | Criterion | Example |
|---|---|---|
| 2 | Last digit even | 42 |
| 3 | Sum of digits divisible by 3 | 123 → 1+2+3=6 |
| 4 | Last two digits divisible by 4 | 236 → 36 |
| 5 | Last digit 0 or 5 | 145 |
| 6 | Divisible by 2 & 3 | 18 |
| 7 | (Double the last digit, subtract from remaining) | 203 → 20−2·3 = 14 (not divisible) |
| 8 | Last three digits divisible by 8 | 1120 → 120 |
| 9 | Sum of digits divisible by 9 | 729 → 7+2+9=18 |
| 10 | Last digit 0 | 310 |
Remember: For 11, alternate sum of digits difference multiple of 11.
D. Number Series & Patterns
- Arithmetic Series: Common difference (d).
- Geometric Series: Common ratio (r).
- Alternating Series: Pattern changes sign or digits.
Check the difference or ratio between consecutive terms.
E. Number Conversion (Base Systems)
| From Base | To Base | Method |
|---|---|---|
| Decimal → Binary | Divide by 2, record remainders | 13 → 1101 |
| Decimal → Octal | Divide by 8 | 13 → 15 |
| Decimal → Hexadecimal | Divide by 16 | 13 → D |
Useful for computer‑related questions.