Number System (संख्या पद्धती)
Overview
The Number System forms the backbone of the entire Mathematics section in Maharashtra Police Bharti examination. Almost every arithmetic problem—whether percentage, ratio, or simplification—requires a solid understanding of different types of numbers and their properties.
Examiners test your ability to classify numbers correctly, apply divisibility rules quickly, and understand properties like even-odd behaviour and prime factorisation.
Master this topic first. Without it, speed and accuracy in other arithmetic areas will suffer. Focus especially on divisibility tests, prime numbers up to 100, and properties of zero and one—these appear repeatedly in competitive exams.
Key Concepts
- Natural Numbers (N): Counting numbers starting from 1. Set = {1, 2, 3, 4, ...}. Does NOT include zero.
- Whole Numbers (W): Natural numbers plus zero. Set = {0, 1, 2, 3, ...}. Every natural number is a whole number, but 0 is whole only.
- Integers (Z): Whole numbers plus negative numbers. Set = {..., -3, -2, -1, 0, 1, 2, 3, ...}. Includes positive, negative, and zero.
- Rational Numbers (Q): Numbers expressible as p/q where p and q are integers and q ≠ 0. Examples: 3/4, -7/2, 5 (which is 5/1), 0.25 (which is 1/4).
- Hierarchy: Natural ⊂ Whole ⊂ Integer ⊂ Rational. Every natural number is rational, but not vice versa.
- Prime Numbers: Numbers greater than 1 with exactly two factors (1 and itself). Examples: 2, 3, 5, 7, 11, 13...
- Composite Numbers: Numbers greater than 1 with more than two factors. Examples: 4, 6, 8, 9, 10...
- Special Cases: 1 is neither prime nor composite. 2 is the only even prime number. 0 is neither positive nor negative.
Formulas / Key Facts
Divisibility Rules (must memorise):
| Divisor | Rule |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Last three digits divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 11 | Difference of sum of alternate digits divisible by 11 |
Important Number Properties:
- Sum of first n natural numbers = n(n+1)/2
- Sum of first n odd numbers = n²
- Sum of first n even numbers = n(n+1)
- Product of two numbers = LCM × HCF
- Number of prime numbers from 1 to 100 = 25
- Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Even-Odd Rules:
- Even ± Even = Even
- Odd ± Odd = Even
- Even ± Odd = Odd
- Even × Any = Even
- Odd × Odd = Odd
Worked Examples
Example 1: Classification Question: Which of the following is NOT a rational number? (a) -7 (b) 0 (c) 22/7 (d) √2
Solution:
- -7 = -7/1 → Rational ✓
- 0 = 0/1 → Rational ✓
- 22/7 is already p/q form → Rational ✓
- √2 = 1.41421... (non-terminating, non-repeating) → Irrational ✗
Answer: (d) √2
Example 2: Divisibility Question: Is 2,34,567 divisible by 9?
Solution: Step 1: Find sum of digits = 2 + 3 + 4 + 5 + 6 + 7 = 27 Step 2: Check if 27 is divisible by 9 → 27 ÷ 9 = 3 (exact)
Answer: Yes, divisible by 9
Example 3: Sum Formula Application Question: Find the sum of all natural numbers from 1 to 50.
Solution: Using formula: Sum = n(n+1)/2 Here n = 50 Sum = 50 × 51 / 2 = 2550 / 2 = 1275
Answer: 1275
Example 4: Divisibility by 11 Question: Check if 918082 is divisible by 11.
Solution: Step 1: Mark alternate positions: 9₁ 1₂ 8₃ 0₄ 8₅ 2₆ Step 2: Sum of odd positions (1st, 3rd, 5th) = 9 + 8 + 8 = 25 Step 3: Sum of even positions (2nd, 4th, 6th) = 1 + 0 + 2 = 3 Step 4: Difference = 25 - 3 = 22 Step 5: 22 is divisible by 11 ✓
Answer: Yes, divisible by 11
Common Mistakes
- Thinking 0 is not a whole number → 0 IS a whole number but NOT a natural number. Remember: Whole = Natural + Zero.
- Calling 1 a prime number → 1 is NEITHER prime NOR composite. Prime numbers must have exactly two distinct factors; 1 has only one factor (itself).
- Confusing divisibility by 4 and 8 → For 4, check last TWO digits. For 8, check last THREE digits. Students often check only the last digit for both.
- Forgetting that 2 is prime → 2 is the ONLY even prime number. All other even numbers have at least three factors (1, 2, and themselves).
- Assuming all decimals are irrational → Terminating decimals (0.25) and repeating decimals (0.333...) are rational. Only non-terminating, non-repeating decimals are irrational.
- Wrong sign rules for integers → Negative × Negative = Positive. Negative × Positive = Negative. Students rush and make sign errors.
Quick Reference
- N ⊂ W ⊂ Z ⊂ Q: Natural inside Whole inside Integer inside Rational
- 1 is neither prime nor composite; 2 is the only even prime
- Divisible by 6 = Divisible by BOTH 2 and 3
- Sum of first n naturals = n(n+1)/2
- For divisibility by 11: (Sum of odd-position digits) − (Sum of even-position digits) must be 0 or divisible by 11
- 25 prime numbers exist between 1 and 100