Number System

Number System is the foundation of quantitative aptitude. It involves understanding the properties of numbers, their classifications, and various operations.

Types of Numbers

1. Natural Numbers (N): (1, 2, 3, 4, …)
2. Whole Numbers (W): (0, 1, 2, 3, …)
3. Integers (Z): (…, -3, -2, -1, 0, 1, 2, 3, …)
4. Rational Numbers (Q): Numbers that can be expressed as p/q, where q != 0.
5. Irrational Numbers: Numbers that cannot be expressed as p/q (e.g., sqrt(2), pi).
6. Real Numbers (R): The set of both rational and irrational numbers.
7. Prime Numbers: Numbers with exactly two factors (1 and itself).
8. Composite Numbers: Numbers with more than two factors.

Important Formulas

• Sum of first n natural numbers: n(n+1) / 2
• Sum of first n even numbers: n(n+1)
• Sum of first n odd numbers: n^2
• Sum of squares of first n natural numbers: n(n+1)(2n+1) / 6
• Sum of cubes of first n natural numbers: [n(n+1) / 2]^2

Shortcuts & Tips

• Units Digit Calculation: Observe the cyclicity of digits.
  • 0, 1, 5, 6: Cyclicity 1 (remain the same)
  • 4, 9: Cyclicity 2
  • 2, 3, 7, 8: Cyclicity 4
• Trailing Zeros: The number of trailing zeros in n! is given by: num = floor(n/5) + floor(n/25) + floor(n/125) + …
• Divisibility by 11: The difference between the sum of digits at odd positions and even positions must be 0 or a multiple of 11.