Number System

Number System

1.1 Types of Numbers

Natural Numbers

• Counting numbers starting from 1: 1, 2, 3, 4, 5, ...
• Denoted by N
• Smallest natural number = 1
• There is no largest natural number (infinite)

Whole Numbers

• All natural numbers including 0: 0, 1, 2, 3, ...
• Denoted by W
• Smallest whole number = 0

Integers

• All whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Denoted by Z (from German Zahlen)
• Positive integers: 1, 2, 3, ...
• Negative integers: -1, -2, -3, ...
• Zero (neither positive nor negative)

Rational Numbers

• Numbers that can be expressed as p/q where p, q are integers and q ≠ 0
• Examples: 1/2, -3/4, 5, 0.75
• Terminating decimals (0.5) and non-terminating repeating decimals (0.333...) are rational

Irrational Numbers

• Numbers that CANNOT be expressed as p/q
• Non-terminating, non-repeating decimals
• Examples: √2 = 1.41421356..., √3, π = 3.14159..., e = 2.71828...

Real Numbers

• All rational and irrational numbers together
• Denoted by R

Prime Numbers

• Numbers greater than 1 that have exactly TWO factors: 1 and themselves
• Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
• 2 is the only even prime number
• 1 is neither prime nor composite

Composite Numbers

• Numbers greater than 1 that have MORE than two factors
• Examples: 4, 6, 8, 9, 10, 12...

Co-prime Numbers

• Two numbers are co-prime if their HCF = 1 (no common factor other than 1)
• Examples: (8, 15), (14, 25), (3, 7)
• Note: Co-prime numbers need not be prime themselves

Twin Primes

• Pairs of primes that differ by 2
• Examples: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43)

1.2 Properties of Numbers

Even and Odd Numbers

Important 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)
• 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

1.3 Face Value and Place Value

• Face Value: The digit itself regardless of position
  • Face value of 7 in 5,730 = 7
• Place Value: Face value × positional value
  • Place value of 7 in 5,730 = 7 × 100 = 700

1.4 Basic Formulae

1.5 Finding Greatest/Least n-digit Number Divisible by a Number

A) Greatest 'n'-digit number exactly divisible by a number 'k':

1. Find the greatest n-digit number (e.g., for 4 digits: 9999)
2. Divide by k
3. Find the remainder r
4. Result = 9999 - r

Example: Greatest 4-digit number divisible by 7:

• 9999 ÷ 7 → remainder = 5
• Required number = 9999 - 5 = 9994

B) Least 'n'-digit number exactly divisible by 'k':

1. Find the least n-digit number (e.g., for 4 digits: 1000)
2. Divide by k
3. Find the remainder r
4. Result = Smallest n-digit number + (k - r) [if r ≠ 0]

Example: Least 4-digit number divisible by 7:

• 1000 ÷ 7 → remainder = 6
• Required number = 1000 + (7 - 6) = 1001

1.6 Number Series and Patterns

• Arithmetic Progression (AP): a, a+d, a+2d, a+3d... (common difference d)
  • nth term: Tn = a + (n-1)d
  • Sum of n terms: Sn = n/2 × [2a + (n-1)d]
• Geometric Progression (GP): a, ar, ar², ar³... (common ratio r)
  • nth term: Tn = ar^(n-1)
  • Sum of n terms: Sn = a(rⁿ - 1)/(r - 1) when r ≠ 1