Lecture on Number System

Introduction

• Focus on creating questions from each line.
• Topic of discussion: संख्‍या पद्धति (Number System).

Types of Numbers

• Complex Numbers (सम्मिश्र संख्याएं)
  • Represented as a + ib where a is the real part and b is the imaginary part.
  • Complex parts: Real part (वास्तविक भाग) and Imaginary part (काल्पनिक भाग).
• Distribution of Real Numbers (वास्तविक संख्याएं)
  • Rational (परिमेय), Irrational (अपरिमेय), National Numbers (प्राकृतिक संख्याएं), Integers (पूर्णांक), Whole Numbers (पूर्ण संख्याएं), Natural Numbers (प्राकृतिक संख्याएं).

Integers (पूर्णांक)

• Defined as whole numbers including negative and positive.
• Examples: -∞ to +∞ along the number line.
• Greatest Integer Function (ग्रेटेस्ट इंटीजर फंक्शन)
  • Example: Greatest integer of 1.2 is 1. -Notation: [x], Greatest Integer is represented by [x].

Properties of Integers

• Positive Integers: Whole positive numbers including zero.
• Negative Integers: Whole negative numbers.
• Zero: Classified as an even integer

Natural Numbers (प्राकृतिक संख्याएं)

• Start from 1 and go up: 1, 2, 3, 4,...
• Do not include zero and negative numbers.

Whole Numbers (पूर्ण संख्याएं)

• Include all natural numbers and zero.
• Examples: 0, 1, 2, 3, 4, 5, ...

Prime Numbers (अभाज्य संख्या)

• Defined as numbers divisible by 1 and itself.
• Smallest prime number: 2 (also only even prime number)
• Examples: 2, 3, 5, 7, 11, 13, 17...

Prime Numbers within Specific Ranges

• 1 to 25: 9 primes.
• 25 to 50: 10 primes.
• 50 to 75: 6 primes.
• 75 to 100: 4 primes.

Composite Numbers (संयोजित संख्याएं)

• Defined as numbers that are not prime.
• Examples: 4, 6, 8, 9, 10...

Twin Primes (द्विक अभाज्य संख्याएं)

• Pairs of prime numbers with a difference of two.
• Examples: 3, 5; 11, 13; 17, 19.

Co-prime Numbers (सम अभाज्य संख्याएं)

• Pairs of numbers having a GCD of 1.
• Examples: 25 and 37.

Perfect Numbers (पूर्णांक)

• Numbers whose proper divisors sum to the number itself.
• Examples: 6, 28, 496, 8128.

Divisibility Rules (भाग महारत नियम)

• 2: Last digit is even.
• 3: Sum of digits is divisible by 3.
• 5: Last digit is either 0 or 5.
• 9: Sum of digits divisible by 9.
• 11: Difference between sum of digits at odd places and sum of digits at even places either 0 or a multiple of 11.

Place Value and Face Value (स्थानीय मान और जातीय मान)

• Place value depends on digit's position in the number.
• Face value is the digit itself.
• Examples:
  • Number 5789:
    • Place Value: 5 in thousands place = 5000
    • Face Value: 5 = 5.

Place Values in Decimals

• Example: 54.321
  • Place Value of 4 in the tens place = 40.
  • Place Value of 3 in the tenths place = 0.3.

Local and Decimal Systems (दशमलव प्रणाली)

• Understanding how each digit represents powers of 10.
• Example: 321.45 = 3x100 + 2x10 + 1x1 + 4x0.1 + 5x0.01.