সংখ্যা পদ্ধতিৰ ধাৰণা

• Mathematics is the study of topics such as quantity, structure, space, and change
• It uses patterns to formulate new conjectures and resolve their truth or falsity by mathematical proof
• Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and social sciences

Number System

• A number system is a system for representing numbers
• It defines a set of values to represent a quantity
• It is also known as the system of numeration
• Numbers are represented with a defined set of symbols
• Number systems enable us to perform arithmetic operations like addition, subtraction, multiplication, and division
• The value of each digit in a number is determined by:
  • The digit
  • The position of the digit in the number
  • The base of the number system

Common Number Systems

• Decimal (Base 10)
  • Uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
  • Each position represents a power of 10
• Binary (Base 2)
  • Uses two digits: 0, 1
  • Each position represents a power of 2
  • Fundamental to digital electronics and computer science
• Octal (Base 8)
  • Uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7
  • Each position represents a power of 8
  • Used in computing for representing binary numbers in a more human-readable form
• Hexadecimal (Base 16)
  • Uses sixteen symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
  • Each position represents a power of 16
  • Commonly used in computing to represent binary data

Decimal Number System

• Base 10 system
• Uses ten digits (0-9)
• The position of each digit is a power of 10
• For example, the number 123 is interpreted as: (1 * 10^2) + (2 * 10^1) + (3 * 10^0) = 100 + 20 + 3 = 123

Binary Number System

• Base 2 system
• Uses two digits (0 and 1)
• Each position represents a power of 2
• For example, the binary number 1011 is interpreted as: (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) = 8 + 0 + 2 + 1 = 11 (in decimal)

Octal Number System

• Base 8 system
• Uses eight digits (0-7)
• Each position represents a power of 8
• For example, the octal number 123 is interpreted as: (1 * 8^2) + (2 * 8^1) + (3 * 8^0) = 64 + 16 + 3 = 83 (in decimal)

Hexadecimal Number System

• Base 16 system
• Uses sixteen symbols (0-9 and A-F)
• The letters A-F represent the decimal numbers 10-15 respectively
• Each position represents a power of 16
• For example, the hexadecimal number 2A is interpreted as: (2 * 16^1) + (10 * 16^0) = 32 + 10 = 42 (in decimal)
• Hexadecimal is used because it provides a human-friendly representation of binary-coded values

Number Base Conversions

• Converting a number from one base to another is a common task

Decimal to Binary Conversion

• Divide the decimal number by 2
• Note the quotient and remainder
• Repeat the division with the quotient until the quotient is 0
• The binary number is the sequence of remainders in reverse order

Binary to Decimal Conversion

• Multiply each digit by 2 raised to the power of its position (starting from 0 on the right)
• Sum the results

Decimal to Hexadecimal Conversion

• Divide the decimal number by 16
• Note the quotient and remainder
• If the remainder is greater than 9, convert it to its hexadecimal symbol (A-F)
• Repeat the division with the quotient until the quotient is 0
• The hexadecimal number is the sequence of remainders in reverse order

Hexadecimal to Decimal Conversion

• Multiply each digit by 16 raised to the power of its position (starting from 0 on the right)
• If the digit is a letter (A-F), convert it to its decimal value (10-15) before multiplying
• Sum the results

Real Numbers

• Real Numbers include both rational and irrational numbers
• Can be positive, negative or zero
• Can be represented on a number line
• Decimal representation terminates, repeats, or is non-repeating and non-terminating

Rational Numbers

• Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0
• Decimal representation either terminates or repeats

Irrational Numbers

• Numbers that cannot be expressed as a fraction of two integers
• Decimal representation is non-repeating and non-terminating
• Examples: π (pi), √2 (square root of 2), e (Euler's number)

Complex Numbers

• Numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1)
• 'a' is the real part, and 'b' is the imaginary part
• If b = 0, then a + bi is a real number
• If a = 0, then a + bi is a purely imaginary number

Integers

• The set of whole numbers and their opposites
• Includes: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Can be positive, negative, or zero

Positive Integers

• Also known as natural numbers or counting numbers: 1, 2, 3, ...

Negative Integers

• The opposites of the positive integers: -1, -2, -3, ...

Zero

• Neither positive nor negative integer
• An integer representing the absence of quantity

Properties of Number Systems

• Closure: A number system is closed under an operation if performing that operation on numbers in the system always results in a number that is also in the system
• Associativity: An operation is associative if the grouping of numbers being operated on does not affect the result
• Commutativity: An operation is commutative if the order of the numbers being operated on does not affect the result
• Identity Element: An element that, when combined with any number under an operation, leaves that number unchanged
• Inverse Element: An element that, when combined with a number under an operation, results in the identity element

Applications of Number Systems

• Computer Science: Binary, octal, and hexadecimal number systems are used extensively in computer programming, data representation, and digital electronics
• Cryptography: Number theory and modular arithmetic, which are aspects of number systems, are essential in cryptographic algorithms
• Engineering: Number systems are used in signal processing, control systems, and various engineering calculations
• Finance: Decimal number system is crucial for financial calculations, accounting, and economic analysis
• Data storage: Number systems define how efficiently data is represented and stored in computer memory