बाइनरी नंबर सिस्टम

• Binary number system is a base-2 numeral system that uses two symbols: 0 and 1.

Basics of Binary Numbers

• Each digit in a binary number represents a power of 2.
• The rightmost digit is the least significant bit (LSB), representing 2^0.
• The leftmost digit is the most significant bit (MSB), representing the highest power of 2.
• Binary numbers are fundamental to computers because digital electronic devices operate using two voltage levels, which are often interpreted as 0 and 1.

Converting Decimal to Binary

• Repeated Division by 2: Divide the decimal number by 2 and note the remainder. Continue dividing the quotient by 2 until the quotient is 0. The remainders, read in reverse order, form the binary number.
• Example: Convert 25 to binary.
  • 25 ÷ 2 = 12, remainder 1
  • 12 ÷ 2 = 6, remainder 0
  • 6 ÷ 2 = 3, remainder 0
  • 3 ÷ 2 = 1, remainder 1
  • 1 ÷ 2 = 0, remainder 1
  • Binary representation: 11001

Converting Binary to Decimal

• Sum of Powers of 2: Multiply each digit by 2 raised to the power of its position (starting from 0 on the right) and sum the results.
• Example: Convert 1101 to decimal.
  • (1 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 8 + 4 + 0 + 1 = 13

Binary Arithmetic

• Addition: Binary addition follows similar rules to decimal addition, but with only two digits.
  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 0 = 1
  • 1 + 1 = 10 (0 with a carry of 1)
• Subtraction: Similar to decimal subtraction, borrowing from the next higher bit when necessary.
  • 0 - 0 = 0
  • 1 - 0 = 1
  • 1 - 1 = 0
  • 0 - 1 = 1 (with a borrow of 1 from the next bit)
• Multiplication: Similar to decimal multiplication.
  • 0 * 0 = 0
  • 0 * 1 = 0
  • 1 * 0 = 0
  • 1 * 1 = 1
• Division: Similar to decimal division

Data Representation

• Data representation refers to how data is stored and represented in a computer system.
• Computers store all types of data (numbers, text, images, audio, video) as binary numbers.

Integer Representation

• Unsigned Integers: Represent positive integers directly in binary. The number of bits determines the range of values that can be represented.
• Signed Integers: Represent both positive and negative integers. Common methods include:
  • Sign-Magnitude: The most significant bit (MSB) represents the sign (0 for positive, 1 for negative), and the remaining bits represent the magnitude.
  • One's Complement: To find the one's complement of a number, invert all the bits (change 0s to 1s and 1s to 0s).
  • Two's Complement: To find the two's complement, invert all the bits (one's complement) and add 1. Two's complement is the most common method for representing signed integers in computers because it simplifies arithmetic operations.

Floating-Point Representation

• Used to represent real numbers (numbers with fractional parts).
• Follows the IEEE 754 standard, which defines how floating-point numbers are stored.
• Components:
  • Sign Bit: Indicates whether the number is positive or negative.
  • Exponent: Represents the power of 2 to scale the number.
  • Mantissa (or Significand): Represents the significant digits of the number.
• Single-Precision (32-bit) and Double-Precision (64-bit) formats are commonly used.

Character Representation

• Characters are represented using encoding schemes that assign a unique numeric code to each character.
• ASCII (American Standard Code for Information Interchange): A 7-bit encoding scheme that represents 128 characters, including uppercase and lowercase letters, digits, punctuation marks, and control characters.
• Extended ASCII: 8-bit encoding schemes that extend ASCII to include additional characters, such as accented letters and symbols.
• Unicode: A more comprehensive encoding standard that supports a much larger range of characters, including characters from various languages around the world.
  • UTF-8 (Unicode Transformation Format - 8-bit): A variable-width encoding that represents Unicode characters using 1 to 4 bytes.
  • UTF-16 (Unicode Transformation Format - 16-bit): A variable-width encoding that represents Unicode characters using 2 or 4 bytes.
  • UTF-32 (Unicode Transformation Format - 32-bit): A fixed-width encoding that represents each Unicode character using 4 bytes.

Image Representation

• Images are represented as a collection of pixels, where each pixel represents a color.
• Bitmap Images: Store the color of each pixel directly. Common formats include BMP, PNG, and GIF.
• Vector Images: Store images as mathematical equations that describe shapes and lines. Common formats include SVG.
• Color Models:
  • RGB (Red, Green, Blue): Represents colors as a combination of red, green, and blue light.
  • CMYK (Cyan, Magenta, Yellow, Key/Black): Used in printing and represents colors as a combination of cyan, magenta, yellow, and black ink.
  • HSL (Hue, Saturation, Lightness): Represents colors based on hue, saturation, and lightness.

Audio Representation

• Audio is represented as a sequence of samples, where each sample represents the amplitude of the sound wave at a particular point in time.
• Sampling Rate: The number of samples taken per second, measured in Hertz (Hz).
• Bit Depth: The number of bits used to represent each sample, which determines the dynamic range of the audio.
• Common Audio Formats:
  • WAV: An uncompressed audio format.
  • MP3: A compressed audio format that uses lossy compression to reduce file size.
  • AAC: A compressed audio format that provides better audio quality than MP3 at the same bit rate.

Video Representation

• Video is represented as a sequence of frames, where each frame is an image.
• Frame Rate: The number of frames displayed per second, measured in frames per second (fps).
• Video Compression: Used to reduce the file size of video files. Common compression techniques include:
  • Lossy Compression: Removes some data from the video to reduce file size, which can result in a loss of quality. Examples include MPEG, H.264, and H.265.
  • Lossless Compression: Compresses the video without losing any data, but typically results in larger file sizes than lossy compression.
• Common Video Formats:
  • MP4: A container format that can store video, audio, and metadata.
  • AVI: An older container format that is less efficient than MP4.
  • MKV: A flexible container format that can store a variety of video and audio codecs.

Importance of Data Representation

• Efficient storage and retrieval of data.
• Accurate processing and manipulation of data.
• Compatibility between different computer systems and applications.
• Effective communication of information.