Binary and Data Representation

Understanding Binary

Binary is the fundamental language of computers, using only two digits (0 and 1) to represent all data. Each binary digit (bit) represents a state that can be either on (1) or off (0).

Key Binary Concepts

  • Binary digits (bits) - The smallest unit of data (0 or 1)
  • Bytes - 8 bits grouped together
  • Binary patterns - Sequences of bits that represent data
  • Base-2 number system - Powers of 2 (1, 2, 4, 8, 16, etc.)

Binary Number System

Converting Between Binary and Decimal

Binary Calculation Decimal
0001 0×8 + 0×4 + 0×2 + 1×1 1
0101 0×8 + 1×4 + 0×2 + 1×1 5
1000 1×8 + 0×4 + 0×2 + 0×1 8
1111 1×8 + 1×4 + 1×2 + 1×1 15

Decimal to Binary Conversion Steps:

  1. Divide the number by 2
  2. Keep track of remainders (0 or 1)
  3. Continue dividing quotient by 2
  4. Read remainders from bottom to top
binary_conversion.txt 
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3  remainder 0
3 ÷ 2 = 1  remainder 1
1 ÷ 2 = 0  remainder 1

Reading from bottom up: 1101

Data Representation

Text Representation (ASCII)

ASCII (American Standard Code for Information Interchange) uses 7 bits to represent characters.

Character ASCII (Decimal) Binary
A 65 1000001
B 66 1000010
a 97 1100001
1 49 0110001

Color Representation (RGB)

Colors are represented using three bytes, one each for Red, Green, and Blue values (0-255).

Color Red Green Blue Hex Code
Red 255 0 0 #FF0000
Purple 128 0 128 #800080
White 255 255 255 #FFFFFF

Image Representation

Digital images are grids of pixels, where each pixel is represented by color values.

Image Storage Components:

  • Resolution (width × height in pixels)
  • Color depth (bits per pixel)
  • Compression method (lossy or lossless)
  • File format (JPEG, PNG, GIF, etc.)

Data Storage Units

Unit Size Example Usage
Bit (b) 1 or 0 Single binary digit
Byte (B) 8 bits Single character
Kilobyte (KB) 1,024 bytes Short text document
Megabyte (MB) 1,024 KB Digital photo
Gigabyte (GB) 1,024 MB Movie file
Terabyte (TB) 1,024 GB Large database

Data Compression

Lossless Compression

Reduces file size while preserving all original data. Commonly used for text and program files.

compression_example.txt 
Original: AAAAAABBBCC
Compressed: 6A3B2C
(Saves space while maintaining all information)

Lossy Compression

Reduces file size by removing some data, typically used for media files where small quality loss is acceptable.

Common Lossy Formats:

  • JPEG - Images
  • MP3 - Audio
  • MP4 - Video

Error Detection and Correction

Parity Bits

A simple error detection method that adds an extra bit to make the total number of 1s even or odd.

Data Count of 1s Even Parity Bit Result
1101 3 1 11011
1001 2 0 10010

Checksum

A more advanced error detection method that creates a value based on the data being transmitted.

checksum_example.txt 
Data: 10110
Checksum calculation: 1+0+1+1+0 = 3
Transmitted: 10110 (data) + 11 (checksum in binary)

Practice Problems

Problem 1: Binary Conversion

Convert the decimal number 25 to binary.

Solution
problem1_solution.txt 
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: 11001

Problem 2: Data Storage

Calculate the storage required for a 1920×1080 image with 24-bit color depth.

Solution
problem2_solution.txt 
Pixels: 1920 × 1080 = 2,073,600
Bits per pixel: 24
Total bits: 2,073,600 × 24 = 49,766,400
Total bytes: 49,766,400 ÷ 8 = 6,220,800
Total MB: 6,220,800 ÷ 1,024,000 ≈ 6.07 MB

Problem 3: Error Detection

Add an even parity bit to the binary number 1011001.

Solution
problem3_solution.txt 
Data: 1011001
Count of 1s: 4 (even)
Even parity bit needed: 0
Result: 10110010

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