Computer Science: Number Systems and Data Representation

Introduction to the Number System

• Conceptual Overview: Modern computing relies on different number systems to represent data, primarily Binary, Denary, and Hexadecimal.

• Denary System (Base 10):     

  * Uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.     

  * Place values are based on powers of 10:         

  * $10^0 = 1$ (Ones)         

  * $10^1 = 10$ (Tens)        

  * $10^2 = 100$ (Hundreds)     

  * Explanation: To find a value, multiply the digit by the place value and sum the results.     

  * Example (365): $(3 \times 100) + (6 \times 10) + (5 \times 1) = 365$.

The Binary System

• Motivation:     

  * Computers consist of millions of tiny switches that can exist in only two states: On or Off.     

  * 1 represents On; 0 represents Off.     

  * Because computers only process electronic signals, all information must be transformed into binary format to be processed.

• Structure (Base 2):     

  * Uses 2 digits: 0 and 1.     

  * Place values are based on powers of 2:         

  * $2^0 = 1$         * $2^1 = 2$         * $2^2 = 4$         * $2^3 = 8$         * $2^4 = 16$         * $2^5 = 32$         * $2^6 = 64$         * $2^7 = 128$

Conversion Procedures: Binary and Denary

• Binary to Denary Conversion:     

  * Method: Multiply the digit value (0 or 1) by its corresponding place value ($2^n$) and sum the results.     * Example 1 (111): $(1 \times 4) + (1 \times 2) + (1 \times 1) = 7$.     

  * Example 2 (1011): $(1 \times 8) + (0 \times 4) + (1 \times 2) + (1 \times 1) = 11$.     

  * DIY Exercise: The binary form of 15 is $1111$ ($8 + 4 + 2 + 1$).     

  * DIY Exercise: The denary form of "1010" is 10 ($1 \times 8 + 1 \times 2$).

• Denary to Binary Conversion (Method 1 - Place Value Subtraction):     

  * Identify the largest power of 2 that fits into the denary number.     

  * Place a '1' in that column and subtract the value from the total; Repeat until the total is 0.     

  * Example (5 to Binary): $5 = 4 + 1$. Place a 1 in column 4, 0 in column 2, and 1 in column 1. Result: $101$.

• Denary to Binary Conversion (Method 2 - Successive Division):     

  * Procedure: Divide the denary number by 2 and record the remainder (0 or 1). Continue dividing the quotient by 2 until the result is 0. Read the remainders from bottom to top.     

  * Example (39 to Binary):         

  * $39 \div 2 = 19$ remainder 1         

  * $19 \div 2 = 9$ remainder 1        

  * $9 \div 2 = 4$ remainder 1         

  * $4 \div 2 = 2$ remainder 0         

  * $2 \div 2 = 1$ remainder 0         

  * $1 \div 2 = 0$ remainder 1         

  * Result: $100111$

The Hexadecimal System

• Motivation:     

  * Hexadecimal is used because it is easier for humans to read/write.     

  * It uses fewer characters and is less error-prone when copying data.

• Structure (Base 16):     

  * Uses 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.     

  * Mapping to Denary:         

  * $A = 10$         

  * $B = 11$        

  * $C = 12$         

  * $D = 13$         

  * $E = 14$         

  * $F = 15$     

  * Place Values: $16^0 = 1$, $16^1 = 16$, $16^2 = 256$, $16^3 = 4096$.

Conversion Procedures: Hexadecimal, Binary, and Denary

• Binary to Hexadecimal Conversion:     

  * Principle: Since $16 = 2^4$, four binary digits (a nibble) are equivalent to one hexadecimal digit.     

  * Procedure: Group the binary string into sets of 4 bits (starting from the right). Convert each group individually.     

  * Example (101111100001):         

  * 1011 = B         

  * 1110 = E         

  * 0001 = 1         

  * Result: BE1.

• Hexadecimal to Binary Conversion:     

  * Procedure: Convert each hex digit into its 4-bit binary equivalent.     

  * Example (F935):         

  * F = 1111         

  * 9 = 1001         

  * 3 = 0011         

  * 5 = 0101         

  * Result: 1111 1001 0011 0101

• Hexadecimal to Denary Conversion:     

  * Method: Multiply the digit value by its place value ($16^n$) and sum.     

  * Example (45A): $(4 \times 256) + (5 \times 16) + (10 \times 1) = 1024 + 80 + 10 = 1114$.     

  * DIY Exercise (BF08): $(11 \times 4096) + (15 \times 256) + (0 \times 16) + (8 \times 1) = 45056 + 3840 + 0 + 8 = 48904$.

• Denary to Hexadecimal Conversion:     

  * Procedure: Successive division by 16. Record remainders.     

  * Example (2004 to Hex):         

  * $2004 \div 16 = 125$ remainder 4         

  * $125 \div 16 = 7$ remainder 13 (D)         

  * $7 \div 16 = 0$ remainder 7         

  * Result: 7D4  

Binary Arithmetic: Addition and Shifting

• Addition Rules:     

  * $0 + 0 = 0$     

  * $0 + 1 = 1$     

  * $1 + 0 = 1$     

  * $1 + 1 = 10$ (0 carry 1)     

  * $1 + 1 + 1 = 11$ (1 carry 1)

• Overflow Condition:     

  * An 8-bit binary register can hold a maximum value of 255 ($2^8 - 1$).     

  * If an addition results in a 9th bit, this is an overflow error.     

  * The sum is too large to be stored in the assigned 8 bits.     

  * Example: $110_{10} + 222_{10} = 322_{10}$. Since 322 > 255, overflow occurs.

• Binary Shifting:     

  * Used by the CPU for rapid multiplication and division.     

  * Multiplication (Left Shift): Shift bits to the left and fill empty spaces with 0.         

  * 1 place shift = $\times 2$         

  * 2 place shift = $\times 4$         

  * $n$ place shift = $\times 2^n$         

  * Data Loss: Shifting a significant bit (1) beyond the register boundary results in overflow and loss of precision.     

  * Division (Right Shift): Shift bits to the right.         

  * 1 place shift = $\div 2$         

  * $n$ place shift = $\div 2^n$         

  * Precision Loss: Shifting the "Least Significant Bit" off the right side results in discarded fractional data (e.g., instead of 5.9, you get 5).

Two's Complement (Negative Binary Representation)

• Definition: A method for processors to represent negative numbers.

• The Sign Bit: In an 8-bit two's complement system, the leftmost bit (8th column) is the sign bit with a value of $-128$.     

  * 0 = Positive     

  * 1 = Negative

• Conversion (Positive to Two's Complement):     

  * Step 1: Convert the positive number to 8-bit binary.     

  * Step 2: Ensure the leftmost bit is 0.     

  * Example (13): Result is $00001101$.

• Conversion (Negative Denary to Two's Complement):     

  * Step 1: Convert the absolute value (positive form) to binary.     

  * Step 2: Invert all bits (0 becomes 1, 1 becomes 0).     

  * Step 3: Add 1 to the result.     

  * Example (-67):         1. $67 = 01000011$         2. Invert = $10111100$         3. Add 1 = $10111101$

• Conversion (Two's Complement to Denary):     

  * Sum the values, treating the leftmost bit as $-128$.     

  * Example binary (10110011): $-128 + 32 + 16 + 2 + 1 = -77$.

Data Representation: Text, Sound, and Image

• Text Representation:     

  * Every character (letters, spaces, punctuation) is assigned a binary code via a Character Set.     

  * ASCII: Uses 7 bits (128 characters). Includes 26 uppercase, 26 lowercase, digits, and punctuation.     * Extended ASCII: Uses 8 bits (256 characters) to include non-English alphabets and graphics.     

  * Unicode: Uses variable length (often 16 bits), enabling over 65,000 characters. Support for worldwide languages, emojis, and symbols.

• Sound Representation:     * Analog signals must be sampled to be converted to digital format.     

  * Sample: A digital recording of the sound wave amplitude at a specific time.     

  * Sampling Resolution (Bit Depth): The number of bits used to represent the amplitude. Higher resolution increases accuracy and dynamic range.     

  * Sample Rate: The number of samples taken per second, measured in Hertz (Hz).     

  * File Size Impact: Higher rates and bit depths lead to better quality and less distortion but produce larger files.     

  * CD Quality: 16-bit sampling resolution and 44.1 kHz sample rate (44,100 samples/sec).

• Image Representation:     

  * Bitmap: A collection of bits that form a grid of pixels (picture elements).     

  * Colour Depth: The number of bits used per pixel to represent color.         

  * 1-bit: 2 colors (0, 1)         

  * 2-bit: 4 colors         

  * 8-bit: 256 colors         

  * 24-bit: Over 16 million colors     

  * Image Resolution: Total number of pixels in the grid (e.g., 1024 x 1080). Higher resolution improves detail but increases file size.

Data Storage and Calculations

• Basic Units:     

  * 1 bit: Smallest unit (0 or 1).     

  * 1 nibble: 4 bits.     

  * 1 byte: 8 bits.

• Binary Prefixes (Base 2):     

  * 1 Kibibyte (KiB): $2^{10} = 1024$ bytes.     

  * 1 Mebibyte (MiB): $2^{20} = 1,048,576$ bytes.     

  * 1 Gibibyte (GiB): $2^{30} = 1,073,741,824$ bytes.     

  * 1 Tebibyte (TiB): $2^{40}$ bytes.     

  * 1 Pebibyte (PiB): $2^{50}$ bytes.

• File Size Formulas:     

  * Image Size (bits): $\text{Resolution (Pixels)} \times \text{Colour Depth (bits)}$     

  * Sound Size (bits):$\text{Sample Rate (Hz)} \times \text{Sample Resolution (bits)} \times \text{Duration (seconds)} \times \text{Channels}$     * Note: For stereo sound, multiply the total by 2.

Data Compression

• Benefits: Saves storage space, reduces streaming time, decreases upload/download times, and reduces costs.

• Lossy Compression:     

  * Eliminates unnecessary data permanently. The original file cannot be reconstructed.     

  * JPEG: Removes color shades humans cannot discern.     

  * MPEG-3 (MP3): Removes sounds outside human hearing range and uses perceptual music shaping.

  * MPEG-4 (MP4): Used for multimedia/video.

• Lossless Compression:     

  * None of the original detail is lost. Necessary for files like spreadsheets or programs where data loss would make the file unusable.     

  * Run-Length Encoding (RLE): Identifies adjacent, identical data items and encodes one value representing the item and another representing the count.         

  * Example: "ssssrrrrkkkjjjjj" becomes "4s4r3k5j".         

  * Limitation: Does not work well without repeated adjacent data.

Practical Applications of Hexadecimal

• MAC Address (Media Access Control):

  * A unique identifier assigned to a Network Interface Card (NIC).   

  * Format: Six pairs of hex digits (e.g., $97\text{-}5C\text{-}E1\text{-}39\text{-}4B\text{-}97$).     

  * First half: Identity of the manufacturer.     

  * Second half: Identity number of the device.

• IP Addresses:     

  * IPv4: 32-bit (often decimal or hex; e.g., 128.65.152.11).    

  * IPv6: 128-bit, divided into 16-bit segments and represented in hexadecimal.

• HTML Colour Codes: Used in web design to define colors using hex triplets (e.g., #FF5733).

• Other Uses: Assembly language, Error codes (referencing memory locations), Memory dumps/locations, and URLs.

Questions & Discussion

• Q: Why use hex for MAC addresses?    

  * A: It is shorter, uses fewer characters, easier to understand/read, and less likely to contain mistakes during manual entry.

• Q: What happens if you add two 8-bit binaries and get a 9-bit result?     

  * A: This is an overflow error; the cumulative value exceeds the capacity of the byte-sized register.

• Q: Is HTML a programming language?     

  * A: No, it is a markup language used for the processing, definition, and presentation of text on web pages.

• Q: Comparison of Lossy vs Lossless for sound?     

  * Advantage (Lossy): The file size is significantly smaller and requires less storage.     

  * Disadvantage (Lossy): Sound quality is reduced and the original state cannot be restored.

• Q: Why use lossless for programs/executable files?    

  * A: Lossy compression would remove actual data/code. Programs need every bit intact to run correctly; otherwise, they will not work or will crash.