Topic A1.2 - Data Representation and Computer Logic

What is a Bit and what are its possible values?

The smallest unit of data in computing, representing a single binary digit that can only have a value of 0 (OFF/FALSE) or 1 (ON/TRUE).

What is a Byte and how many bits does it contain?

A group of 8 bits used as a fundamental unit of digital storage.

Why do computers use Binary (Base-2) for processing, storage, and transmission?

Because electronic circuits and transistors operate using two distinct electrical states (ON/OFF or high/low voltage), making binary reliable and straightforward to process logic and math.

Convert the 8-bit binary number 01011101 into base 2 (Decimal).

93 (Calculated as 64 + 16 + 8 + 4 + 1 = 93).

How many binary bits does a single Hexadecimal (Base-16) digit represent?

4 binary bits (1 nibble).

Convert the Hexadecimal value 2A5 into Denary (Decimal).

677 (Calculated as 2*(16^2) + 10*(16^1) + 5*(16^0) = 512 + 160 + 5 = 677).

Convert the Hexadecimal value 3B into Denary (Decimal) and Binary.

Denary: 59 (3*16 + 11*1). Binary: 00111011.

Explain the step-by-step method to convert a Denary number (e.g., 495) into Hexadecimal.

1. Find the highest power of 16 that fits into the number (16^2 = 256). 2. Divide by 256 to get the first hex digit. 3. Take the remainder and divide by 16 to get the second digit. 4. The final remainder gives the last hex digit.

Explain the bit structure and character capacity of standard ASCII.

Uses 8 bits total per character slot (7 bits store character data giving 2^7 = 128 unique characters, while 1 bit is used as a parity bit). Covers uppercase, lowercase, numbers, symbols, spaces, and punctuation.

Compare ASCII (and Extended-ASCII) with Unicode in terms of bit size and character support.

ASCII uses 7-8 bits per character (max 128-256 characters, limited to Western alphabets). Unicode uses 16-32 bits per character, supporting millions of characters including non-Western scripts, historical symbols, and emojis.

Contrast Decimal (SI Metric) storage prefixes with Binary (IEC) storage prefixes.

Decimal (SI Metric): Based on powers of 10 (1 kB = 1,000 bytes = 10^3). Binary (IEC): Based on powers of 2 (1 KiB = 1,024 bytes = 2^10, 1 MiB = 2^20, 1 GiB = 2^30, 1 TiB = 2^40).

State typical storage requirements for standard data types: ASCII char, Word 'Monday', Plain-text email, 3-min MP3, 1-hr MPEG4 video.

1 Extended-ASCII char: 1 byte. Word 'Monday': 6 bytes. Plain-text email: ~2 KB. 3-min MP3 audio: ~3 MB. 1-hr MPEG4 video: ~4 GB.

What is a Pixel and how does it function in digital displays?

The smallest controllable element in a display grid, representing a single colored square. Each pixel displays one solid color at a time formed by combining red, green, and blue light (RGB).

What is a Bitmap Image?

An image composed of a grid of individual pixels, where each pixel has a specific binary value defining its color and intensity (e.g., JPEG, PNG, GIF, BMP).

How is 24-bit True Color represented in pixels using Hexadecimal?

Each pixel color is stored as a 6-digit Hexadecimal code (#RRGGBB). The first two digits represent Red, middle two represent Green, and last two represent Blue (giving 256^3 = 16,777,216 color combinations).

Define Color Depth (Bit Depth) in bitmap graphics.

The number of bits used to represent the color of a single pixel. Formula: $2^n$ unique colors, where $n$ is the bit depth (e.g., 3 bits = 8 colors, 24 bits = 16.7M colors).

What factors affect Bitmap Image Quality?

1. Number of pixels (more pixels mean higher detail and smoother transitions); 2. Image resolution (higher PPI creates finer detail); 3. Screen resolution (higher screen PPI displays images sharper); 4. Color/Bit depth (higher depth prevents color banding and increases realism).

What factors affect Bitmap File Size?

1. Higher Image Resolution (more total pixels increase file size); 2. Higher Screen Resolution (indirectly increases size as images are rendered at higher detail); 3. Higher Color/Bit Depth (more bits stored per pixel increase file size).

Explain the 3 steps of Analog-to-Digital Audio Conversion (ADC).

1. Sampling: Measuring the continuous sound wave's amplitude at discrete time intervals (Sample Rate in Hz); 2. Quantization: Rounding sampled amplitudes to the nearest binary value defined by the bit depth; 3. Encoding: Converting quantized values into binary data.

Define Sample Rate and Bit Depth in digital audio encoding.

Sample Rate: The number of audio samples taken per second, measured in Hertz (e.g., 44.1 kHz for CD quality, 48 kHz for professional audio). Bit Depth: The number of bits used per sample to record amplitude accuracy (e.g., 16-bit, 24-bit).

How is digital audio played back through speakers or headphones?

The stored digital binary data is decoded and converted back into a continuous analog electrical signal using a Digital-to-Analog Converter (DAC).

What audio file formats correspond to uncompressed, lossy compressed, and lossless compressed audio?

Uncompressed: WAV. Lossy Compressed: MP3. Lossless Compressed: FLAC.

What is a Video Frame and how is digital video represented?

A video is a sequence of still images called frames played rapidly in sequence. Each frame is a bitmap image made of pixels storing RGB color values.

How does Inter-Frame Video Compression (e.g., H.264) reduce file size?

Instead of saving every full frame, it saves keyframes and stores only the differences/changes between consecutive frames over time.

What is the formula to calculate the raw (uncompressed) file size of a video?

Raw File Size = (Width × Height) × Bit Depth × Frame Rate (fps) × Duration (seconds).

Solve this IB calculation: A 20-second video recorded at 30 fps, 1280×720 resolution, 24-bit color, compressed at a 20:1 ratio. What is the final file size in MB?

Raw Size = (1280 × 720) × 24 bits × 30 fps × 20 s = 13,271,040,000 bits. Convert to Bytes = / 8 = 1,658,880,000 Bytes. Apply 20:1 Compression = / 20 = 82,944,000 Bytes. Convert to MB = / 1,000,000 = 82.94 MB (or ~79.1 MiB using binary IEC).

What is a Logic Gate and how does it relate to physical CPU hardware?

An electronic switch/circuit in a processor that takes 1 or 2 binary inputs (0 = low voltage, 1 = high voltage) and produces a single binary output based on a Boolean operation.

What are Logic Gates used for in computer systems?

1. Performing basic Boolean operations (AND, OR, NOT); 2. Building complex circuits (adders, multiplexers, memory units); 3. Processing binary data to perform calculations and control tasks.

State the logic rules and outputs for AND, OR, and NOT gates.

AND: Output is 1 only if BOTH inputs are 1. OR: Output is 1 if AT LEAST ONE input is 1. NOT: Inverts the single input (0 becomes 1, 1 becomes 0).

State the logic rules and outputs for NAND, NOR, and XOR (Exclusive OR) gates.

NAND: Output is 0 only if BOTH inputs are 1 (inverse of AND). NOR: Output is 1 only if BOTH inputs are 0 (inverse of OR). XOR: Output is 1 if inputs are DIFFERENT (one is 1, the other is 0).

State the logic rule for XNOR (Exclusive NOR) gates.

XNOR: Output is 1 if inputs are identical (both 0 or both 1).

What are the fundamental laws of Boolean Algebra for simplification?

Identity (1A=A, 0+A=A); Null (0A=0, 1+A=1); Idempotent (AA=A, A+A=A); Inverse (A·Ā=0, A+Ā=1); Double Negation (Ā̄=A); Commutative (AB=BA, A+B=B+A); Associative ((AB)C=A(BC)); Distributive (A+BC=(A+B)(A+C), A(B+C)=AB+AC).

State the Absorption Laws and De Morgan's Laws in Boolean Algebra.

Absorption Laws: A(A+B) = A and A + AB = A. De Morgan's Laws: NOT(A AND B) = NOT A OR NOT B (ĀB̄ = Ā + B̄) and NOT(A OR B) = NOT A AND NOT B (Ā+B̄ = Ā·B̄).

What is a Karnaugh Map (K-Map) and what are its key grouping rules?

A visual method used to simplify Boolean expressions. Rules: 1. Group adjacent 1s into powers of 2 ($2^n$: 1, 2, 4, 8); 2. Groups must be rectangular or square (no diagonals); 3. Make groups as large as possible; 4. Use the fewest groups possible; 5. Groups can overlap or wrap around edges.

What is a Half-Adder circuit, its components, and its Boolean expressions?

A digital circuit used to add two 1-bit binary numbers (A and B). Uses one XOR gate for the Sum and one AND gate for the Carry. Expressions: Sum = A ⊕ B; Carry = A · B.

State the complete Truth Table for a Half-Adder.

Inputs (A, B) -> Outputs (Sum, Carry): (0, 0) -> (0, 0); (0, 1) -> (1, 0); (1, 0) -> (1, 0); (1, 1) -> (0, 1).

What is a Full-Adder circuit and how does it differ from a Half-Adder?

A digital circuit that adds three 1-bit binary inputs (A, B, and Carry-In / $C_{in}$) to produce a Sum ($S$) and Carry-Out ($C_{out}$). Unlike a half-adder, it accounts for an incoming carry bit from a previous stage.

What are the Boolean expressions for the Sum (S) and Carry-Out (C_out) of a Full-Adder?

Sum ($S$) = $A \oplus B \oplus C_{in}$; Carry-Out ($C_{out}$) = $(A \cdot B) + (C_{in} \cdot (A \oplus B))$.

State the key Truth Table outputs for a Full-Adder.

(A, B, Cin) -> (Sum, Cout): (0,0,0)->(0,0); (0,0,1)->(1,0); (0,1,0)->(1,0); (0,1,1)->(0,1); (1,0,0)->(1,0); (1,0,1)->(0,1); (1,1,0)->(0,1); (1,1,1)->(1,1).