Number Systems

Why Computers Use Binary

• Computers process data using **logic gates** that have only **two states** (on/off).
• **Binary** (base-2) uses digits **1** (on) and **0** (off) to match these states.
• All data must be converted to binary before a computer can process it.
• Examples: magnetic hard drives use North/South polarity; optical disks use land/pit.
• Binary allows fast processing and efficient storage.

Number Systems Overview

• **Denary** (base-10): digits 0–9, each column is a power of 10.
• **Binary** (base-2): digits 0–1, each column is a power of 2.
• **Hexadecimal** (base-16): digits 0–9 and letters A–F (A=10, B=11, C=12, D=13, E=14, F=15).
• One hex digit represents exactly **4 bits** (a **nibble**).
• Largest 16-bit denary value: 65,535 (binary 1111111111111111).

Converting Between Binary & Denary

• **Denary → Binary**: write column headings (powers of 2) from largest ≤ number down to 1; for each column, put 1 if the value fits, subtract it, else 0.
• **Binary → Denary**: write column headings, multiply each 1-bit by its heading, sum the results.
• If a binary number ends in **1**, the denary result is **odd** (quick check).
• Pad with leading zeros if a specific bit-length is required (e.g., 8-bit).

Converting Between Hexadecimal & Denary

• **Denary → Hex**: Method 1: convert denary to binary, split into nibbles, convert each nibble to hex. Method 2: divide by 16; quotient = first hex digit, remainder = second hex digit.
• **Hex → Denary**: Method 1: convert each hex digit to a nibble, join to 8-bit binary, convert to denary. Method 2: multiply first digit by 16, add second digit.
• Exam is non-calculator; use binary method if unsure of multiplication/division by 16.

Converting Between Hexadecimal & Binary

• **Binary → Hex**: split 8-bit binary into two nibbles (4 bits each); convert each nibble to its hex digit (0–F).
• **Hex → Binary**: convert each hex digit to its 4-bit binary nibble; join the nibbles.
• Always write out the full 8-bit binary result for clarity.

Uses of Hexadecimal

• Hex is used because it is **shorter** than binary and **easier for humans to read** with **fewer copying errors**.
• **MAC addresses**: 12 hex digits (e.g., AA:BB:CC:DD:EE:FF) instead of 48 binary digits.
• **Colour codes**: 6 hex digits (e.g., #66FF33) instead of 24 binary digits.
• **URLs**: non-standard characters are encoded as hex preceded by **%** (e.g., %20 for space).

Binary Addition

• Rules: 0+0=0, 0+1=1, 1+0=1, 1+1=0 carry 1, 1+1+1=1 carry 1.
• Add from rightmost bit to left, carrying over when sum ≥ 2.
• **Overflow** occurs when the result exceeds the available bits (e.g., 8-bit sum > 255).
• Always show **carry bits** clearly; marks are awarded for correct carries even if final answer is wrong.

Binary Shifts

• **Logical left shift** multiplies by 2 for each shift (e.g., shift left 1 → ×2).
• **Logical right shift** divides by 2 for each shift (e.g., shift right 1 → ÷2).
• Bits shifted out are lost; empty positions are filled with **0**.
• **Overflow** occurs when a **1** is shifted out of the most significant bit (MSB) on a left shift.

Two's Complement

• Two's complement represents **signed** (positive and negative) binary numbers.
• The **MSB** has a negative weight (−128 in 8-bit); remaining bits are positive.
• To convert a negative denary number: write the positive binary, then copy bits from right until the first 1, then **invert** all remaining bits.
• Example: −76 → positive 01001100 → copy rightmost bits (100) → invert rest → 10110100.

Atomic structure of sodium (Na) – analogy for binary place values: each shell corresponds to a power of 2 (2^0, 2^1, 2^2...).

Particle arrangement in solid, liquid, and gas – analogous to binary states (ordered/disordered).