Data Representation (Copy)

Cheat Sheet: A Level Computer Science – Number Systems, Representations & Encodings

1. Binary Magnitudes and Prefixes

Binary vs Decimal Prefixes

• Use: Decimal prefixes often used for storage capacity (e.g. hard drives), while binary prefixes are used in memory size (e.g. RAM).

2. Number Systems

Bases

• Binary (Base 2): Only digits 0 and 1
• Denary/Decimal (Base 10): Digits 0–9
• Hexadecimal (Base 16): Digits 0–9 and letters A–F
• BCD (Binary Coded Decimal): Each digit of a denary number is represented separately in 4-bit binary

3. Conversions Between Number Bases

Binary ↔ Decimal

• Binary to Decimal: Multiply each bit by 2ⁿ (n = position from right, starting at 0), then add.
• Decimal to Binary: Divide by 2 repeatedly, record remainders, read bottom to top.

Hexadecimal ↔ Decimal

• Hex to Decimal: Convert each digit using hex values and multiply by powers of 16.
• Decimal to Hex: Divide by 16 repeatedly, record remainders, convert to hex digits.

Binary ↔ Hexadecimal

• Group binary in 4s (from right), convert each group to one hex digit.

Decimal ↔ BCD

• Convert each decimal digit into its 4-bit binary form separately.

4. One’s and Two’s Complement

One’s Complement

• Flip all bits (0 ↔ 1)
• Used to represent negative numbers

Two’s Complement

• Take one’s complement and add 1
• Allows representation of both positive and negative integers
• Range for 8-bit two’s complement: -128 to +127

5. Binary Addition and Subtraction

Binary Addition Rules

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (carry 1)

Binary Subtraction Using Two’s Complement

• Convert subtrahend to two’s complement
• Add to the minuend
• If overflow bit, ignore it (only when operands have different signs)

6. Overflow in Binary

• Occurs when: Result > max representable number in fixed number of bits
• E.g. adding two 8-bit numbers resulting in a 9-bit answer

7. Practical Applications of BCD and Hexadecimal

BCD Applications

• Used in digital clocks, calculators, and financial systems where exact decimal representation is needed

Hexadecimal Applications

• Color codes in web design (e.g., #FF5733)
• Memory addressing and debugging (compact and easy to read vs long binary)

8. Internal Binary Representation of Character Data

ASCII (7-bit)

• Represents 128 characters (including control codes and standard English characters)
• E.g., A = 65 = 01000001

Extended ASCII (8-bit)

• 256 characters – adds symbols and foreign language characters

Unicode

• Supports 1.1+ million characters
• Encodes characters from almost all written languages
• UTF-8 (variable length), UTF-16, UTF-32 are common encoding formats

Important Notes

• Character set: A defined list of characters recognized by the computer system with a corresponding binary code
• You do not need to memorize the binary for each character (e.g., A = 01000001), just understand how encoding works

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