Hexadecimal Numbers

Overview

Hexadecimal (or "hex") is a base-16 number system that uses sixteen symbols: 0–9 and A–F. In computing, hexadecimal is often preferred for representing binary data in a more readable format, as it is compact and easy to interpret. Understanding hexadecimal notation and converting between hex, binary, and decimal (denary) is essential, as hex is widely used in memory addresses, colour codes, and low-level programming.

Hexadecimal System

Base-16 Representation: Hexadecimal uses sixteen symbols:
- 0 to 9 represent values 0 to 9.
- A to F represent values 10 to 15.
Compact Form: Each hexadecimal digit represents four binary bits (a nibble). This allows large binary numbers to be represented with fewer characters, making hex both compact and more human-readable.

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Example: The hexadecimal number 2F is equivalent to 0010 1111 in binary.

Benefits of Hexadecimal Over Binary

Readability: Hexadecimal is easier for humans to read and write than long binary sequences.
Compact Representation: Four binary bits are represented by one hex digit, making hex notation shorter.
Common Uses:
- Memory Addresses: Hex is often used to represent memory locations in computing.
- Colour Codes: Web colours are represented in hex (e.g., #FF5733).
- Assembly Language and Machine Code: Hexadecimal provides an easier-to-read representation of binary-coded machine instructions.

Conversion Processes

Converting Between Binary and Hexadecimal

Binary to Hexadecimal

Split the binary number into groups of four bits (starting from the right).
Convert each group to its corresponding hex digit.

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Example: Convert 11010110 to hexadecimal.

Group into nibbles: 1101 0110
Convert each nibble: 1101 = D, 0110 = 6

Result: D6

Hexadecimal to Binary

Replace each hex digit with its four-bit binary equivalent.

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Example: Convert 3B to binary.

3 in hex = 0011 in binary
B in hex = 1011 in binary

Result: 00111011

Converting Between Decimal (Denary) and Hexadecimal

Decimal to Hexadecimal

Repeatedly divide the decimal number by 16, recording the remainders.
Write down the remainders in reverse order to get the hex representation.

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Example: Convert 254 to hexadecimal.

$254 \div 16 = 15$ remainder 14 (E)
$15 \div 16 = 0$ remainder 15 (F)

Result: FE

Hexadecimal to Decimal

Multiply each hex digit by its positional value (powers of 16) and sum the results.

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Example: Convert 4A to decimal.

4 in the $16^1$ place = $4 \times 16 = 64$
A (which is 10) in the $16^0$ place = $10 \times 1 = 10$
Sum: 64 + 10 = 74

Result: 74

Examples

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Example 1: Binary to Hexadecimal Convert 11101100 to hexadecimal.

Split into nibbles: 1110 1100
Convert each nibble: 1110 = E, 1100 = C

Result: EC

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Example 2: Hexadecimal to Binary Convert 5A to binary.

5 in hex = 0101 in binary
A (10) in hex = 1010 in binary

Result: 01011010

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Example 3: Decimal to Hexadecimal Convert 345 to hexadecimal.

$345 \div 16 = 21$ remainder 9
$21 \div 16 = 1$ remainder 5

Result: 159

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Example 4: Hexadecimal to Decimal Convert 7F to decimal.

7 in the $16^1$ place = $7 \times 16 = 112$
F (15) in the $16^0$ place = $15 \times 1 = 15$
Sum: 112 + 15 = 127

Result: 127

Note Summary

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Common Mistakes

Grouping Errors in Binary-Hex Conversion: Forgetting to start grouping from the right when converting binary to hex.
Incorrect Remainder Order in Decimal to Hex Conversion: Reversing the order of remainders incorrectly.
Confusing Hex Values for Decimal Values: Misinterpreting letters in hex (e.g., treating A as 1 rather than 10).

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Key Takeaways

Hexadecimal (Base-16): A compact, human-readable format, with values ranging from 0–9 and A–F.
Conversion Skills:
Binary to Hex: Group into nibbles, and convert each to a hex digit.
Decimal to Hex: Use division by 16, reverse remainders.
Hex to Decimal: Multiply each digit by powers of 16 and sum.
Practical Uses: Hex is commonly used for readability in computing contexts like memory addresses, colours, and machine code.

Hexadecimal Numbers Simplified Revision Notes for A-Level OCR Computer Science

Hexadecimal Numbers

Overview

Hexadecimal System

Benefits of Hexadecimal Over Binary

Conversion Processes

Converting Between Binary and Hexadecimal

Binary to Hexadecimal

Example: Convert 11010110 to hexadecimal.

Hexadecimal to Binary

Example: Convert 3B to binary.

Converting Between Decimal (Denary) and Hexadecimal

Decimal to Hexadecimal

Example: Convert 254 to hexadecimal.

Hexadecimal to Decimal

Example: Convert 4A to decimal.

Examples

Note Summary

Common Mistakes

Key Takeaways

500K+ Students Use These Powerful Tools to Master Hexadecimal Numbers For their A-Level Exams.

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