Hexadecimal to Binary Conversion

Hexadecimal number is a number expressed in the base 16 numeral system. Hexadecimal number's digits have 16 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Each digit of a hexadecimal number counts a power of 16.

Hexadecimal number example:

62C₁₆ = 6×16² + 2×16¹ + 12×16⁰ = 1580₁₀

Binary number is a number expressed in the base 2 numeral system. Binary number's digits have 2 symbols: zero (0) and one (1). Each digit of a binary number counts a power of 2.

Binary number example:

1101₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 13₁₀

How to Convert Hexadecimal to Binary

Convert every hex digit (start lowest digit) to 4 binary digits, with this table:

Example

Understanding Hexadecimal-to-Binary Conversion

Hexadecimal-to-binary conversion involves converting hexadecimal numbers (base 16) into their corresponding binary representations (base 2). Hexadecimal uses digits 0-9 and letters A-F to represent values from 0 to 15, while binary uses only 0 and 1.

The general approach to converting hexadecimal numbers to binary includes:

Write down the hexadecimal number.
Convert each hexadecimal digit into its 4-bit binary equivalent.
Combine the binary groups to get the final binary representation.

Steps for Hexadecimal-to-Binary Conversion

Step 1: Write down the hexadecimal number.

Step 2: Convert each hexadecimal digit into its 4-bit binary equivalent using a conversion table.

Step 3: Combine all binary values to obtain the complete binary number.

Example: Converting Hexadecimal to Binary

Convert \( 2F \) to binary:

Step 1: Write the hexadecimal number \( 2F \).
Step 2: Convert each digit:
- \( 2 \) → \( 0010 \)
- \( F \) → \( 1111 \)
Step 3: Combine the binary values: \( 00101111 \).
Final binary result: \( 00101111 \).

Conversion Table for Quick Reference

Here is a quick reference for hexadecimal digits and their binary equivalents:

0 → 0000
1 → 0001
2 → 0010
3 → 0011
4 → 0100
5 → 0101
6 → 0110
7 → 0111
8 → 1000
9 → 1001
A → 1010
B → 1011
C → 1100
D → 1101
E → 1110
F → 1111

Applications of Hexadecimal-to-Binary Conversion

Hexadecimal-to-binary conversion is commonly used in:

Digital systems and computing, where hexadecimal provides a compact way of representing binary numbers.
Programming, especially in debugging and working with low-level machine code.
Representing memory addresses, color codes, and other binary data in an easier-to-read format.

Practice Problem

Convert \( 3A \) to binary:

Solution:
- \( 3 \) → \( 0011 \)
- \( A \) → \( 1010 \)
Combine the binary values: \( 00111010 \).
Final binary result: \( 00111010 \).

Hexadecimal-to-Binary Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a hexadecimal digit to its binary equivalent.	Write down the hexadecimal number. Convert each hexadecimal digit to its 4-bit binary equivalent using a conversion table. Combine the binary equivalents to form the complete binary number.	For `2F`: ‘2’ in binary: `0010`. ‘F’ in binary: `1111`. Combine: `00101111`. Binary: `00101111`.
Handling Larger Numbers	Converting hexadecimal numbers with more digits to binary.	Follow the same method of converting each hexadecimal digit to its 4-bit binary equivalent. Combine the binary values in the same order as the hexadecimal digits.	For `3A7`: ‘3’ in binary: `0011`. ‘A’ in binary: `1010`. ‘7’ in binary: `0111`. Combine: `001110100111`. Binary: `001110100111`.
Verifying Conversion	Checking the accuracy of hexadecimal-to-binary conversion.	Convert the binary number back to hexadecimal. Ensure the hexadecimal result matches the original number.	For `00101111` (binary for `2F`): Group the binary into 4-bit segments: `0010`, `1111`. Convert each group back to hexadecimal: `2`, `F`. Hexadecimal: `2F`, matches the original input.
Handling Special Cases	Converting special hexadecimal numbers (e.g., letters A-F).	Use a hexadecimal-to-binary conversion table to quickly convert letters to their binary equivalents.	For `FE`: ‘F’ in binary: `1111`. ‘E’ in binary: `1110`. Combine: `11111110`. Binary: `11111110`.
Applications	Understanding where hexadecimal-to-binary conversion is used.	Hexadecimal simplifies binary representation in computing systems. Commonly used in programming, debugging, memory addressing, and color codes.	Example: Converting hexadecimal color codes (e.g., `#FF5733`) into binary for digital display.