Hexadecimal to Decimal 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

Decimal number is a number expressed in the base 10 numeral system. Decimal number's digits have 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each digit of a decimal number counts a power of 10.

Decimal number example:

653₁₀ = 6×10² + 5×10¹ + 3×10⁰ = 653₁₀

How to Convert Hexadecimal to Decimal

A regular decimal number is the sum of the digits multiplied with power of 10.

Hex numbers are read the same way, but each digit counts power of 16 instead of power of 10.

For hex number with n digits:

d_n-1 ... d₃ d₂ d₁ d₀

Multiply each digit of the hex number with its corresponding power of 16 and sum:

decimal = d_n-1×16^n-1 + ... + d₃×16³ + d₂×16² + d₁×16¹ + d₀×16⁰

Example

Understanding Hexadecimal-to-Decimal Conversion

Hexadecimal-to-decimal conversion involves converting numbers in hexadecimal (base 16) into their equivalent decimal (base 10) values. Hexadecimal uses digits 0-9 and letters A-F to represent values from 0 to 15, while decimal uses digits 0-9.

The general approach to converting hexadecimal numbers to decimal includes:

Write down the hexadecimal number.
Assign powers of 16 to each digit based on its position (rightmost digit is \(16^0\), next is \(16^1\), and so on).
Multiply each digit by \(16^n\), where \(n\) is the position of the digit, starting from 0.
Add the results to get the decimal equivalent.

Steps for Hexadecimal-to-Decimal Conversion

Step 1: Write down the hexadecimal number.

Step 2: Assign powers of 16 to each digit starting from the rightmost digit.

Step 3: Multiply each digit by \(16^n\), where \(n\) is the positional power.

Step 4: Add all the results to get the decimal equivalent.

Example: Converting Hexadecimal to Decimal

Convert \( 2F \) to decimal:

Step 1: Write the hexadecimal number \( 2F \).
Step 2: Assign powers of 16:
- \( F = 15 \) at \(16^0\)
- \( 2 = 2 \) at \(16^1\)
Step 3: Multiply each digit by \(16^n\):
- \( F \times 16^0 = 15 \times 1 = 15 \)
- \( 2 \times 16^1 = 2 \times 16 = 32 \)
Step 4: Add the results: \( 32 + 15 = 47 \).
Final decimal result: \( 47 \).

Conversion Table for Quick Reference

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

0 → 0
1 → 1
2 → 2
3 → 3
4 → 4
5 → 5
6 → 6
7 → 7
8 → 8
9 → 9
A → 10
B → 11
C → 12
D → 13
E → 14
F → 15

Applications of Hexadecimal-to-Decimal Conversion

Hexadecimal-to-decimal conversion is commonly used in:

Understanding memory addresses in computing systems.
Converting color codes (e.g., in HTML and CSS).
Interpreting machine-level data in programming and debugging.

Practice Problem

Convert \( 3A \) to decimal:

Solution:
- \( A = 10 \) at \(16^0\)
- \( 3 = 3 \) at \(16^1\)
Multiply and add:
- \( A \times 16^0 = 10 \times 1 = 10 \)
- \( 3 \times 16^1 = 3 \times 16 = 48 \)
- \( 48 + 10 = 58 \).
Final decimal result: \( 58 \).

Hexadecimal-to-Decimal Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a single hexadecimal digit to its decimal equivalent.	Write down the hexadecimal digit. Refer to the hexadecimal-to-decimal conversion table to find the equivalent decimal value.	For `A`: ‘A’ in decimal: `10`. Decimal: `10`.
Handling Larger Numbers	Converting a multi-digit hexadecimal number to decimal.	Write the hexadecimal number as a sum of its digits multiplied by powers of 16. Evaluate each term and add them to get the decimal value.	For `3A7`: ‘3’ × 16² = 768 ‘A’ × 16¹ = 160 ‘7’ × 16⁰ = 7 Sum: `768 + 160 + 7 = 935`. Decimal: `935`.
Verifying Conversion	Checking the accuracy of hexadecimal-to-decimal conversion.	Convert the decimal value back to hexadecimal by repeatedly dividing by 16. Ensure the resulting hexadecimal matches the original input.	For `935`: Divide by 16: 935 ÷ 16 = 58 remainder `7`. Divide 58 by 16: 58 ÷ 16 = 3 remainder `A`. Combine: `3A7`. Hexadecimal: `3A7`, matches the original input.
Handling Special Cases	Converting hexadecimal numbers with letters (A-F) to decimal.	Replace each letter with its decimal equivalent (A=10, B=11, ..., F=15). Multiply each value by the corresponding power of 16.	For `FE`: ‘F’ × 16¹ = 15 × 16 = 240 ‘E’ × 16⁰ = 14 × 1 = 14 Sum: `240 + 14 = 254`. Decimal: `254`.
Applications	Understanding where hexadecimal-to-decimal conversion is used.	Hexadecimal is often used to simplify the representation of large decimal numbers in computing. Commonly applied in memory addressing, color codes, and data representation.	Example: Converting hexadecimal memory address `1A3F` to decimal for clarity: ‘1’ × 16³ = 4096 ‘A’ × 16² = 2560 ‘3’ × 16¹ = 48 ‘F’ × 16⁰ = 15 Sum: `4096 + 2560 + 48 + 15 = 6719`. Decimal: `6719`.