Hexadecimal to Decimal Conversion: A Comprehensive Guide

Learning how to convert hexadecimal to decimal can be essential for anyone working with digital systems, programming, or computer science. In this guide, we will explore the process step by step and provide a detailed explanation using place values and a practical example with a graphing calculator.

Understanding Hexadecimal and Decimal Systems

Before diving into the conversion, it's important to understand the two systems involved:

Decimal (Base-10) System: A numeral system where each place value is a power of 10. For instance, in the number 1234, the '4' is in the ones place, the '3' is in the tens place, and so on. Hexadecimal (Base-16) System: A numeral system where each place value is a power of 16. The digits used in hexadecimal are 0-9 for the first ten values and A-F for the remaining values, representing 10-15.

Converting Hexadecimal to Decimal

To convert a hexadecimal number to its decimal equivalent, you can follow these steps:

Assign decimal values to each hexadecimal digit (0-9 for the first ten and 10-15 for A-F). Multiply each digit by 16 raised to the power of its position, starting from the right with 0. Sum the results to obtain the decimal value.

Let's take the example of converting the hexadecimal number 1A3 to decimal:

1A3 in hexadecimal 1 16^2 10 16^1 3 16^0 1 256 10 16 3 1 256 160 3 419 in decimal

Using Place Values in Conversion

The place values in hexadecimal are the powers of 16. For the integer part of a number, the place values are:

16^n 16^(n-1) 16^(n-2) · · · 16^3 16^2 16^1 16^0 . 4096quad256quad16quad1 .

For the fractional part, the place values are increasing negative powers of 16:

.16^(-1) 16^(-2) 16^(-3) 16^(-4) . 1/16 1/256 1/4096 1/65536 · · ·

Let's convert a hexadecimal number with a decimal part, for instance, 10AF.5B, to decimal:

Identify each digit and its place value: Multiply each digit by its place value: Add up the results: 1 × 16^3 4096 0 × 16^2 0 A × 16^1 10 × 16 160 F × 16^0 15 × 1 15 0.5 × 16^(-1) 0.5 × 1/16 1/32 0.5 × 16^(-2) 0.5 × 1/256 1/512 B × 16^(-3) 11 × 1/4096

Summing up the results, the decimal equivalent of 10AF.5B is approximately 4271.86.

Using a Graphing Calculator for Conversion

Graphing calculators can facilitate this conversion process. Here's how to do it with a TI-84 CE Python graphing calculator:

Define the variables A through F to represent their hexadecimal values (10 to 15). Create a list (L1) containing the hexadecimal digits of the number. Create another list (L2) containing the place values of each digit. {16^3 16^2 16^1 16^0} → L2 {1 10 11 15} → L1 seq16 n, n, 3, 0 -1 → L2 For 10AF: {4096 256 16 1} → L2 {1 10 11 15} → L1 L1 * L2 → L3 sum(L3) → Decimal Value

This method will yield the decimal value of the hexadecimal number. For numbers with a decimal part, the result may be approximated due to the calculator's precision limits.

Here's an example using 10AF.5B:

A × 16^1 10 × 16 160 F × 16^0 15 × 1 15 0.5 × 16^(-1) 1/32 0.5 × 16^(-2) 1/512 B × 16^(-3) 11 × 1/4096

The decimal equivalent of 10AF.5B can be calculated to be approximately 4271.8671875.

Conclusion

Whether you're a programmer, a mathematician, or someone interested in digital systems, understanding how to convert hexadecimal to decimal is a valuable skill. By using place values and a graphing calculator, you can simplify this process and achieve precise results.