Hex to Decimal Converter

Understanding Hexadecimal and Decimal Conversion (Bi-Directional)

The Hexadecimal (Base-16) and Decimal (Base-10) systems are the two most fundamental number bases used in computing. This calculator provides a bi-directional tool to switch between these systems quickly and accurately, offering a step-by-step breakdown of the underlying mathematical logic.

Hexadecimal to Decimal Conversion Logic

To convert from Hex to Decimal, we use positional notation, where each hex digit (\(d\)) is multiplied by the corresponding power of \(16\) based on its position (\(i\)), starting from \(i=0\) on the right.

Formula (Hex \(\rightarrow\) Dec): \[\text{Decimal} = \sum_{i=0}^{n-1} (d_i \times 16^i)\]

For instance, to convert A4: \(A = 10\), \(4 = 4\). The calculation is: \((10 \times 16^1) + (4 \times 16^0) = 160 + 4 = 164\). The chart illustrates the contribution of each digit to this final sum.

Decimal to Hexadecimal Conversion Logic

The reverse conversion, from Decimal to Hex, is typically performed using the remainder method, which involves repeatedly dividing the decimal number by \(16\) and noting the remainder. The Hex result is derived by reading the remainders from the bottom (last remainder) up.

The process continues until the quotient is zero. Each remainder is then converted to its Hex equivalent (e.g., a remainder of \(12\) becomes \(C\)).

Method (Dec \(\rightarrow\) Hex): Repeated Division by 16, using the remainders in reverse order.

Applications in Computing

Hexadecimal is vital in areas like web development (for color codes like #FF0000), memory addressing (representing 64-bit addresses), and data representation (as four binary bits equal one hex digit, allowing for compact display of binary data). Understanding the conversion process is crucial for working with low-level data and hardware interfaces.