Binary to Decimal Converter

The Binary to Decimal Converter is a powerful tool designed to convert numbers from the binary system (base-2) to the decimal system (base-10). This calculator is essential for various applications in computer science, digital electronics, and mathematics.

What is Binary to Decimal Conversion?

Binary to decimal conversion is the process of transforming a number represented in the binary numeral system (using only 0s and 1s) to its equivalent in the decimal numeral system (using digits 0-9). This conversion is fundamental in computing, as it bridges the gap between the machine's internal binary representation and the decimal system we use in everyday life.

Conversion Formula

The mathematical formula for converting a binary number to decimal is:

\[D = \sum_{i=0}^{n-1} b_i \cdot 2^i\]

Where:

• \(D\) is the decimal equivalent
• \(b_i\) is the \(i\)-th bit in the binary number (0 or 1)
• \(n\) is the number of bits
• \(i\) is the position of the bit, starting from right to left (0-indexed)

Calculation Steps

To convert a binary number to decimal, follow these steps:

1. Identify each bit in the binary number, starting from the rightmost bit.
2. Multiply each bit by 2 raised to the power of its position (0-indexed from right).
3. Sum all the resulting products.

Example with Visual Representation

Let's convert the binary number 1101 to decimal:

\[ \begin{align*} 1101_2 &= (1 \cdot 2^3) + (1 \cdot 2^2) + (0 \cdot 2^1) + (1 \cdot 2^0) \\ &= (1 \cdot 8) + (1 \cdot 4) + (0 \cdot 2) + (1 \cdot 1) \\ &= 8 + 4 + 0 + 1 \\ &= 13_{10} \end{align*} \]

This visual representation illustrates how each bit in the binary number contributes to the final decimal value. The subscript 2 denotes binary, while 10 denotes decimal.

Understanding binary to decimal conversion is crucial for anyone working with digital systems or computer programming. It forms the basis for more complex numerical operations and data representations in computing.