Binary Converter

The Binary Converter is a versatile tool designed to convert numbers between different number bases, including binary (base 2), decimal (base 10), and other bases up to 36. This calculator is particularly useful for computer science and digital logic applications.

What is Binary Conversion?

Binary conversion is the process of changing a number's representation from one base to another, most commonly between binary and decimal. Binary (base 2) uses only two digits, 0 and 1, while decimal (base 10) uses ten digits, 0 through 9.

Conversion Formula

To convert a number from binary to decimal, we use the following formula:

\[ (a_n a_{n-1} ... a_1 a_0)_2 = \sum_{i=0}^n a_i \cdot 2^i \]

Where:

• \(a_i\) represents each binary digit (0 or 1)
• \(i\) is the position of the digit from right to left, starting at 0
• \(n\) is the number of digits minus 1

Calculation Steps

Let's convert the binary number 1101 to decimal:

1. \(1 \cdot 2^3 = 1 \cdot 8 = 8\)
2. \(1 \cdot 2^2 = 1 \cdot 4 = 4\)
3. \(0 \cdot 2^1 = 0 \cdot 2 = 0\)
4. \(1 \cdot 2^0 = 1 \cdot 1 = 1\)

Sum the results: \(8 + 4 + 0 + 1 = 13\)

Therefore, \((1101)_2 = (13)_{10}\)

Example with Visual Representation

Let's visualize the conversion of (1101)₂ to decimal:

This visual representation illustrates how each bit in the binary number corresponds to a power of 2. The sum of the values where the bits are 1 gives us the decimal equivalent.

The Binary Converter simplifies these calculations, allowing you to quickly and accurately convert between binary, decimal, and other bases, making it an invaluable tool for students, professionals, and enthusiasts in computer science and related fields.