Binary to Decimal Converter

What is the Binary to Decimal Converter?

The Binary to Decimal Converter is an online utility designed to translate binary values (base-2) into standard decimal integers (base-10). Binary is the foundation of modern digital computing, where all machine instructions and data are represented as strings of 0s and 1s. This tool makes it simple to read, convert, and understand binary values instantly.

How to Convert Binary to Decimal

Converting binary to decimal is based on the concept of positional weights. Since binary is base-2, each position represents a power of 2, starting from the rightmost position (index 0).

To convert a binary number to a decimal number, multiply each binary digit (0 or 1) by its positional weight (\(2^n\)), and sum all the results.

Formula: \[ \text{Decimal} = (d_{n-1} \times 2^{n-1}) + (d_{n-2} \times 2^{n-2}) + \dots + (d_0 \times 2^0) \]

Example: Convert binary 1101 to decimal:

• Rightmost digit (index 0): \(1 \times 2^0 = 1 \times 1 = 1\)
• Second digit (index 1): \(0 \times 2^1 = 0 \times 2 = 0\)
• Third digit (index 2): \(1 \times 2^2 = 1 \times 4 = 4\)
• Leftmost digit (index 3): \(1 \times 2^3 = 1 \times 8 = 8\)
• Sum: \(8 + 4 + 0 + 1 = 13_{10}\)

Tool Advantages

• Arbitrary Precision (BigInt): Unlike standard JavaScript calculators which suffer from precision loss for values over \(2^{53} - 1\), this converter uses native Web BigInt logic, allowing you to convert binary strings of hundreds or thousands of bits flawlessly.
• Line-by-Line Mode: Convert a list of multiple binary values in one go, separated by spaces, commas, or custom delimiters.
• Step-by-Step Breakdown: Provides exact mathematical positional expansion in real time so you can learn the math behind the conversion.