# Binary to Decimal Converter

> Convert binary numbers to decimal with step-by-step calculation breakdown showing positional values

**Category:** Conversion
**Keywords:** binary, decimal, convert, base 2, base 10, number system, programming
**URL:** https://complete.tools/binary-to-decimal-converter

## How it calculates

The conversion from binary to decimal uses the formula: Decimal = Σ (b_i × 2^i), where 'b_i' is the ith binary digit (0 or 1) and 'i' is the position of the digit from the right, starting at 0. Each binary digit is multiplied by 2 raised to the power of its position index. For example, in the binary number 1011, the calculation is as follows: 1×2^3 + 0×2^2 + 1×2^1 + 1×2^0 = 1×8 + 0×4 + 1×2 + 1×1 = 8 + 0 + 2 + 1 = 11. This formula highlights the mathematical relationship where each binary digit's value is determined by its position, contributing to the total decimal value based on powers of two.

## Who should use this

Software developers converting binary data for compatibility in applications, Computer scientists analyzing binary algorithms, Data analysts interpreting binary-coded decimal (BCD) data formats in databases, and Electronics engineers designing digital circuits that require binary-to-decimal conversions for display.

## Worked examples

Example 1: Convert binary 1101 to decimal. Step 1: Identify positions: 1 (2^3) + 1 (2^2) + 0 (2^1) + 1 (2^0). Step 2: Calculate values: 1×8 + 1×4 + 0×2 + 1×1 = 8 + 4 + 0 + 1 = 13. Thus, binary 1101 equals decimal 13. Example 2: Convert binary 101010 to decimal. Step 1: Identify positions: 1 (2^5) + 0 (2^4) + 1 (2^3) + 0 (2^2) + 1 (2^1) + 0 (2^0). Step 2: Calculate values: 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 0×1 = 32 + 0 + 8 + 0 + 2 + 0 = 42. Therefore, binary 101010 equals decimal 42.

## Limitations

This tool has several limitations. First, it can only convert binary strings up to a certain length due to precision limits, typically 32 bits or 64 bits, depending on the implementation. Second, it does not handle negative binary numbers or fractions, which are represented in other forms (like two's complement for negatives). Third, the tool assumes input is valid binary; invalid characters (e.g., letters) will produce errors. Lastly, for very large binary numbers, the converted decimal value might exceed standard data type limits, leading to inaccuracies.

## FAQs

**Q:** How does the tool handle leading zeros in binary numbers?


**A:** Leading zeros in binary numbers do not affect the decimal value; for instance, binary 00101 equals decimal 5, just like binary 101.


**Q:** Can this tool convert floating-point binary numbers?


**A:** No, this tool is designed for integer binary numbers only. Floating-point binary requires a different conversion approach.


**Q:** What happens if I input a non-binary string?


**A:** The tool will return an error message since it only accepts strings composed of the digits 0 and 1.


**Q:** How does the conversion process differ for signed binary numbers?


**A:** Signed binary numbers use two's complement representation, which requires additional steps to determine the decimal value, and this tool does not accommodate that format.

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*Generated from [complete.tools/binary-to-decimal-converter](https://complete.tools/binary-to-decimal-converter)*