# Base Converter (Radix)

> Convert numbers between Binary, Octal, Decimal, Hexadecimal, and custom bases (2-36).

**Category:** Dev
**Keywords:** base, radix, binary, hex, octal, decimal, converter, math
**URL:** https://complete.tools/base-converter

## How it calculates

To convert a number from base 'b1' to base 'b2', first convert the number to decimal (base 10), then from decimal to the desired base. The formula for converting a number 'N' in base 'b' to decimal is: 
N_decimal = Σ (d_i × b^i) 
where: 
d_i = the digit at position i (from the right, starting at 0) 
b = the base of the original number 
i = the position of the digit. After obtaining the decimal equivalent, convert it to the target base 'b2'. The formula for converting from decimal to another base involves repeatedly dividing the decimal number by the new base and recording the remainders. The final value in base 'b2' is then read from the last remainder to the first.

## Who should use this

Software developers working on algorithms that require base conversions in programming languages. Cryptographers analyzing data in different numeral systems for security protocols. Data scientists interpreting large datasets that involve various numbering systems for analytics. Educators teaching numeral systems and their applications in computer science curricula.

## Worked examples

Example 1: Convert the binary number 1011 to decimal. 
Using the formula: N_decimal = Σ (d_i × b^i) 
Here, d_0 = 1, d_1 = 1, d_2 = 0, d_3 = 1, and b = 2. 
N_decimal = (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0) = 8 + 0 + 2 + 1 = 11. 
Thus, the binary number 1011 equals 11 in decimal. 

Example 2: Convert the decimal number 45 to hexadecimal. 
First, divide 45 by 16 (the base for hexadecimal). 45 ÷ 16 = 2 with a remainder of 13. The remainder 13 corresponds to 'D' in hexadecimal. Next, divide the quotient 2 by 16, which is 0 with a remainder of 2. Thus, reading from last remainder to first, 45 in decimal is 2D in hexadecimal.

## Limitations

The Base Converter may have precision limits when converting very large numbers due to floating-point representation constraints in computing. Additionally, edge cases such as negative numbers or fractions may not be handled correctly, as the tool focuses primarily on whole numbers. The conversion process assumes that the input is a valid number in the specified base; invalid inputs may result in errors or undefined behavior. It is also important to note that the tool may not account for leading zeros in the input, which can affect the final result when interpreting the number in certain contexts.

## FAQs

**Q:** How does the tool handle fractional values in base conversions?


**A:** The Base Converter currently does not support fractional values; it is designed for whole numbers only.


**Q:** What happens if I input a number with invalid digits for a specific base?


**A:** The tool will return an error message indicating invalid input, as it only accepts valid digits according to the numeral system.


**Q:** Can the tool convert very large numbers, like those used in scientific applications?


**A:** The tool can convert large numbers, but precision may be limited due to floating-point representation, which could affect the accuracy of very large values.


**Q:** Is there a limitation on the number of digits I can input?


**A:** While there is no strict limit, extremely large inputs may lead to performance issues or inaccuracies due to the tools’ computational limits.

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*Generated from [complete.tools/base-converter](https://complete.tools/base-converter)*