The Base Conversion Calculator converts numbers from one numbering system to another, such as binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16). This tool solves the problem of translating values across bases—a frequent task in computer science, digital electronics, networking, and mathematics. Programmers, students, and engineers use it to understand binary or hexadecimal data, translate colour codes or file permissions, and verify base conversions quickly and accurately.

Required Inputs & Typical Use Cases

• Number

  Enter the number you want to convert. Ensure the digits are valid for its source base (e.g., 1011 is valid in binary, but 1021 is not).

• Source Base (b)

  Specify the base of the original number (e.g., 2 for binary, 8 for octal, 10 for decimal, 16 for hexadecimal).

• Target Base

  Choose the base you want to convert the number into (e.g., decimal to binary, hex to decimal).

Common scenarios: (1) Convert a binary string such as 1011₂ into its decimal equivalent; (2) Translate a hexadecimal value like 0x1A3 into decimal or binary for programming tasks; (3) Check octal file permissions (e.g., 755) by converting them to binary; (4) Convert decimal numbers into base 7 or base 9 for number theory exercises; (5) Translate RGB colour codes from hex to decimal values.

The Formula

V = Σ(dᵢ × bⁱ) for i = 0 to n−1

• V

  Resulting value in decimal (base 10)

• b

  Source base of the number (e.g., 2, 8, 10, 16)

• n

  Number of digits in the original number

• dᵢ

  The i-th digit from the right, where 0 ≤ dᵢ < b

Worked Example

1. Step 1: Identify digits and values

   For '1A3'₁₆, the digits are 1, A, and 3. In hexadecimal, A represents 10.

2. Step 2: Determine positional weights

   The powers of 16 for each position (from right) are: 16² = 256, 16¹ = 16, and 16⁰ = 1.

3. Step 3: Multiply digits by weights

   Compute (1 × 256) + (10 × 16) + (3 × 1) = 256 + 160 + 3.

4. Step 4: Sum the products

   Total = 256 + 160 + 3 = 419. Therefore, 1A3₁₆ equals 419₁₀.

Tips, Assumptions & Limitations

• Ensure each digit is valid for its source base (e.g., no digit beyond 1 in binary, beyond 7 in octal, etc.).
• For bases above 10, letters represent values: A = 10, B = 11, up to F = 15 in hexadecimal.
• Fractional numbers require converting both the integer and fractional parts separately using similar positional principles.
• Some simple converters may not handle negative numbers or bases greater than 16; check the tool’s supported range.
• To convert from decimal to another base manually, use the repeated division and remainder method: divide the number by the target base, record the remainder, divide the quotient again, and read remainders in reverse order.