Binary Hex Decimal Octal Converter
Solve binary hex decimal octal problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Formula
N = d(n)*b^n + d(n-1)*b^(n-1) + ... + d(1)*b^1 + d(0)*b^0
Any number N in base b is the sum of each digit d multiplied by the base raised to its positional power. For example, hex 1A3 = 1*16^2 + 10*16^1 + 3*16^0 = 256 + 160 + 3 = 419 in decimal.
Worked Examples
Example 1: Convert Decimal 255 to All Bases
Problem: Convert the decimal number 255 to binary, hexadecimal, and octal.
Solution: Decimal 255:\nBinary: 255 / 2 repeatedly gives 11111111 (8 ones)\nHexadecimal: 255 / 16 = 15 remainder 15, so FF\nOctal: 255 / 8 = 31 r7, 31/8 = 3 r7, 3/8 = 0 r3, so 377\nBCD: 0010 0101 0101\nBits: 8 | Bytes: 1 | All bits set
Result: Binary: 11111111 | Hex: FF | Octal: 377
Example 2: Convert Binary 10110100 to Other Bases
Problem: Convert the binary number 10110100 to decimal, hexadecimal, and octal.
Solution: Binary: 10110100\nDecimal: 1*128 + 0*64 + 1*32 + 1*16 + 0*8 + 1*4 + 0*2 + 0*1 = 180\nHex: group into nibbles: 1011 0100 = B4\nOctal: group into 3-bit: 010 110 100 = 264\nASCII: not printable (180 > 126)
Result: Decimal: 180 | Hex: B4 | Octal: 264
Frequently Asked Questions
How do you convert binary to hexadecimal?
Converting binary to hexadecimal is straightforward because each hexadecimal digit represents exactly 4 binary digits (bits). Group the binary number into groups of 4 bits starting from the right, padding with leading zeros if needed. Then convert each group to its hex equivalent. For example, binary 11010110 becomes 1101 0110, which converts to D6 in hex (1101 = 13 = D, 0110 = 6). This direct mapping makes hex a preferred shorthand for binary in computing. An 8-bit byte is always represented by exactly 2 hex digits, a 16-bit word by 4 hex digits, and a 32-bit value by 8 hex digits. This is why memory addresses and color codes use hexadecimal notation.
What is the octal number system and where is it used?
The octal system (base-8) uses digits 0 through 7. Each octal digit represents exactly 3 binary bits, making octal-to-binary conversion simple. For example, octal 375 equals binary 011 111 101. Octal was widely used in early computing when word sizes were multiples of 3 bits (such as 12-bit, 24-bit, and 36-bit architectures). Today, octal is still used in Unix/Linux file permissions, where each permission set (read, write, execute) is represented by 3 bits or one octal digit. For instance, chmod 755 sets permissions to rwxr-xr-x. Octal also appears in some programming languages for representing character codes and escape sequences.
How do you perform arithmetic in binary?
Binary arithmetic follows the same rules as decimal but with only two digits. For addition: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 carry 1). For example, 1011 + 0110 = 10001 (11 + 6 = 17). Subtraction uses borrowing: 10-1=1, 0-1 requires borrowing. Multiplication works like decimal long multiplication but simpler since you only multiply by 0 or 1. Binary division follows long division with the same simplification. Computers perform all arithmetic in binary at the hardware level using logic gates. Addition is implemented with full-adder circuits, while multiplication is often done through repeated addition and bit-shifting. Understanding binary arithmetic helps debug low-level software issues and optimize code.
Why do programmers prefer hexadecimal over binary?
Programmers prefer hexadecimal because it provides a compact, human-readable representation of binary data. A single hex digit replaces 4 binary digits, so an 8-bit byte becomes just 2 hex characters instead of 8 binary digits. A 32-bit address becomes 8 hex characters versus 32 binary characters. This compactness makes hex much easier to read, type, and remember. For example, the binary value 11111111101011001101111000010010 is hard to parse at a glance, but its hex equivalent FFACDe12 is manageable. Hex also makes bit manipulation more intuitive: setting the high nibble is simply changing one hex digit. Memory dumps, network packet captures, and debug logs all use hex for readability.
Is Binary Hex Decimal Octal Converter free to use?
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Can I use the results for professional or academic purposes?
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.