Binary Converter
Convert binary numbers to decimal, hexadecimal, and octal simultaneously or Decimal to Binary.
How Binary Numbers Work
Binary (base-2) is the number system that underpins all digital computing. It uses only two digits — 0 and 1 — called bits. Each position in a binary number represents a power of 2, just as each position in a decimal number represents a power of 10. The rightmost bit is 2⁰ (1), the next is 2¹ (2), then 2² (4), 2³ (8), and so on. A group of 8 bits is called a byte and can represent values from 0 (00000000) to 255 (11111111).
Binary Conversion Formulas
Binary to Decimal: Decimal = dₙ×2ⁿ + dₙ₋₁×2ⁿ⁻¹ + ... + d₁×2¹ + d₀×2⁰. Binary to Hexadecimal: Group binary digits into sets of 4 from right to left, then convert each group (0000=0, 0001=1, ..., 1010=A, ..., 1111=F). Binary to Octal: Group binary digits into sets of 3 from right to left, then convert each group (000=0, 001=1, ..., 111=7). Pad with leading zeros if the leftmost group is incomplete.
Worked Example: Convert 11111111 (Binary)
To Decimal: 1×2⁷ + 1×2⁶ + 1×2⁵ + 1×2⁴ + 1×2³ + 1×2² + 1×2¹ + 1×2⁰ = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255. To Hexadecimal: Split into groups of 4: 1111 1111 → F F → FF. To Octal: Split into groups of 3: 11 111 111 → pad to 011 111 111 → 3 7 7 → 377. So 11111111₂ = 255₁₀ = FF₁₆ = 377₈.
Binary Conversion Reference Table
0000 = 0 = 0x0 · 0001 = 1 = 0x1 · 0010 = 2 = 0x2 · 0011 = 3 = 0x3 · 0100 = 4 = 0x4 · 0101 = 5 = 0x5 · 0110 = 6 = 0x6 · 0111 = 7 = 0x7 · 1000 = 8 = 0x8 · 1001 = 9 = 0x9 · 1010 = 10 = 0xA · 1011 = 11 = 0xB · 1100 = 12 = 0xC · 1101 = 13 = 0xD · 1110 = 14 = 0xE · 1111 = 15 = 0xF. Key byte values: 10000000 = 128, 11000000 = 192, 11111111 = 255.
Technical Details
This converter accepts any valid binary string (digits 0 and 1 only, spaces allowed for readability) and simultaneously outputs the decimal, hexadecimal, and octal equivalents. Binary is the native language of CPUs — every instruction, memory address, and data value is ultimately stored as binary. Common bit widths: 8-bit (byte, 0–255), 16-bit (0–65,535), 32-bit (0–4,294,967,295), and 64-bit (0–18,446,744,073,709,551,615). JavaScript uses 64-bit IEEE 754 floating-point internally, so this tool accurately handles integers up to 2⁵³ − 1 (9,007,199,254,740,991).
Frequently Asked Questions
What is 11111111 in decimal? 11111111₂ = 255₁₀. This is the maximum value of a single byte. How do I convert binary to hex quickly? Group the binary digits into sets of 4 from the right and convert each group: 1010 = A, 1111 = F, 0011 = 3, etc. What is the largest binary number this tool handles? It handles integers up to JavaScript's safe integer limit of 2⁵³ − 1, which is a 53-bit binary number. Why is binary important? All digital computers process data in binary because electronic circuits have two stable states (on/off, high/low voltage), which map directly to 1 and 0.