Binary & Hex Calculator

Group binary digits into sets of 4 (from right to left), then convert each group: 0000=0, 0001=1, …, 1111=F. For example, 11011011₂ = 1101 1011 = DB₁₆. Going the other way, each hex digit becomes 4 binary digits. With this calculator, set From = Binary, To = Hex and the bit width to pad the output to the desired width.

Two's complement is the standard way computers represent signed integers. The most significant bit is the sign bit (0 = positive, 1 = negative). To negate a number: invert all bits and add 1. For example, +5 in 8-bit is 00000101₂. Invert: 11111010₂. Add 1: 11111011₂ = -5. The beauty is that addition hardware works identically for signed and unsigned numbers — the same binary adder handles both, and the programmer just decides how to interpret the result. A CPU's overflow flag detects signed overflow, while the carry flag detects unsigned overflow.

8-bit: good for learning and practice (range -128 to 127 signed). 16-bit: common in embedded systems and older hardware (range -32,768 to 32,767). 32-bit: standard for bitwise operations in most programming languages (range -2.1×10⁹ to 2.1×10⁹). 64-bit: modern CPU architecture, memory addresses (range -9.2×10¹⁸ to 9.2×10¹⁸). 128-bit: IPv6 addresses, UUIDs, cryptography (huge range). Choose the width that matches your target system or the data type you are working with.

Because the result (128) exceeds the maximum signed 8-bit value (127). In binary: 01111111₂ + 00000001₂ = 10000000₂. The most significant bit flips from 0 to 1, which the signed interpretation reads as -128. This is "signed overflow" — the CPU sets its overflow flag. The same bit pattern interpreted as unsigned gives 128, which is correct. This is why understanding both signed and unsigned interpretations matters when working with fixed-width arithmetic.

Each left shift by 1 position multiplies the number by 2. So x << 3 = x × 2³ = x × 8. The bits move into higher-value positions. However, bits shifted beyond the bit width are lost (truncated). In 8-bit: 11111111₂ (255) << 1 = 11111110₂ (254). Mathematically 255×2=510, but 510 & 0xFF = 254. Right shift (>>) divides by 2, rounding toward negative infinity for signed integers. Both behaviors match real CPU operation.

AND: each bit is 1 only if BOTH input bits are 1. Used for masking (isolating specific bits). OR: each bit is 1 if EITHER input bit is 1. Used for setting bits. XOR: each bit is 1 if the input bits are DIFFERENT. Used for toggling bits. These three operations, combined with shifts, can implement any boolean function. They are fundamental to cryptography, graphics (color channel extraction), networking (subnet masks), and hardware register manipulation.

Endianness describes the byte order of multi-byte values in computer memory. Big endian stores the most significant byte first (at the lowest memory address), like how we write numbers — 0x1234 is stored as [0x12, 0x34]. Little endian stores the least significant byte first — 0x1234 is stored as [0x34, 0x12]. x86 and x64 processors use little-endian, while network protocols (TCP/IP, HTTP) use big-endian (also called network byte order). Some architectures like ARM are bi-endian and can switch. Endianness only matters for multi-byte values — 8-bit values are always the same. When reading binary file formats or network packets, knowing the endianness is essential for correct interpretation.