Binary Calculator

A register value of 0x3F in a microcontroller datasheet means the same thing as 00111111 in binary or 63 in decimal — and fluency in converting between these representations is fundamental to embedded programming, assembly language, memory addressing, and digital circuit debugging. The binary calculator performs all conversions and arithmetic operations across all four number systems that dominate computing.

The Four Number Systems: When Each Is Used

Each number base has a natural application domain in computing:

• Binary (base 2): the hardware layer — logic gates, flip-flops, memory cells operate in binary; assembly language and microcontroller register documentation uses binary for bitmask clarity
• Decimal (base 10): human interface layer — addresses, counts, and values displayed to users; most high-level programming uses decimal
• Hexadecimal (base 16, digits 0–9, A–F): the programmer's shorthand for binary — each hex digit represents exactly 4 bits, so a 32-bit value is expressed as 8 hex digits vs. 32 binary digits; used for memory addresses, color codes (RGB), byte values in protocols, hash function outputs
• Octal (base 8): Unix file permissions (chmod 755 = rwxr-xr-x = 111 101 101 binary); older minicomputer word architectures; each octal digit represents exactly 3 bits

Conversion Methodology

Converting between number systems:

Binary → Decimal: multiply each bit by its positional power of 2 and sum. 10110₂ = 1×16 + 0×8 + 1×4 + 1×2 + 0×1 = 22.

Decimal → Binary: repeatedly divide by 2 and record remainders (read remainders bottom-up). 22 ÷ 2 = 11 R0 → 11 ÷ 2 = 5 R1 → 5 ÷ 2 = 2 R1 → 2 ÷ 2 = 1 R0 → 1 ÷ 2 = 0 R1 → reading remainders: 10110₂.

Binary ↔ Hexadecimal: group binary digits in sets of 4 from right; convert each group to a single hex digit. 10110₂ = 0001 0110 = 16₁₆. This grouping relationship makes binary-hex conversion trivial and explains why hex is used as a compact binary representation.

Use this online calculator for any base conversion. The binary arithmetic calculator handles step-by-step arithmetic operations. The binary to decimal converter focuses specifically on base-2 to base-10 conversion.

Two's Complement: Signed Integer Representation

For signed integers, the binary representation depends on the two's complement convention: the most significant bit is the sign bit (0 = positive, 1 = negative). Converting a two's complement negative binary to decimal: if the sign bit is 1, invert all bits and add 1 to get the magnitude, then negate. 11110110₂ in 8-bit signed: sign bit = 1 (negative); invert = 00001001; +1 = 00001010 = 10 decimal; result = −10. This is the representation used by all modern programming languages for signed integer types (int8_t, int16_t, int32_t, int64_t in C). The two's complement calculator and arithmetic calculators cover the complete number system toolkit.

IEEE 754 Floating-Point: Binary Representation of Decimal Fractions

Integer values have exact binary representations; decimal fractions generally do not. The IEEE 754 double-precision standard represents floating-point numbers as: (−1)^sign × 1.mantissa₂ × 2^(exponent−1023). The fraction 0.1 decimal = 0.000110011001100... in binary — a repeating pattern that cannot be represented exactly in finite bits. This is why 0.1 + 0.2 ≠ 0.3 in binary floating-point arithmetic — the stored approximations of 0.1 and 0.2 sum to a value slightly different from the stored approximation of 0.3. Understanding this is essential for writing correct numerical comparisons in code (never use == for floating-point equality; use abs(a−b) < epsilon).