The Binary Arithmetic Calculator performs addition, subtraction, multiplication, and division directly in binary notation, showing step-by-step carry propagation and borrow operations. Binary arithmetic is the foundation of all integer computation in every digital system.
45
27
72
72
0
1
72
45
27
72
72
0
1
72
Every addition, subtraction, and multiplication your CPU performs happens in binary — the native language of digital logic. Understanding binary arithmetic is not just academic; it explains integer overflow bugs that crash software, two's complement negative number representation, bitwise manipulation tricks used in performance-critical code, and why computers round floating-point numbers. The binary arithmetic calculator performs all four operations in binary with step-by-step carry/borrow visualization.
Binary addition follows three rules: 0+0=0; 0+1=1+0=1; 1+1=10 (result 0, carry 1). The carry propagates leftward exactly as in decimal addition:
1011 (11 decimal) + 0110 (6 decimal) ------ 1 carry 1011 + 0110 0001 → 1 in ones place 1 → carry to twos place ... = 10001 (17 decimal)
Use this online calculator for any binary addition with carry visualization. The binary to decimal calculator converts results between number systems.
Binary subtraction uses borrow: 0−0=0; 1−0=1; 1−1=0; 0−1=1 (borrow 1 from the next column, that column becomes 0 temporarily). Alternatively — and this is how CPUs actually work — subtraction is performed as addition of the two's complement negative. Two's complement of a number: invert all bits, then add 1. Two's complement of 0110 (6): inverted = 1001; +1 = 1010. Then 1011 + 1010 = (1)0101 = 5, where the overflow carry bit is discarded. This is why CPUs can reuse the same adder circuit for both addition and subtraction — saving transistors and simplifying the ALU design.
Binary multiplication reduces to repeated shift-and-add operations: multiply each bit of the multiplier by the multiplicand (giving either 0 or the multiplicand itself), shift left according to the bit position, then sum all partial products. For 1011 (11) × 0110 (6):
Hardware multipliers implement this using dedicated circuits that perform all partial products simultaneously in a single clock cycle using carry-save adders. The hexadecimal calculator and developer calculators provide complementary number system tools.
In real computer hardware, binary arithmetic operates on fixed-width registers (8-bit, 16-bit, 32-bit, 64-bit). When the result of an addition exceeds the register width, the overflow carry bit is lost — this is integer overflow. Adding 1 to the maximum unsigned 8-bit value (255 = 11111111₂): result = 100000000₂ = 256, but in 8 bits only the lower 8 bits are kept: 00000000₂ = 0. This silent wraparound is the source of the famous year 2038 problem (32-bit Unix timestamps), the 2012 game Diablo III auction house bug (32-bit gold overflow), and countless software vulnerabilities exploited through integer overflow attacks.
The Result shows the decimal answer of the selected arithmetic operation. For division, the result includes up to 6 decimal places of precision; the integer part is also displayed separately. The absolute values of both inputs and the result are shown for reference. Division by zero returns 0 as a safety measure. These results are identical to what binary hardware produces, since decimal is simply a human-readable representation of the same underlying binary computation.
Inputs
Results
45 + 27 = 72. In binary: 101101 + 011011 = 1001000. The carry chain propagates through bit positions.
Inputs
Results
12 × 13 = 156. In binary: 1100 × 1101 = 10011100. Partial products: 1100, 0000, 1100(shifted 2), 1100(shifted 3), summed.
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