Binary Addition Calculator

Binary addition combines base-2 numbers using the same place-value principle as decimal addition. Each column represents a power of two rather than a power of ten, and every digit is either zero or one. The four basic cases are simple: zero plus zero gives zero, zero plus one gives one, one plus zero gives one, and one plus one gives binary ten. In the last case, write zero in the current column and carry one into the next column to the left. To add longer values by hand, align their rightmost digits and work from right to left. Include any carry from the preceding column. A column containing one plus one plus a carry totals decimal three, represented as binary eleven, so write one and carry one. Continue until every column and final carry have been processed. This procedure is exactly what a ripple-carry adder performs in basic digital hardware, although modern processors use faster carry-lookahead and parallel-prefix circuits. For example, adding 1011 and 0101 starts on the right. One plus one produces zero with a carry. The next column contains one plus zero plus the carry, producing zero with another carry. The following column contains zero plus one plus the carry, again producing zero with a carry. The left column contains one plus zero plus the carry, producing binary ten. Reading the result gives 10000. In decimal, the same check is eleven plus five equals sixteen. Binary addition is fundamental to computer arithmetic. Subtraction can be implemented by adding a two's-complement value, multiplication can be built from shifted additions, and address calculations repeatedly add offsets to base locations. Checksums, counters, timers, graphics operations, and cryptographic algorithms all rely on addition at the bit level. Learning the carry rule makes many other low-level operations easier to understand. This calculator treats inputs as unsigned whole numbers and preserves arbitrary practical lengths through exact integer arithmetic. Leading zeros do not change the numerical result, so 00101 and 101 both represent five. Fractional binary numbers and signed two's-complement interpretation are outside this calculator's scope. For fixed-width hardware work, remember that a result may need one more bit than the larger operand. If that extra carry is discarded, the stored result wraps modulo the selected word size. The decimal value shown with the binary result is a convenient independent interpretation, not a separate approximation. You can verify any answer by converting each input to decimal, adding them, and converting the sum back to base two. Powers of two are especially useful checks: adding one to a run of ones turns those ones into zeros and creates a new leading one.