adds

The term adds denotes the operation of addition in mathematics. The result of adding two numbers a and b is the sum, written a + b. Addition is one of the basic arithmetic operations and is defined for many number systems, including integers, rationals, reals, and complex numbers. Properties include commutativity (a + b = b + a), associativity (a + (b + c) = (a + b) + c), and the identity element 0, since a + 0 = a. Addition can be extended to any finite sequence: a1 + a2 + ... + an. In digital systems and computer science, addition is implemented with circuits called adders. A half-adder adds two single bits, producing a sum bit and a carry bit. A full-adder adds three bits (two operands plus a carry-in). Cascading full-adders yields a ripple-carry adder, and more sophisticated designs include carry-lookahead and carry-save adders to improve speed. Multi-operand addition uses pairwise combining or specialized algorithms.

In different number representations, addition can produce overflow or carry out. In signed arithmetic, two's complement representation allows straightforward addition with overflow detection by sign-bit rules. In floating-point arithmetic, addition requires aligning exponents and rounding, with potential loss of precision. In modular arithmetic, addition is performed modulo a fixed base, important in cryptography and error detection.

Common examples: 7 + 5 = 12; -3 + 8 = 5. Additions are foundational in mathematics, computing, finance, and science, forming the basis for more complex operations and algorithms.