Principles of Binary Addition and Logic Gates

Key Insight

Addition is the foundational arithmetic operation in computing. While conceptually similar to decimal addition, binary addition uses a simpler table for single digits. Key operations are: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 0 with a 'carry the 1'. Each binary digit addition generates two outputs: a 'sum bit' and a 'carry bit'. For example, 1 + 1 yields a sum bit of 0 and a carry bit of 1. Breaking down addition into sum and carry operations simplifies circuit design, as all circuit elements (switches, lightbulbs, wires) inherently represent binary digits.

Logic gates directly implement these sum and carry functions. The 'carry bit' for adding two binary digits is calculated by an AND gate, where the output is 1 only if both inputs are 1. The 'sum bit' is derived from an Exclusive OR (XOR) gate. An XOR gate's output is 1 if one input is 1 but not both. This functionality can be constructed by connecting an OR gate and a NAND gate to the same inputs, then feeding their outputs into an AND gate. This composite circuit precisely matches the required 'sum bit' behavior.

These logic gates form the building blocks of binary adders. A 'Half Adder' combines an XOR gate and an AND gate to add two binary digits (A and B), producing a sum and a carry output. However, for multi-digit addition, a 'Half Adder' cannot incorporate a carry-in from a previous column. To address this, a 'Full Adder' is required, which adds three binary numbers: A, B, and a 'Carry In'. A 'Full Adder' is constructed using two 'Half Adders' and an OR gate, totaling 18 relays. The 'Carry Out' from the first 'Half Adder' and the 'Carry Out' from the second 'Half Adder' are combined by an OR gate to produce the final 'Carry Out'.