Digital Logic

Digital logic is where this series stops talking about computing in the abstract and starts building the actual physical machinery underneath it, starting from the same true/false logic the discrete-mathematics topic already covered and ending at the arithmetic circuits that do real binary addition. Boolean algebra supplies the three core operations — AND (true only if both inputs are true), OR (true if either input is true), and NOT (flips true to false) — and every other logical function, no matter how complex, can be built by combining just these three. A logic gate is the physical circuit that implements one Boolean operation, and on a real chip it is built from the exact CMOS transistor pairs this site's own process coverage already describes: a transistor acts as a voltage-controlled switch, and arranging a handful of them in specific patterns produces an AND gate, an OR gate, or an inverter (NOT gate) directly in silicon — this is the literal bridge from "0s and 1s" in the data-representation topic to physical voltage levels on a real die. Wiring individual gates together builds a circuit: a combinational circuit's output depends only on its current inputs, with no memory of anything before, while a sequential circuit adds a clock and feedback so its output also depends on what happened in previous steps — this is exactly what a register is, a small set of flip-flops (one-bit memory cells built from feedback-looped gates) that hold a fixed number of bits, forming the fastest, closest-to-the-processor storage tier this series' hardware coverage has already referenced above DRAM and cache. An arithmetic unit is a combinational circuit purpose-built for math: a half-adder combines two single bits using an XOR gate for the sum and an AND gate for the carry, and chaining several full-adders together (each one also accepting a carry-in from the previous bit) builds a complete binary adder capable of adding two multi-bit numbers exactly the way the data-representation topic's two's-complement integers get added in hardware. Every layer here builds strictly on the one below it — Boolean algebra defines the rules, gates realize them in silicon, circuits wire gates into either memory or computation, and arithmetic units specialize circuits for math — which is exactly the layered picture the next topic, computer architecture, scales up into a full CPU: an arithmetic logic unit, registers, and a control path, all built from nothing more than the gates introduced here.

Boolean algebra
AND, OR, NOT — the three operations everything else is built from
Logic gates
CMOS transistors wired to implement each operation in silicon
Combinational circuits
output depends only on current inputs
Sequential circuits / registers
clock + feedback adds memory of past state
Arithmetic units
half-adders and full-adders chained into a binary adder
Three Gates, Built From Boolean Algebra Every logic gate on a real die is CMOS transistors wired this way AND gate A·B true only if both inputs true OR gate A+B true if either input is true NOT gate flips true to false on a real die, each gate is a handful of CMOS transistors — voltage-controlled switches arranged to produce one of these three shapes every more complex logical function is built by combining AND, OR, and NOT XOR, NAND, and NOR are themselves just useful combinations of these three
From Half-Adder to a Real Binary Adder Chaining the same circuit per bit adds entire numbers Half-adder: adds bit A + bit B sum = A XOR B carry = A AND B 1 + 1 = sum 0, carry 1 exactly like adding by hand Full-adder: adds A + B + carry-in takes the previous bit's carry as a third input this is what actually lets addition chain across bits Chained bit 0 adder's carry-out feeds bit 1 adder's carry-in repeat per bit a register is the same idea applied to storage instead of math: a flip-flop is a small feedback loop of gates that holds one bit chain enough flip-flops together and a register holds a full binary number

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