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 exact AND/OR truth table discrete mathematics already covered. Boolean algebra supplies three core operations — AND, true only if both inputs are true; OR, true if either input is true; and NOT, which flips true to false — and every other logical function, no matter how complex, can be built by combining just these three, the same operations behind the truth table worked through earlier in this series. A logic gate is the physical circuit that implements one Boolean operation, and it is built directly from the hardware this entire series has been running on top of: 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 directly in silicon — this is the literal bridge from a true/false proposition to a physical voltage level on a real chip. 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, which is exactly what a register is — a small set of flip-flops, one-bit memory cells built from feedback-looped gates, holding a fixed number of bits as the fastest, closest-to-the-processor storage tier, even faster than the stack covered earlier in this series. 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 binary integers from data representation get added together in real 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 the layered picture that scales all the way up into the full CPU this entire series has been describing as running on top of exactly these gates.

Boolean algebra
AND, OR, NOT, the same truth table from discrete mathematics
Logic gates
transistors wired to implement each operation in silicon
Combinational circuits
output depends only on current inputs
Sequential circuits and registers
a clock plus feedback adds memory of past state
Arithmetic units
half-adders and full-adders chained into a binary adder
Three Gates, Built From Boolean Algebra The same AND/OR truth table, now realized in physical silicon AND gate A and B true only if both inputs true OR gate A or B true if either input is true NOT gate flips true to false on a real die, each gate is a handful of 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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