Part 03 of 12

Arithmetic, signs, and overflow

Build the mathematics that later becomes a physical arithmetic circuit.

Six connected lessonsA diagram per chapterStart from first principles

The story before the details

Arithmetic does not require a mysterious calculator hidden inside the processor. The basic problem is to add columns with weights 1, 2, 4, 8, and so on. Carrying is simply what happens when a column contains two or three units but is allowed to keep only zero or one. That carry enters the next position, where the weight is doubled.

Negative numbers are more subtle because there is no physical minus-sign wire attached to an ordinary register. Two’s complement gives each fixed-width pattern a consistent signed interpretation while preserving the same addition circuitry. Inversion plus one produces the modular additive inverse; subtraction can therefore reuse the adder rather than needing a separate arithmetic engine.

Finite range is not a footnote. Eight-bit unsigned storage wraps after 255, while signed eight-bit storage represents −128 through +127. The circuits still output bits when a mathematical result cannot fit. Your job is to understand the interpretation and the status flags, rather than assume the hardware stops operating.

Four-bit ripple adder, diagram of connected labeled components — Four-bit ripple adder. Follow the arrows to identify where information travels and what each block contributes.

What you will discover

These six lessons are arranged so each one answers a question created by the previous one. Do not rush to the last lesson. If a concept feels opaque, use the checkpoint before continuing.

Keep this in mind

Build the mathematics that later becomes a physical arithmetic circuit. You should be able to explain each concept using only an input, a rule, an output, and any state that must be retained.

Start this part →← Previous part Next part →