Overview

Prerequisites

• Prior topics: 4 · Just Enough Python -- every script in this topic is fully type-annotated Python, and you should already be comfortable reading functions, classes, and list/dict/set literals the way that primer taught them; 7 · Data Structures & Algorithms Essentials gives the stack/heap and Big-O vocabulary this topic deepens into the theory underneath it.
• Tools & environment: a macOS/Linux terminal; Python 3.13.12; a REPL for following along with the number/representation demos. Every example is standard-library-only (struct, sys, itertools, re, heapq, zlib, hashlib, array, time, math) -- no pip install is required anywhere in this topic.
• Assumed knowledge: reading and writing typed Python functions and classes; comfort with arithmetic and simple algebra. No formal computer-science or discrete-math background is assumed -- this topic builds that background from first principles.

Why this exists -- the big idea

The problem before the solution: skip the bedrock and you hit a ceiling you can't explain -- why 0.1 + 0.2 != 0.3, why one loop runs 10x faster than an equivalent-looking one, why some problems have no fast solution at all no matter how clever the code. The one idea worth keeping if you forget everything else: everything you write runs on a finite machine built in layers from logic gates upward -- knowing the layers below lets you explain the anomaly you hit instead of just fearing it or copy-pasting a fix you don't understand.

Cross-cutting big ideas, taught here and then reused for the rest of this curriculum: layering-and-leaks -- the whole computing stack is abstraction stacked on abstraction over physical gates, and its leaks (float rounding error, cache-unfriendly access patterns, a stack that overflows) are exactly what this topic demystifies; abstraction-and-its-cost -- every layer you add (a number system, a data structure, a Turing-complete language) buys convenience at a real, sometimes surprising, cost that only becomes visible once you understand the layer beneath it.

This topic teaches the CS bedrock a self-taught-style engineer usually skips: data representation (number systems, two's complement, IEEE-754 floats, endianness, Unicode), digital logic (boolean algebra, gates, combinational vs. sequential circuits), discrete math (sets, propositional and predicate logic, combinatorics, graph theory, induction), machine organization (the fetch-decode- execute cycle, registers, the memory hierarchy, the stack and heap), automata and computability (finite automata, regular languages, context-free grammars, the Chomsky hierarchy, Turing machines, the halting problem), complexity (P vs. NP, NP-completeness), and information theory (entropy, compression, checksums and hashing) -- all at intuition depth, grounded in 55 small, runnable Python demonstrations plus an intra-topic capstone. Deep dives into specific areas are spread across later topics in this curriculum; this topic builds the mental model those later topics hang on.

How this topic is organized

• Learning -- 55 runnable, heavily annotated examples across Beginner (Examples 1-18: number systems, two's complement, IEEE-754 floats, endianness, Unicode, boolean algebra, gates, combinational vs. sequential circuits, sets, relations, and propositional logic), Intermediate (Examples 19-40: predicate logic, combinatorics, graph theory, induction, the fetch-decode-execute cycle, the memory hierarchy, the stack and heap, and finite automata through the Chomsky hierarchy), and Advanced (Examples 41-55: Turing machines, the halting problem, P vs. NP, NP-completeness, Shannon entropy, compression, and checksums/hashing) -- plus an intra-topic capstone that builds a small "CS foundations toolkit."

Every example is a complete, self-contained, fully type-annotated (DD-39) runnable file colocated under learning/code/, actually executed to capture its documented output -- every printed value, bit pattern, and timing number on this topic's pages is a genuine, captured transcript, never a fabricated one.

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