Beyond Bits: An Introduction to Computation over the Reals
Tillmann Miltzow

TL;DR
This paper provides an accessible introduction to computation over the real numbers, emphasizing core concepts, techniques, and their relevance, suitable for educational purposes and broad audiences.
Contribution
It offers a simplified, classroom-friendly exposition of real computation concepts, proofs, and examples, bridging existing literature with modern complexity theory.
Findings
Highlights core techniques in real computation
Provides proofs and examples suitable for teaching
Positions approach relative to existing literature
Abstract
We introduce a lightweight and accessible approach to computation over the real numbers, with the aim of clarifying both the underlying concepts and their relevance in modern research. The material is intended for a broad audience, including instructors who wish to incorporate real computation into algorithms courses, their students, and PhD students encountering the subject for the first time. Rather than striving for completeness, we focus on a carefully selected set of results that can be presented and proved in a classroom setting. This allows us to highlight core techniques and recurring ideas while maintaining an approachable exposition. In some places, the presentation is intentionally informal, prioritizing intuition and practical understanding over full technical precision. We position our exposition relative to existing literature, including Matousek's lecture notes on…
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