Data Representation and Computation

Theory and Practice

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What is computation? In its broadest sense, computation is the process of using a defined procedure to transform input data into output data. Imagine waking up in the morning and deciding to make a cup of coffee. You gather the necessary ingredients-water, coffee grounds, and sugar-and follow a specific set of steps: measure the water, heat it, mix it with the coffee. The result is a freshly brewed cup of coffee. In this analogy, making coffee reflects a computational process: starting with raw inputs, applying a series of procedures, and producing a final output. In digital computation, this process happens within a computer, using data and algorithms to transform inputs such as numbers or instructions into outputs such as decisions, results, or actions. This book explores digital computation with the aim of showing how modern computer systems actually work. It begins at the most fundamental level: the bit. These tiny binary units are combined in precise ways to represent the multimedia data we use every day-music, videos, images, text, and numbers. The computer is designed not only to store and interpret these bits but also to compress, share, and protect them. When data is shared-whether across programs or networks-the computer ensures its integrity and security, making sure it arrives intact and as intended. To understand how computers carry out such complex processes, we examine the foundational components of digital circuits. At the heart of these are logic gates: simple devices that perform basic computational tasks by taking bits as input and producing new bits as output based on logical operations. By combining these gates, computers perform operations such as addition, subtraction, and comparison. The central processing unit (CPU), often considered the brain of the computer, is essentially a complex assembly of logic gates and circuits. What makes these devices even more powerful is their programmability-they can be reused to solve a vast array of problems simply by changing the instructions they receive. The book concludes by investigating the limits of computation-the problems that computers cannot solve, and those that are theoretically solvable but practically intractable. It is here that readers come to appreciate not just the power of computation, but also its boundaries. Understanding what computers cannot do is just as important as understanding what they can. The theory throughout the textbook is presented through more than 80 theorems, propositions, and definitions, supported by dozens of hand-drawn figures and clear tables. Over 400 rigorous explanations and detailed discussions guide readers through complex topics, and more than 300 practical examples, mathematical proofs, and case studies provide tangible applications. This is all designed to help readers grasp both the "how" and the "why" behind computing theory. To reinforce learning, the book includes 250 original exercises aligned with the core concepts, with full solutions available. This combination of theory, practice, and reflection equips readers with a comprehensive and accessible understanding of computation in modern systems.

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