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Section C.2 R1: Computer Arithmetic

Synopsis.

  • Recap Positional number systems.

  • Understand that sequences of digits are just sequences of symbols and that they require an embedding into the naturals.

  • Understand that the positional systems make sense because there is a bijection between digit sequences of length \(k\) and naturals between \(0\) and \(2^k-1\text{.}\) This is an opportunity to recap simple induction proofs.

  • Understand binary and hexadecimal numbers and the appropriate terminology.

  • Introduce and, or, xor as binary operations on bits that can be easily implemented in hardware.

  • Understand that computers operate on binary numbers of a fixed length and that this permeates the type systems of many imperative programming languages for performance reasons, so eventually people have to be aware of this and the consequences (overflows).

Sections Covered.

Section 1.1, Section 1.2