## SectionC.2R1: 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).