What are the key take-aways about computer arithmetic?

It's deterministic — never random — but finite representation means it doesn't obey all the math rules you'd expect (such as associativity).

The practical takeaway is to know which familiar properties still hold and which break: fixed-width integer operations form a ring but can overflow, and floating-point preserves ordering but loses associativity — and attackers exploit exactly these gaps.

Does NOT generate random values:

• Arithmetic has well-defined (if surprising) behavior
• Results are deterministic, just not what math predicts

Cannot assume "usual" mathematical properties:

• Due to finite representation
• Integers: satisfy ring properties (commutativity, associativity, distributivity)
• Floating-point: satisfy ordering properties (monotonicity, sign values)
• But NOT all mathematical properties!

Important observations:

• Need to understand which abstractions apply in which contexts
• Critical for compiler writers and serious programmers
• Security vulnerabilities often exploit these gaps

Go deeper:

• Modular arithmetic (Wikipedia) — machine integers form the ring Z/2^W Z, so arithmetic wraps and ordering can't be assumed.
• Floating-point arithmetic (Wikipedia) — the flip side: finite precision breaks associativity and exactness for reals.