Module 01: Mathematics for Systems

Why Math for Backend Engineers?

Backend performance analysis requires math. Not advanced math — but specific applied math that most engineers never learned formally:

• Big-O analysis tells you whether your algorithm will work at 100 rows or 100 million rows
• Probability explains cache hit rates, failure probabilities, and expected values
• Statistics is how you interpret benchmark results without being fooled
• Queueing theory is the formal model behind connection pool sizing and load balancing
• Numerical stability matters when you compute money, time deltas, or coordinates

Contents

Key Insight

Every performance equation in later modules traces back to this module.

N+1 query cost  = O(N) queries × RTT      ← Big-O + networking
Pool wait time  = L / λ = W               ← Little's Law (queueing)
Cache hit rate  = Zipf distribution       ← Probability
p99 vs p50 gap = heavy-tailed distribution ← Statistics