functools and math

lru_cache: the decorator you already understand

In lesson 5-2 you sketched a caching decorator. The standard library ships a production-grade one, functools.lru_cache. Stick it on any pure function (same inputs, same output, no side effects) and repeat calls become dictionary lookups:

from functools import lru_cache

@lru_cache(maxsize=None)
def fib(n):
    if n < 2:
        return n
    return fib(n - 1) + fib(n - 2)

Without the cache, fib(35) recomputes the same subproblems millions of times. With it, each fib(k) is computed once. This technique, memoization, reappears as dynamic programming in the Algorithms course.

Caching is the rare optimization that is both free to add and safe, provided the function really is pure, which is why lru_cache shows up in virtually every serious Python codebase.

The math module in sixty seconds

You used + - * / // % ** in Python for Beginners. math adds the rest:

import math

math.sqrt(2)       # 1.4142135623730951
math.floor(3.7)    # 3, round down
math.ceil(3.2)     # 4, round up
math.gcd(12, 18)   # 6, greatest common divisor
math.pi            # 3.141592653589793
math.inf           # infinity, a useful "worse than anything" start value

math.inf deserves a note: when hunting for a minimum, initialize best = math.inf and any real value will beat it. You will use exactly that trick in the unit 10 capstone.