Algebra in Complexity Analysis

When analyzing algorithms, you write time complexity as a formula in terms of input size $n$. Algebra helps you simplify these formulas.

For example, if you have nested loops where the outer runs $n$ times and the inner runs $n$ times, the total is $n \times n = n^2$ operations. You write this as $O(n^2)$.

If you have $3n^2 + 5n + 7$ operations, you drop constants and lower-order terms to get $O(n^2)$. Algebra tells you which term dominates as $n$ grows.