Students completing a year of Calculus can find novel and practical challenges in mathematics using numerical analysis. This text follows the mathematician's process of examining a problem and deciding which algebraic or calculus techniques lead to reliable solutions. Each algorithm includes all developmental steps, illustrating the beauty of reducing complex systems to streamlined formulae. Accessible proofs using techniques beyond those in geometry are fully explained. All algorithms contain detailed examples of success and situational pitfalls. An exhaustive set of practice problems was meticulously developed by hand, not AI-generated, ensuring deviations at each stage of a particular algorithm. Many exercises highlight advantages and disadvantages of algorithms through comparison.