Mathematics is a powerful tool for understanding the world, but many problems you encounter in science and engineering cannot be solved exactly using algebra or calculus alone. These problems are called analytically unsolvable or intractable. For instance, you might want to predict the weather, simulate the airflow over an airplane wing, or determine how a drug disperses in the human body. The mathematical models for these scenarios often involve complex equations — such as nonlinear differential equations or systems with many interacting variables — that cannot be solved with a neat formula.

In these cases, you turn to numerical methods: systematic techniques for finding approximate solutions using computers. Numerical methods allow you to tackle problems that are impossible or impractical to solve by hand. For example, you can use numerical integration to estimate the area under a curve when no exact antiderivative exists, or use iterative algorithms to find the roots of equations that cannot be factored.

Real-world examples abound. Engineers use numerical simulations to design safer cars and airplanes. Physicists rely on numerical methods to model the behavior of stars and galaxies. Biologists use computational models to understand the spread of diseases. In finance, numerical methods help estimate the value of complex financial derivatives. Whenever an exact solution is unavailable or too costly to compute, numerical methods step in to provide practical answers.