Numerical methods are techniques used to solve mathematical problems by approximating solutions with numerical values. These methods are particularly useful for equations that cannot be solved analytically, such as complex integrals or differential equations. They rely on algorithms and computational processes to provide approximate solutions, making them essential in fields like engineering, physics, and finance. Common numerical methods include Newton's method for finding roots of equations, Euler's method for solving ordinary differential equations, and Monte Carlo simulations for statistical analysis. By using these methods, researchers and professionals can analyze data and model real-world scenarios more effectively.