What Are Numerical Methods?

Numerical Methods are a set of techniques used to find approximate solutions to mathematical problems that cannot be solved exactly or easily using standard algebraic methods. When equations are too complex for analytic solutions, numerical approaches let us use computers and calculators to get answers close enough for practical use.

Why Use Numerical Methods?

• Many real-world problems have no exact solution.
• Computers can process numbers quickly, making even complex calculations feasible.
• They allow us to model natural phenomena, design machines, and solve engineering problems.

Key Concepts

• Approximation: Results are close to, but not exactly, the true answer.
• Algorithm: A step-by-step procedure to perform calculations.
• Error: The difference between the true value and the numerical result.

Common Scenarios

• Solving equations that don't have an explicit solution.
• Calculating integrals or derivatives where formulas are too hard to use.
• Simulating physical systems where exact answers are impossible.

When Do We Need Them?

Whenever you use a calculator to find a square root or a computer to simulate weather, you're using numerical methods!