Approximation is a general term that refers to the representation of a value or set of values using an alternative value or set of values that is simpler, faster to compute, or easier to work with. In mathematics and engineering, approximations are often used to facilitate calculations and to make complex problems more tractable. There are many different techniques for approximating values, and these techniques can be applied to a wide range of problems in many different fields.
DEFINITIONS
- Approximation: The act or process of representing a value or set of values using an alternative value or set of values that is simpler, faster to compute, or easier to work with.
- Numerical approximation: The use of numerical techniques to approximate a value or set of values.
- Analytic approximation: The use of mathematical techniques to approximate a value or set of values.
- Accuracy: The degree to which an approximation matches the true value or set of values it is intended to represent.
- Precision: The degree to which an approximation is consistent and reproducible.
EXAMPLES
- Trigonometric functions: Trigonometric functions such as sine, cosine, and tangent can be approximated using Taylor series expansions or other mathematical techniques.
- Numerical integration: The process of calculating the area under a curve can be approximated using numerical integration techniques such as the trapezoidal rule or Simpson’s rule.
- Approximation of pi: The value of pi can be approximated using various methods, including the use of infinite series or Monte Carlo simulations.
- Machine learning: In machine learning, approximations are often used to simplify complex models or to make them more efficient. For example, decision trees can be approximated using random forests, which are collections of decision trees that are combined to make predictions.
- Approximation of derivatives: The derivative of a function can be approximated using finite difference techniques, which involve evaluating the function at a set of points and using these values to estimate the derivative.
10 QUESTION QUIZ
- What is approximation?
- What is numerical approximation?
- What is analytic approximation?
- What is accuracy?
- What is precision?
- How can trigonometric functions be approximated?
- What is numerical integration?
- How can the value of pi be approximated?
- How are approximations used in machine learning?
- How can derivatives be approximated?
