Analytical methods use mathematical principles to fully predict the implications of a theory, while numerical methods can only achieve an approximate prediction. Analytical methods are preferred for simple equations, while numerical methods are used for complex equations. Differential equations must be solved using calculus and often have only approximate results. Numerical methods are used for practical problems, while analytical predictions require one or more equations relating different variables.
Analytical methods use mathematical principles to fully predict the implications of a theory. They can be used to solve an equation in its entirety without any degree of estimation. They are in contrast to numerical methods, which can only achieve an approximate prediction. Analytical methods are the preferred way to determine the outcome of a hypothesis when the associated equations are simple and an accurate answer is desired. Numerical methods are used when the equations are too complex to solve completely.
Mathematicians use analytic methods when they use the basic principles of algebra to solve an equation. If the equation is simple enough, a complete solution can be obtained by manipulating the equation in symbolic form. In this case, there is no room for approximations or guesses: the principles of mathematics always govern what operations can be performed. If the variable in question can be successfully isolated, analysis is the tool to make it possible.
In the equation y = 2x, one can solve for x using analytical methods. To isolate the variable x, both sides of the equation must be divided by the number 2. For any value of y, x can be completely determined with relative ease.
In a simple real-world application of this equation, one could assume that the length of a human foot was twice its width: length = 2*width. This equation necessarily implies that width = ½*length. Practical application of the equation may not constitute accurate theory, but manipulation of the equation is done using analytical methods. That is, the equation can predict foot width without introducing approximations.
There are some equations that no one can solve analytically. For example, many differential equations have no known solutions. A differential equation relates the rate of change of a quantity to its value. Unlike an algebraic equation, differential equations must be solved using calculus. Often, their results can only be approximate.
Numerical methods are used to solve a variety of practical problems. Many companies try to optimize sales using numerical methods to approximate market conditions. They may try to predict the outcome of various business strategies, but generally cannot use analytics. Making analytical predictions, such as the size of the human foot, would require one or more equations relating different variables. The market is generally very complicated and has too many variables to be modeled this way.
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