In mathematics, making an approximation is the act or process of finding a number acceptably close to an exact value; that number is then called an approximation or approximate value. Approximating has always been an important process in the experimental sciences and engineering, in part because it is impossible to make perfectly accurate measurements. Approximation also arises because some numbers can never be expressed completely in decimal notation. In these cases approximations are used. For example, irrational numbers, such as pi (π), are nonterminating, nonrepeating decimals. Every irrational number can be approximated by a rational number, simply by truncating it. Thus, π can be approximated by 3.14, or 3.1416, or 3.141593, and so on, until the desired accuracy is obtained.
Another application of the approximation process occurs in iterative procedures. Iteration is the process of solving equations by finding an approximate solution, then using that approximation to find successively better approximations, until a solution of adequate accuracy is found. Iterative methods, or formulas, exist for finding square roots, solving higher order polynomial equations, solving differential equations, and evaluating integrals.
The limiting process of making successively better approximations is also an important ingredient in defining some very important operations in mathematics. For instance, both the derivative and the definite integral come about as natural extensions of the approximation process. The derivative arises from the process of approximating the instantaneous rate of change of a curve by using short line segments. The shorter the segment the more accurate the approximation, until, in the limit that the length approaches zero, an exact value is reached. Similarly, the definite integral is the result of approximating the area under a curve using a series of rectangles. As the number of rectangles increases, the area of each rectangle decreases, and the sum of the areas becomes a better approximation of the total area under investigation. As in the case of the derivative, in the limit that the area of each rectangle approaches zero, the sum becomes an exact result.
Every function can be expressed as a series, the indicated sum of an infinite sequence. For instance, the sine function is equal to the sum: sin(x) = x — x3/3! + x5/5! — x7/7!+.... In this series the symbol (!) is read "factorial" and means to take the product of all positive integers up to and including the number preceding the symbol (3! = 1 × 2 × 3, and so on). Thus, the value of the sine function for any value of x can be approximated by keeping as many terms in the series as required to obtain the desired degree of accuracy. With the growing popularity of digital computers, the use of approximating procedures has become increasingly important. In fact, series like this one for the sine function are often the basis upon which handheld scientific calculators operate.
An approximation is often indicated by showing the limits within which the actual value will fall, such as 25 ± 3, which means the actual value is in the interval from 22 to 28. Scientific notation is used to show the degree of approximation also. For example, 1.5 × 106 means that the approximation 1,500,000 has been measured to the nearest hundred thousand; the actual value is between 1,450,000 and 1,550,000. But 1.500 × 106 means 1,500,000 measured to the nearest thousand. The true value is between 1,499,500 and 1,500,500.