Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. However some known form may exist and may be able to represent the real form so that no significant deviation can be found. It also is used when a number is not rational, such as the number π, which often is shortened to 3.14159, or √2 to 1.414. Numerical approximations sometimes result from using a small number of significant digits. Approximation theory is a branch of mathematics, a quantitative part of functional analysis. Diophantine approximation deals with approximations of real numbers by rational numbers.
Related to approximation of functions is the asymptotic value of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large. For example, the sum (k/2)+(k/4)+(k/8)+...(k/2^n) is asymptotically equal to k. Unfortunately no consistent notation is used throughout mathematics and some texts will use ≈ to mean approximately equal and ~ to mean asymptotically equal whereas other texts use the symbols the other way around.
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