This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)

Defining Numerical Values

If you make a definition such as , Mathematica will use the value you give for any function it encounters. In some cases, however, you may want to define a value that is to be used specifically when you ask for numerical values.
expr=valuedefine a value to be used whenever possible
N[expr]=valuedefine a value to be used for numerical approximation

Defining ordinary and numerical values.

This defines a numerical value for the function .
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Defining the numerical value does not tell Mathematica anything about the ordinary value of .
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If you ask for a numerical approximation, however, Mathematica uses the numerical values you have defined.
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You can define numerical values for both functions and symbols. The numerical values are used by all numerical Mathematica functions, including NIntegrate, FindRoot, and so on.
N[expr]=valuedefine a numerical value to be used when default numerical precision is requested
N[expr,{n,Infinity}]=valuedefine a numerical value to be used when n-digit precision and any accuracy is requested

Defining numerical values that depend on numerical precision.

This defines a numerical value for the symbol , using terms in the product for -digit precision.
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Here is the value of , computed to 30-digit precision using the value you specified.
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Mathematica treats numerical values essentially like upvalues. When you define a numerical value for f, Mathematica effectively enters your definition as an upvalue for f with respect to the numerical evaluation operation N.