Defining Numerical Values

If you make a definition such as f[x_]:=value, the Wolfram Language will use the value you give for any f 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 f.
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Defining the numerical value does not tell the Wolfram Language anything about the ordinary value of f.
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If you ask for a numerical approximation, however, the Wolfram Language 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 Wolfram Language 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 ndigit precision and any accuracy is requested

Defining numerical values that depend on numerical precision.

This defines a numerical value for the symbol const, using 4n+5 terms in the product for ndigit precision.
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Here is the value of const, computed to 30digit precision using the value you specified.
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The Wolfram Language treats numerical values essentially like upvalues. When you define a numerical value for f, the Wolfram Language effectively enters your definition as an upvalue for f with respect to the numerical evaluation operation N.

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