Numerical Calculations

Exact symbolic results are usually very desirable when they can be found. In many calculations, however, it is not possible to get symbolic results. In such cases, you must resort to numerical methods.

N[expr]	numerical value of an expression
NIntegrate[f,{x,x_min,x_max}]	numerical approximation to
NSum[f,{i,i_min,Infinity}]	numerical approximation to
FindRoot[lhs==rhs,{x,x₀}]	search for a numerical solution to an equation, starting with
NSolve[lhs==rhs,x]	numerical approximations to all solutions of an equation
FindMinimum[f,{x,x₀}]	search for a minimum of f, starting with
NMinimize[f,x]	attempt to find the global minimum of f

Basic numerical operations.

The Wolfram Language maintains this expression in an exact, symbolic form:

In[1]:=1

✖

https://wolfram.com/xid/0b3etlgqihsj-haa

Out[1]=1

You can even use standard symbolic operations on it:

In[2]:=2

✖

https://wolfram.com/xid/0b3etlgqihsj-vu5

Out[2]=2

N[expr] gives you a numerical approximation:

In[3]:=3

✖

https://wolfram.com/xid/0b3etlgqihsj-oj9

Out[3]=3

Functions such as Integrate always try to get exact results for computations. When they cannot get exact results, they typically return unevaluated. You can then find numerical approximations by explicitly applying N. Functions such as NIntegrate do the calculations numerically from the start, without first trying to get an exact result.

There is no exact formula for this integral, so the Wolfram Language returns it unevaluated:

In[4]:=4