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 Documentation / Mathematica / The Mathematica Book / Practical Introduction / Numerical Calculations  /

1.1.2 Exact and Approximate Results

A standard electronic calculator does all your calculations to a particular accuracy, say ten decimal digits. With Mathematica, however, you can often get exact results.

  • Mathematica gives an exact result for

    , even though it has 31 decimal digits.
  • In[1]:= 2 ^ 100


    You can tell Mathematica to give you an approximate numerical result, just as a calculator would, by ending your input with //N. The N stands for "numerical". It must be a capital letter. Section 2.1.3 will explain what the // means.

  • This gives an approximate numerical result.
  • In[2]:= 2 ^ 100 //N


  • Mathematica can give results in terms of rational numbers.
  • In[3]:= 1/3 + 2/7


  • //N always gives the approximate numerical result.
  • In[4]:= 1/3 + 2/7 //N


    Getting numerical approximations.

    When you type in an integer like 7, Mathematica assumes that it is exact. If you type in a number like 4.5, with an explicit decimal point, Mathematica assumes that it is accurate only to a fixed number of decimal places.

  • This is taken to be an exact rational number, and reduced to its lowest terms.
  • In[5]:= 452/62


  • Whenever you give a number with an explicit decimal point, Mathematica produces an approximate numerical result.
  • In[6]:= 452.3/62


  • Here again, the presence of the decimal point makes Mathematica give you an approximate numerical result.
  • In[7]:= 452./62


  • When any number in an arithmetic expression is given with an explicit decimal point, you get an approximate numerical result for the whole expression.
  • In[8]:= 1. + 452/62