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.
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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. "Special Ways to Input Expressions" will explain what the 
means.
This gives an approximate numerical result.
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Mathematica can give results in terms of rational numbers.
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//N always gives the approximate numerical result.
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| expr//N | give an approximate numerical value for expr |
Getting numerical approximations.
When you type in an integer like 
, Mathematica assumes that it is exact. If you type in a number like 
, 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.
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Whenever you give a number with an explicit decimal point,
Mathematica produces an approximate numerical result.
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Here again, the presence of the decimal point makes
Mathematica give you an approximate numerical result.
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When any number in an arithmetic expression is given with an explicit decimal point, you get an approximate numerical result for the whole expression.
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