# 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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