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

Out[1]=

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

Out[2]=

Mathematica can give results in terms of rational numbers.

In[3]:= 1/3 + 2/7

Out[3]=

//N always gives the approximate numerical result.

In[4]:= 1/3 + 2/7 //N

Out[4]=

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

Out[5]=

Whenever you give a number with an explicit decimal point, Mathematica produces an approximate numerical result.

In[6]:= 452.3/62

Out[6]=

Here again, the presence of the decimal point makes Mathematica give you an approximate numerical result.

In[7]:= 452./62

Out[7]=

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

Out[8]=