## Simple Mathematica calculations

In your notebook, type

**(1 + 2) * 3**

and press **Enter**. You'll get the result:
9

You can omit the multiplication sign, and Mathematica is
smart enough to know that it is implied:
**(1 + 2) 3**
9

But Mathematica can do a lot more than simple integer math. Try

**102 / 9**

You get the result
34 / 3

You see that Mathematica reduced the fraction to its simplest
form by cancelling out common factors in the numerator and
denominator. Also, notice that the result is still expressed
as a fraction, not as a decimal number. This is because the
fraction is more exact than any decimal approximation to it,
and Mathematica tries to maintain as much accuracy as possible
unless you specify otherwise.
To get a numerical approximation to this result, type

**N[102 / 9]**

You'll get the result
11.3333

The **N[...]** command tells Mathematica to evaluate the
quantity in brackets **numerically**. If we want more
decimal places, we can ask Mathematica to calculate the same
thing to 20 digits:
**N[102 / 9, 20]**
11.333333333333333333

At this point, it's useful to introduce a Mathematica
shortcut that lets us take the previous output cell and
include it in the current input cell. Enter

**102 / 9**

and you'll get the standard Mathematica result:
34 / 3

Then enter in a new input cell
**N[%, 20]**

and you'll get the result
11.333333333333333333

just as before. The **%** symbol tells Mathematica to
insert the output of the previous calculation.
Mathematica knows the basic mathematical functions like
sine and cosine. Typically these functions have the
name you would expect, but with their first letter
capitalized. To take the sine of 34/3 radians, enter

**Sin[%]**

and you'll get the result
-0.9434996270154848971

Notice that Mathematica has kept 20 significant figures;
it remembers that that was the accuracy of the previous
output cell and passes that information along to the
current cell.
If we try to compute the sine of 34/3 radians from
scratch, we don't get the same result:

**Sin[102 / 9]**
Sin[34 / 3]

Here Mathematica has left both the fraction and the
sine function unevaluated. To evaluate this numerically,
we can use the **N[...]** command:
**N[%, 20]**
-0.9434996270154848971

Notice that Mathematica uses **square**
brackets to delimit the argument of a function, instead
of the more conventional parentheses. This is because
parentheses are used exclusively for **grouping**
in Mathematica, as in the calculation we entered at the
beginning of this section:

**(1 + 2) * 3**

Other parts of the Mathematica tutorial: