In[1]:=
2+ 11 * 8
Out[1]=
90
Try other expressions to get used to the syntax used in Mathematica.
Here are
a few examples.
In[2]:=
2^67 -1
Out[2]=
147573952589676412927
In[3]:=
(11^23-6^23)/5
Out[3]=
179085890705002863728743
In[4]:=
17/29
Out[4]=
17 -- 29
Notice that the result of 17/29 is again 17/29. This is an important feature of
Mathematica.
Mathematica represents mathematical values exactly, without any
approximations. The
value of 17/29 is 17/29, so that is the result given by Mathematica. If
you wish to see
a decimal approximation, then you can use the N function of
Mathematica. The
command
N[ expression, places] gives a approximation of the expression to the
specified number
of places.
In[5]:=
N[17/29, 8]
Out[5]=
0.5862069
Exercise: Compute the value of square root of 2 to 50
decimal places. The Mathematica function for evaluating the square root
is
Sqrt[ expression].
In[6]:=
Mathematical constants like e, pi, and i are represented
by E, Pi, I in Mathematica. Mathematica has all built-in
variables and functions starting with a
capital letter, hence it is a good idea to start all your variables and
functions with a lower
case letter to avoid any confusion with an existing name.
Another feature of Mathematica is worth paying attention. Recall that we
use the symbol
^ for exponentiation. Let us compute the cube root of -1.
In[7]:=
(-1)^(1/3)
Out[7]=
1/3 (-1)
In[8]:=
N[ (-1)^(1/3), 6]
Out[8]=
0.5 + 0.866025 I
Observe that the approximation of the cube root of -1 is a complex number. The
number
-1 has three cube roots, two of them complex. Mathematica works
internally with complex
numbers, so it has selected a primitive complex cube root. To be able to
compute
a real cube root of a negative real number, we have to define a function
ourselves. We
will see how to accomplish this in a later section.
Exercise: Determine sin(pi/12) to 8 decimal places. Compute sin(x)/x for
values of
x close to zero to observe the limit as x approaches 0. The Mathematica
function for
evaluating sin(x) is Sin[x].
Here are examples of a few more useful functions. Quotient[a,b] gives the
quotient
when a is divided by b and Mod[a,b] gives the remainder that has the same sign
as b. The
two functions satisfy the relation
a= b Quotient[a,b] + Mod[a,b].
In[9]:=
Quotient[ -34, 7]
Out[9]=
-5
In[10]:=
Mod[ -34, 7]
Out[10]=
1
In[11]:=
Quotient[ 34, -7]
Out[11]=
-5
In[12]:=
Mod[ 34, -7]
Out[12]=
-1
Exercise: Verify that the relation between the Quotient and Remainder that
is
specified above is valid.
Another useful function is Floor[ x], that returns the largest integer less
than x. Here x has to
be an expression that evaluates to a real number, otherwise Floor doesn't do
anything.
Related functions are Ceiling and Round; Ceiling returns the smallest integer
greater
than the expression and Round returns the nearest integer. Here are a few
examples.
In[13]:=
Floor[ -4.5]
Out[13]=
-5
In[14]:=
Ceiling[-4.5]
Out[14]=
-4
In[15]:=
Round[ -4.6]
Out[15]=
-5
In[16]:=
Round[ -4.4]
Out[16]=
-4
Exercise: What does Round do for numbers of the form integer + 1/2?
In[17]:=
Floor[1/3]
Out[17]=
0
In[18]:=
Floor[Sqrt[2]]
Out[18]=
Floor[Sqrt[2]]
Paranthesis: These are used for grouping expressions. Careful use of
parenthesis
is necessary to make clear the meaning of arithmetical expression. This is
because multiplication
and division have a higher order of precedence than addition and
subtraction.
For example, 3 + 7* 8 is not (3+7) *8 but 3 + (7*8). The expression
21/7-5 is (21/7) -5 and not 21/(7-5).
In[19]:=
21/7-5
Out[19]=
-2
In[20]:=
21/(7-5)
Out[20]
In Mathematica (as in other programming languages) it is essential to
use parenthesis
Brackets: Square brackets are used for specifying arguments of functions. For
example,
In[21]:=
Braces: These are used for specifying lists. A list is an order collection of
objects. Lists are
In[22]:=
Out[22]=
Notice the semicolon at the end of the definition of the list named numbers
above. The
Up to Tutorial
to make the meaning of arithemetical expressions transparent.
Exercise: What does a/b/c represent in Mathematica; (a/b)/c or a/(b/c)?
What about
a/b*c?
Floor takes a single argument, Floor[ x]. Square brackets cannot be used for
grouping terms
in arithmetical functions as done in written mathematical work. Try it in a
few examples to
see what happens.
2*[3+4]
Syntax::sntxf:
"2*" cannot be followed by
"[3+4]".
;[o]
Syntax::sntxf:
"2*" cannot be followed by
"[3+4]".
used for representing vectors, matrices and other mathematical collections.
Many
functions will return lists as their result and other functions manipulate
lists. List
manipulation is one of the most powerful features of Mathematica and is
discussed in more
detail in a later section. Here are a few examples of their use.
We can compute the remainder obtained by the first few Fibonacci numbers using
the list
manipulation features of Mathematica.
numbers= { 1,1,2,3,5,8,13,21,34};
Mod[ numbers, 3]
{1, 1, 2, 0, 2, 2, 1, 0, 1}
purpose of the semicolon is suppress the output of the expression. Try this in
a few of
your own computations.