If statement

The If statement is used to perform actions based on the results of tests. The general
format of the statement is

If[ test, statements1, statements2]

where statesments1 is performed if the result of the test is True and statements2
is performed if the result of the test is False. If nothing is to be done when the test
is False then statements2 can be omitted. It is important to separate the statements
with a comma. The statements can be a series of expression separated by semi-colons.

In[58]:=

  Clear[ realcuberoot]
  realcuberoot[x_]:= If[ x>= 0,
                           x^(1/3),    (* True value *)
                           -(-x)^(1/3) (*False value *)
                        ]

In[59]:=

  realcuberoot[-1]

Out[59]=

-1

The function returns the real cube root of a negative number. Observe that we
placed a comment about the result in the If statement. Mathematica comments
are placed between the expressions (* and *).

Another example of an If statement is the following in which we check if a number
is a perfect square.

In[60]:=

  squareQ[ n_]:= If[ Floor[ N[Sqrt[n]]]^2== n,
                          True,  (* Yes, aperfect square *)
                          False (* not a square *)
                    ]

In[61]:=

  squareQ[196]

Out[61]=

  True

In[62]:=

  squareQ[255]

Out[62]=

  False

The function can be written without any If statements by writing
squareQ[n_]:= Floor[ N[ Sqrt[n]]]^2 == n;

This is because the result of the test for equality is a True or False. The following are
the relational operators that can be used. These can be combined with logical operators,
And, Or , or Not to create all the tests that we need.

x = = y Is x equal to y?

x!= y Is x not equal to y?

x> y Is x greater than y?

x< y Is x less than y.

Similarly, the relations >= and <= are self-explanatory. The only issue to keep in mind is
that two equal signs are necessary for a test of equality. A single = is an assignment of
one to the other and not a test. The boolean AND operator is represented by
&& and the logical OR by ||. The complement is represented by the exclamation !.
For example, to test if n is a prime that is not of the form 5k+3, we can use

PrimeQ[n] && !( Mod[ n,5]==3).

Exercise: Write a function that returns True if a number is either a prime or a perfect square,
and False otherwise..

Write another function that returns True if a number is a prime or the square of a
prime, and False otherwise.

Up to Loops and Conditionals