DEFINITION. A real number r is said to be rational
if it is the ratio m/n of two
integers.
EXAMPLE. 2/3, (−6)/5.
Definition. A real number r is said to be algebraic
if it is a solution to a polynomial
equation where the coefficients of the polynomial are integers.
EXAMPLE. 
is
algebraic because it is a solution to p(x) = −2x^2 + 4 = 0.
FACT. Every rational number is algebraic. Show it!
(Hint: a linear polynomial will
do.)
DEFINITION. A real number r is said to be
transcendental if it is not algebraic.
We have that rational numbers are algebraic, algebraic
numbers are not necessarily
rational (see below), and irrational numbers can be either algebraic or
transcendental, but
not both!
The numbers e and π are transcendental. This is not easy
to prove.
It is not known whether 
is transcendental.
In a precise sense : “most real numbers” are transcendental; however there are
enough
rational numbers to al low us to compute!
PROBLEM. Show that is
not rational. (Hint: argue by contradiction; choose
the integers m and n to have no common factors ; use that all positive integers
can be
factored as a product of prime numbers)
PROBLEM. What can you say about the rationality of
when t is a positive
integer?
