Definition. A rational number is a number that can
be written as a quotient of two integers ,
where b≠ 0.
Proposition. Every rational number can be written as an infinite
repeating decimal. E.g.
The bar over 3, indicates that the number 3 repeats indefinitely.
Definition. A perfect square is an integer whose square root is another
integer. E.g.
so,
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, . . .
are perfect squares .
Definition. An irrational number is a number that can not be written as a
quotient of two integers
,
where b≠ 0.
Proposition. In particular, a square root that is not equal to an integer
is an irrational number.
Proposition. An irrational number is a number that has a nonterminating,
nonrepeating decimal represen-
tation.
Problem. Is a rational or
irrational number?
9 <10 < 16
calculating the square root to the previous expression ,
Since there is no integer between 3 and 4,
is not equal to an integer, so it is an irrational number .
Procedure. You may obtain a rational number between other two taking the
average of these two numbers: