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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:

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