Special Problem. 1.5 # 10
Prove that √ is not a rational number.
Definitions:
 A rational number is any number that can be expressed as the quotient or fraction
of two integers , with the denominator q 0.

An irrational number is a number that cannot be expressed as a fraction for any
integers p and q. Irrational numbers have decimal expansions that neither terminate
nor become periodic (repeating decimal).
√
Proof by Contradiction:
Where:
o
o
o gcd (m,n) = 1
Assume √ is a rational number.
(By Definition of Rational Number)
√
( )
(Square Both Sides)
(By Distribution)
Prove that if 5 is a multiple of
, then 5 is a multiple of m. Proof by contrapositive
Assume that if 5 is not a multiple of m, then 5 is not a multiple
.
(
)
Let
. If 5 is not a multiple of m, then when 5
divides m, there is a remainder r, such that
(
)
(
(
)
)
True, there is a remainder when 5 divides
.
, and
is true, therefore if
, then
.
(Proved by contrapositive.)
(By Substitution where
as proved above.)
(By Distribution)
(Reduced)
( )
(By substitution)
5 divides m and 5 divides n, which is a contradiction that the greatest common
denominator of m and n is 1. It proves that 5 cannot be written as the quotient of two
integers, 5 is an irrational number.