If 7 is a prime number then how to prove that √7 is irrational?

1 Answer
MattyMatty
Feb 22, 2018

#"See explanation"#

Explanation:

#"Suppose "sqrt(7)" is rational."#
#"Then we can write it as the quotient of two integers a and b : "#
#"Now suppose the fraction a/b is in simplest form so it cannot"#
#"be simplified anymore (no common factors)."#

#sqrt(7) = a/b#

#"Now square both sides of the equation."#

#=> 7 = a^2/b^2#
#=> 7 b^2 = a^2#
#=> " a is divisible by 7"#
#=> a = 7 m" , with m an integer also"#
#=> 7 b^2 = (7 m)^2 = 49 m^2#
#=> b^2 = 7 m^2#
#=> " b is divisible by 7"#
#"So both a and b is divisible by 7 so that the fraction is not"#
#"in simplest form, which gives a contradiction with our"#
#"assumption."#

#"So our assumption that "sqrt(7)" is rational is wrong."#
#=> sqrt(7) " is irrational."#

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