Saturday, 23 September 2017

radicals - Are the square roots of all non-perfect squares irrational?




I was asked to define a non-perfect square. Now obviously, the first definition that comes to mind is a square that has a root that is not an integer. However, in the examples, 0.25 was considered a perfect square. And the square itself + its root were both not integers.



Is it that all non-perfect squares have irrational roots, e.g. 2?



Answer



In the integers, a perfect square is one that has an integral square root, like 0,1,4,9,16, The square root of all other positive integers is irrational. In the rational numbers, a perfect square is one of the form ab in lowest terms where a and b are both perfect squares in the integers. So 0.25=14 is a perfect square in the rationals because both 1 and 4 are perfect squares in the integers. Any rational that has a reduced form where one of the numerator and denominator is not a perfect square in the integers is not a perfect square. For example, 12 is not a perfect square in the rationals. 1 is a perfect square in the integers, but 2 is not, and there is no rational that can be squared to give 12


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