Difference between revisions of "Proof that the square root of any nonperfect square positive integer is irrational"
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| − | + | ==== Proof that any nonperfect square positive integer is irrational ==== | |
Let us assume that <math>\sqrt{n}</math> is rational where <math>n</math> is a nonperfect square positive integer. Then it can be written as <math>\frac{p}{q}=\sqrt{n} \Rightarrow \frac{p^2}{q^2}=n \Rightarrow (q^2)n=p^2</math>. But no perfect square times a nonperfect square positive integer is a perfect square. Therefore <math>\sqrt{n}</math> is irrational. | Let us assume that <math>\sqrt{n}</math> is rational where <math>n</math> is a nonperfect square positive integer. Then it can be written as <math>\frac{p}{q}=\sqrt{n} \Rightarrow \frac{p^2}{q^2}=n \Rightarrow (q^2)n=p^2</math>. But no perfect square times a nonperfect square positive integer is a perfect square. Therefore <math>\sqrt{n}</math> is irrational. | ||
Latest revision as of 14:17, 19 June 2019
Proof that any nonperfect square positive integer is irrational
Let us assume that 
is rational where 
is a nonperfect square positive integer. Then it can be written as 
. But no perfect square times a nonperfect square positive integer is a perfect square. Therefore is irrational.
