I have been looking at various proofs of this statement, for example Proof 1 on this page : http://www.cut-the-knot.org/proofs/sq_root.shtml(adsbygoogle = window.adsbygoogle || []).push({});

I'd like to know if the following can be considered as a valid and rigorous proof:

Given ##y \in \mathbb{Z}##, we are looking for integers m and n ##\in \mathbb{Z}## such that ##(m/n)^2=y##.

We can write ##m/n=x## where ##x \notin \mathbb{Z}##.

Thus ##(m/n)^2=x^2=y##, or

##m^2=x^2n^2##

##m^2-x^2n^2=0##

##(m-xn)(m+xn)=0##

Taking only the positive value,

##m=xn##

Now ##x \notin \mathbb{Z}##,

which means ##xn \notin \mathbb{Z}## which implies ##m \notin \mathbb{Z}##

which contradicts our requirement that ##m\in \mathbb{Z}##

So no pair of m, n can be found as specified.

===

Edit : We could just go directly to ##m=xn## from ##(m/n)^2=x^2=y## , perhaps without loss of rigor?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# B Proof that non-integer root of an integer is irrational

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - Proof integer root | Date |
---|---|

B Did President Garfield really come up with an alternate proof? | Mar 7, 2018 |

I How to know if a polynomial is odd or even? | Jul 7, 2016 |

Alternative Proof to show any integer multiplied with 0 is 0 | Jan 31, 2014 |

Proof about relatively prime integers. | Apr 22, 2013 |

Number of integer solutions to x^2 + y^2 <= n? [simple proof] | Dec 18, 2011 |

**Physics Forums - The Fusion of Science and Community**