# proof that $\sqrt{2}$ is irrational

Assume that the square root of $2$ (http://planetmath.org/SquareRootOf2) is rational. Then we can write

$$\sqrt{2}=\frac{a}{b},$$ |

where $a,b\in \mathbb{N}$ and $a$ and $b$ are relatively prime. Then $2={(\sqrt{2})}^{2}={\left({\displaystyle \frac{a}{b}}\right)}^{2}={\displaystyle \frac{{a}^{2}}{{b}^{2}}}$. Thus, ${a}^{2}=2{b}^{2}$. Therefore, $2\mid {a}^{2}$. Since $2$ is prime, it must divide $a$. Then $a=2c$ for some $c\in \mathbb{N}$. Thus, $2{b}^{2}={a}^{2}={(2c)}^{2}=4{c}^{2}$, yielding that ${b}^{2}=2{c}^{2}$. Therefore, $2\mid {b}^{2}$. Since $2$ is prime, it must divide $b$.

Since $2\mid a$ and $2\mid b$, we have that $a$ and $b$ are not relatively prime, which contradicts the hypothesis^{}. Hence, the initial assumption^{} is false. It follows $\sqrt{2}$ is irrational.

With a little bit of work, this argument can be generalized to any positive integer that is not a square. Let $n$ be such an integer. Then there must exist a prime $p$ and $k,m\in \mathbb{N}$ such that $n={p}^{k}m$, where $p\nmid m$ and $k$ is odd. Assume that $\sqrt{n}=a/b$, where $a,b\in \mathbb{N}$ and are relatively prime. Then ${p}^{k}m=n={(\sqrt{n})}^{2}={\left({\displaystyle \frac{a}{b}}\right)}^{2}={\displaystyle \frac{{a}^{2}}{{b}^{2}}}$. Thus, ${a}^{2}={p}^{k}m{b}^{2}$. From the fundamental theorem of arithmetic^{}, it is clear that the maximum powers of $p$ that divides ${a}^{2}$ and ${b}^{2}$ are even. Since $k$ is odd and $p$ does not divide $m$, the maximum power of $p$ that divides ${p}^{k}m{b}^{2}$ is also odd. Thus, the same should be true for ${a}^{2}$. Hence, we have reached a contradiction^{} and $\sqrt{n}$ must be irrational.

The same argument can be generalized even more, for example to the case of nonsquare irreducible fractions and to higher order roots.

Title | proof that $\sqrt{2}$ is irrational |
---|---|

Canonical name | ProofThatsqrt2IsIrrational |

Date of creation | 2013-03-22 12:39:13 |

Last modified on | 2013-03-22 12:39:13 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 11 |

Author | Wkbj79 (1863) |

Entry type | Proof |

Classification | msc 11J72 |

Related topic | Irrational |

Related topic | Surd |