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I have read a few proofs that $\sqrt{2}$ is irrational.

I have never, however, been able to really grasp what they were talking about.

Is there a simplified proof that $\sqrt{2}$ is irrational?

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I can't edit yet, could someone edit "grash" to "grasp"? Thanks. –  Coltin Jul 20 '10 at 19:53
    
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$\sqrt{2}$ isn't integer (it's strictly in between 1 and 2). So if it's rational, it's equal to an irreducible fraction $p/q$. Then the fraction $p^2 / q^2$ is also irreducible, but it is equal to 2, which is an integer! –  Alexei Averchenko Apr 3 '11 at 1:16
    
Plato said I can't call myself human unless I can prove this. –  Mike Jones May 7 '11 at 19:48
    
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8 Answers 8

up vote 47 down vote accepted

You use a proof by contradiction. Basically, you suppose that $\sqrt{2}$ can be written as $p/q$. Then you know that $2q^2 = p^2$. However, both $q^2$ and $p^2$ have an even number of factors of two, so $2q^2$ has an odd number of factors of 2, which means it can't be equal to $p^2$.

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Thanks! This is quite clear. –  John Gietzen Jul 22 '10 at 14:51
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This is by far my favorite proof of $\sqrt{2}$ irrational. I believe it is due to Chaitin (at least I think it's his -- it's in his book Meta Math! (p. 98), and he does not attribute it to anyone else). Of course, this depends on unique prime factorization, but it's still quite elementary. The descent method in the standard proof is, of course, hidden in the prime factorization proof, but that's a fine place for it. Note that the original poster couldn't grasp the popular proof, and I bet the descent with contradiction is the obstacle -- I've seen that with many students. –  David Lewis Apr 3 '11 at 10:53
    
This proof easily generalizes to any exponent k and ratio b >= 2 which is not a perfect power of k, as follows (not in Chaitin's book, but it ain't so hard)... Assume $m^k = b n^k$ Then the unique prime factorizations of $m^k$ and $n^k$ must have all exponents that are multiples of k, and that must also therefore be true of b. But that means b is a perfect k-th power, $b = c^k$ for some integer c. The case k = b = 2 is the classical theorem, with 2 not a perfect square. –  David Lewis Apr 3 '11 at 10:54
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It is interesting that the Greeks (and I guess most everybody since) missed this proof, because they had unique prime factorization (cf. Euclid's algorithm), and this proof makes clear that that the irrationality of non-perfect roots is intimately related to it. –  David Lewis Apr 3 '11 at 10:55
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@Bill -- thanks. Makes sense that this proof is not due to Chaitin. It's just that I had never seen it, and it is so attractive (IMHO) that I assumed it must be recent, or it would be better known. He does not attribute it, but I guess that is normal with ancient, folklore proofs. I wonder if anyone does know where or from whom it did originate -- did the Greeks know it? As for not requiring unique prime factorization, you are correct, mathematically. But I was thinking more pedagogically -- it's feasible to introduce this proof in school when prime factorization has been taught. –  David Lewis Mar 14 '12 at 17:27
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Consider this proof by contradiction:

Assume that $\sqrt{2}$ is rational. Then there exists some rational $R=\sqrt{2}=\frac{Q}{D}$, where $Q$ and $D$ are positive integers and relatively prime (since $R$ can be expressed in simplified form).

Now consider $R^2 = 2 = \frac{Q^2}{D^2}$. Since $Q$ and $D$ are relatively prime, this means that only $Q^2$ can have $2$ in its prime decomposition, and the exponent must be one. Thus, $Q^2 = 2^1 x$, for some odd integer $x$. But $Q^2$ is a square, and thus the exponents for all of its prime factors must be even. Here we have a contradiction.

Thus, $\sqrt{2}$ must be irrational.

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If you are implicitly using uniqueness of prime factorizations then you need to explicitly state that, and state how it applies to yields your deduction. This is essential for proofs at this level. –  Bill Dubuque Apr 14 '12 at 0:19
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Another method is to use continued fractions (which was used in one of the first proofs irrationality of $\displaystyle \pi$).

Instead of $\displaystyle \sqrt{2}$, we will consider $\displaystyle 1 + \sqrt{2}$.

Now $\displaystyle v = 1 + \sqrt{2}$ satisfies

$$v^2 - 2v - 1 = 0$$

i.e

$$v = 2 + \frac{1}{v}$$

This leads us to the following continued fraction representation

$$1 + \sqrt{2} = 2 + \cfrac{1}{2 + \cfrac{1}{2 + \dots}}$$

Any number with an infinite simple continued fraction is irrational and any number with a finite simple continued fraction is rational and has at most two such simple continued fraction representations.

Thus it follows that $\displaystyle 1 + \sqrt{2}$ is irrational, and so $\displaystyle \sqrt{2}$ is irrational.

Exercise: Show that the Golden Ratio is irrational.

More information here: http://en.wikipedia.org/wiki/Continued_fraction

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The continued fraction proof in Aryabhata's answer can be recast into an elementary form that requires no knowledge of continued fractions. Below is a variant of such that John Conway (JHC) often mentions, followed by my (WGD) reinterpretation of it to highlight the key role played by the principality of (denominator) ideals in $\:\mathbb Z\:$ (which I call unique fractionization).


THEOREM (JHC) $\quad \rm r = \sqrt{n}\ \:$ is integral if rational,$\:$ for $\:\rm n\in\mathbb{N}$

Proof $\ \ \ $ Put $\ \ \displaystyle\rm r = \frac{A}B ,\;$ least $\rm\; B>0\:.\;$ $\ \displaystyle\rm\sqrt{n}\; = \frac{n}{\sqrt{n}} \ \Rightarrow\ \frac{A}B = \frac{nB}A.\ \:$ Taking fractional parts yields $\rm\displaystyle\ \frac{b}B = \frac{a}A\ $ for $\rm\ 0 \le b < B\:.\ $ But $\rm\displaystyle\ B\nmid A\ \Rightarrow\:\ b\ne 0\ \:\Rightarrow\ \frac{A}B = \frac{a}b\ $ contra $\rm B $ least. $\:$ QED

Abstracting out the Euclidean descent at the heart of the above proof yields the following


THEOREM (WGD) $\quad \rm r = \sqrt{n}\ \:$ is integral if rational,$\:$ for $\:\rm n\in\mathbb{N}$

Proof $\ \ $ Put $\ \ \displaystyle\rm r = \frac{A}B ,\;$ least $\rm\; B>0\:.\;$ $\ \displaystyle\rm\sqrt{n}\; = \frac{n}{\sqrt{n}} \ \Rightarrow\ \frac{A}B = \frac{nB}A\ \Rightarrow\ B\:|\:A\ $ by this key result:


Unique Fractionization $\ $ The least denominator $\rm\:B\:$ of a fraction divides every denominator.

Proof $\rm\displaystyle\ \ \frac{A}B = \frac{C}D\ \Rightarrow\ \frac{D}B = \frac{C}A \:.\ $ Taking fractional parts $\rm\displaystyle\ \frac{b}B = \frac{a}A\ $ where $\rm\ 0 \le b < B\:.\ $ But

$\rm\displaystyle\ \:B\nmid D\ \Rightarrow\ b\ne 0\ \Rightarrow\ \frac{A}B = \frac{a}b\ \ $ contra leastness of $\rm\:B\:.\quad\quad $ QED


Thus JHC's proof essentially "inlines" the above proof - which is better viewed as principality of (denominator) ideals in $\mathbb Z\:,$ cf. my post here. See also this discussion between John Conway and I.

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or...suppose gcd(a,b) = 1, and $\sqrt{n} = \frac{a}{b}$ for integers a and b. then $a^2 = b^2n$. since $b^2$ divides $b^2n$, it must be the case that $b^2|a^2$. if p is a prime that divides b then p divides $a^2$, hence p divides a contradicting gcd(a,b) = 1. thus, there can be no such prime that divides b, so b = -1 or b = 1, that is, a/b is an integer. –  David Wheeler Mar 14 '12 at 9:36
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If $\sqrt 2$ were rational, we could write it as a fraction $a/b$ in lowest terms. Then $$a^2 = 2 b^2.$$ Look at the last digit of $a^2$. It has to be $0$, $1$, $4$, $5$, $6$ or $9$. Now look at the last digit of $2b^2$. It has to be $0$, $2$ or $8$. As $a^2$ and $2b^2$ are the same number, its last digit must be $0$. But that's only possible if $a$ ends in $0$ and $b$ ends in $0$ or $5$. Either way both $a$ and $b$ are multiples of $5$ contradicting $a/b$ being in lowest terms.

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Can you explain how you got the numbers, 0, 1, 4, 5, 6, or 9 –  Tyler Hilton Aug 10 '10 at 19:59
    
The last digit of the square of a number depends only on the last digit of the number. To see this, just think about how you usually multiply two numbers (by hand) and focus on what can contribute to the 1's column. From here, you just compute 0^2, 1^2, 2^2,..., 9^2 and record the last digits to get 0,1,4,9,6,5,6,9,4,1, which, not counting multiples, is 0,1,4,5,6, or 9. –  Jason DeVito Aug 16 '10 at 2:20
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You can also use the rational root test on the polynomial equation $x^2-2=0$ (whose solutions are $\pm \sqrt{2}$). If this equation were to have a rational solution $\frac{a}{b}$, then $a \vert 2$ and $b \vert 1$, hence $\frac{a}{b}\in \{\pm 1, \pm 2\}$. However, it's straightforward to check that none of $1,-1,2,-2$ satisfy the equation $x^2-2=0$. Therefore the equation has no rational roots and $\sqrt{2}$ is irrational.

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There is also a proof of this theorem that uses the well ordering property of the set of positive integers, that is in a non empty set of positive integers there is always a least element. The proof follows the approach of proof by contradiction but uses the well ordering principle to find the contradiction :) -

Let us assume $\sqrt{2}$ is rational, hence it can be written down in the form $\sqrt{2}=a/b$ assuming that both $a$ and $b$ are positive integers in that case if we look at the set $S = \{k\sqrt{2} \mid k, k\sqrt{2}\text{ are integers}\}$ we find that it's a non empty set of positive integers, it's non empty because $a = b\sqrt{2}$ is in the above set. Now using the Well ordering principle we know that every set of positive integers which is non-empty has a minimum element, we assume that smallest element to be $s$ and let it equal to $s =t\sqrt{2}$. Now an interesting thing happens if we take the difference between the following quantities $s\sqrt{2} - s = (s-t)\sqrt{2} = s(\sqrt{2} - 1)$ which is a smaller element than $s$ itself, hence contradicting the very existence of $s$ being the smallest element. Hence we find that $\sqrt{2}$ is irrational.

I know the proof but I am still amazed at how the author came up with the set assumption. Sometimes such assumptions make you feel kinda dumb :). If anyone has some insight regarding how to come up with such assumptions kindly post your answer in the comment, otherwise I would just assume that it was a workaround.

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Here are some of my favorite (sketches) of proofs for the irrationality of $\sqrt{2}$.

  • Using Newton's method to approximate roots of the polynomial $f(x) = x^2 - 2$, then showing that the sequence does not converge to a rational number.
  • Proof by contradiction, assume that $\sqrt{2} = \frac{n}{m}$ for some $n,m \in \mathbb{Z}$ with $m \neq 0$, then $2m^{2} = n^2$, hence $n$ must be even and we can let $n = 2k$ for some $k \in \mathbb{Z}$, but then $m^2 = 2k^2$ will also be even, which is impossible if $\frac{n}{m}$ is reduced. Therefore, $\sqrt{2}$ cannot be expressed as a ratio of integers.
  • Since $f(x) = x^2 -2$ is irreducible over $\mathbb{Q}[x]$, its roots must lie in some finite extension field $\mathbb{Q}(\sqrt{2})$ over the rationals.

[Reposted from closed topicProve the square root of 2 is irrational

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It would help to give further details about the first and third methods. I know a lot about these topics yet I cannot be sure precisely what you have in mind. –  Bill Dubuque Nov 7 '12 at 20:15
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