How can you prove that the square root of two is irrational?

Question

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?

Answer 1

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

Answer 2

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.

Answer 3

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.

Answer 4

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

Answer 5

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) reformulation that highlights the key role played by the principality of (denominator) ideals in $\:\mathbb Z\:$ (which I call unique fractionization) in order to emhpasize its very close relationship unique factorization).

---

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 $\rm\:B.\,$ Thus $\rm\,B\mid D\,$ as claimed $\quad $ QED

---

Thus JHC's proof essentially "inlines" the above proof - which can be more conceptually viewed as the principality of (denominator) ideals in $\mathbb Z,\,$ cf. my post here. See also this sci.math discussion between John Conway and I (click "plain text" to get correct equation formatting).

See here for how to view the proof more conceptually as a denominator descent by the division algorithm (here $\,\rm B < D$ denoms of $\,\rm r\Rightarrow \underbrace{D\bmod B}_{\large\rm b}\,$ denom of $\,\rm r$), where we use the language: $\rm\,0\neq d\,$ denom of $\,\rm r\,$ to mean $\rm \,dr = n\,$ is an integer, i.e. $\,\rm\,r = n/d\,$ is writable with denom $\,\rm d$.

Answer 6

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.

Answer 7

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

Answer 8

Let me give a proof based on (partly) Newton's method and (mainly) Pell's equation. The proof technique is also similar to that of continued fractions.

Consider the iterations: $x_{n+1} = \frac{x_n}{2}+\frac{1}{x_n}$. This is the formula given by Newton's method for the function $f(x) = x^2 -2$. Let's use a special initial value: $x_0 = \frac{3}{2}$.

Denote $x_n = \frac{p_n}{q_n}$. We immediately get the formula: \begin{equation} p_{n+1} = p_n^2 + 2q_n^2, q_{n+1} = 2p_n q_n. \end{equation}

Furthermore, you might notice that $p_n, q_n$ are all the solutions of the Pell's equation $x^2-2y^2 = 1$. But we only need the following calculation: \begin{equation} p_{n+1}^2-2q_{n+1}^2 = (p_n^2 -2q_n^2)^2, \end{equation} \begin{equation} p_0^2-2q_0^2 = 3^2-2\cdot 2^2 = 1. \end{equation}

From the above formula, we have the relation: $\frac{2}{x_n}< \sqrt{2} < x_n$. Now assume $\sqrt{2} = \frac{s}{t}$. We can get the estimate: \begin{equation} |\frac{p_n}{q_n}-\frac{s}{t}| = |x_n - \sqrt{2}| < |x_n - \frac{2}{x_n}| = \frac{1}{p_n q_n}. \end{equation}

So $|tp_n-sq_n| < \frac{t}{p_n}$. Because $p_n$ grows to infinity, we have $|tp_n - sq_n|<1$ for some $n$. Then $tp_n = sq_n$ and $x_n = 2$. Contradiction!

If you know more about Pell's equation ($x^2-ny^2 = 1$ has solution for all positive nonsquare integer $n$), you can prove that $\sqrt{n}$ is irrational.

Answer 9

The irrationality of $\sqrt{2}$ is equivalent to that of $1+\sqrt{2}$, which is equivalent to there being no length $\ell$ such that the hypotenuse and perimeter of an isosceles right triangle are both whole-number multiples of $\ell$. Suppose there were such a length. We say that the hypotenuse (indicated schematically by the black path in the triangle on the left) and the perimeter (indicated by the brown path in the triangle on the left) are measured by $\ell$. To find $\ell$, we apply the Euclidean algorithm, measuring off the perimeter in units of the hypotenuse, and observing that any remainder is measured by $\ell$. In any right triangle, the perimeter contains at least two hypotenuse-lengths (since the legs together are greater than the hypotenuse) but never three (since the hypotenuse is greater than each leg separately). (The isosceles right triangle, in fact maximizes the ratio of perimeter to hypotenuse, this ratio being $1+\sqrt{2}\approx 2.4$.) The two hypotenuse-lengths are indicated schematically by the black diagonal path and the green L-shaped path in the middle figure. The remainder is the short black vertical path. We claim that the brown triangular path in the middle figure is also a hypotenuse-length path, and therefore also measured by $\ell$. But then this small isosceles right triangle, which is less than half as big (in linear dimensions) as the original isosceles right triangle, has hypotenuse and perimeter that are measured by $\ell$. Which is a contradiction: the Euclidean algorithm can be iterated, always producing a new isosceles right triangle of less than half the size of the previous one whose hypotenuse and perimeter are measured by $\ell$. Such repeated reductions to less than half must eventually result in a figure whose size is less than any conceivable $\ell$.

To establish the claim that the brown triangular path is hypotenuse-length, note that if three congruent isosceles right triangles are arranged like the gray triangles in the middle figure, and the rectangular gap in the middle filled in, the result is an isosceles right triangle. Then the black, green, and brown paths all have length $2s+d$, using the notation of the figure on the right.

For a more organic derivation, let $\triangle ABC$ be an isosceles right triangle. Draw a circle with center $C$ and radius $\overline{CB}$ intersecting $\overline{AC}$ at $G$. The length of $\overline{AG}$ is the amount by which the length of the hypotenuse exceeds that of a leg, which is the length that the short segment of the green L in the middle figure needs to have. Now construct a segment perpendicular to $\overline{AC}$ at $G$, intersecting $\overline{AB}$ at $D$. Triangle $AGD$ is an isosceles right triangle, so $\overline{DG}\cong\overline{AG}$. But also $\overline{DB}\cong\overline{DG}$ as required, which can be seen by observing that the right triangles $CGD$ and $CBD$ share a hypotenuse and have corresponding legs $\overline{CG}$ and $\overline{CB}$ congruent, and are therefore congruent. The points $F$ and $E$ can be constructed by a symmetric procedure.

One final image: if the hypotenuse and perimeter of the largest triangle in the fractal below are both measured by $\ell$, then the hypotenuse and perimeter of every similar triangle in the fractal, and the perimeter of every similar rectangle in the fractal, are measured by $\ell$ as well.

Remarks: This proof is essentially the same as the continued fraction proof of Aryabhata and the algebraic proof of Thomas Andrews. The only added feature is the geometric interpretation and the pictures.

To relate it to continued fractions, let the hypotenuse be $1$, so that the perimeter is $1+\sqrt{2}$. Taking out two hypotenuse lengths gives $$ 1+\sqrt{2}=2+(\sqrt{2}-1). $$ The quantity in parentheses is the length of the short black vertical segment in the middle diagram, which is the hypotenuse of the original triangle scaled down by a factor of $1+\sqrt{2}$. So $$ 1+\sqrt{2}=2+\frac{1}{1+\sqrt{2}}. $$ Iterating gives $$ 1+\sqrt{2}=2+\frac{1}{2+\frac{1}{1+\sqrt{2}}}=2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\ldots}}}. $$

To relate it to the algebraic proof of Thomas Andrews, refer to the rightmost diagram and let $$ \sqrt{2}=\frac{p}{q}=\frac{2s+d}{s+d}, $$ where $p$ has been interpreted as the length of the hypotenuse and $q$ as the length of a leg. But the small right triangles in the diagram make it clear that also $$ \sqrt{2}=\frac{d}{s}=\frac{2q-p}{p-q}. $$ If $\frac{p}{q}$ was in lowest terms, this is a contradiction, which is another way of expressing the infinite descent idea of the geometric proof.

Answer 10

Proof of a more general statement:-

I am going to prove the following statement using proof by contradiction:

Square root of a non-perfect square number is irrational.

So let's start,

Let $N$ be a non-perfect square and let $a,b\in\mathbb{Z}$. So the statement becomes

$\sqrt{N}=\frac{a}{b}$ and assume $gcd(a,b)=1$

$\implies N=\frac{a^2}{b^2}$

$\implies a^2=Nb^2$

Which means that $N$ is a perfect square which contradicts our assumption.

$\therefore \sqrt{N}\neq\frac{a}{b}$.QED

Answer to your question:-

By the proof above, If we let $N=2$

Then $\forall a,b\in\mathbb{Z},\sqrt{2}\neq\frac{a}{b}$

Answer 11

The claim is that $\sqrt 2$ is irrational. The opposite of this statement is that $\sqrt 2$ is rational, that is $\sqrt 2 = p \div q$, for two integers p, q >= 1. We will show that p and q are both even integers. That shows that p, q cannot exist: If there was a pair p, q then there would be a pair with the smallest possible value of q. But then p, q would be both even, say p = 2p’, q = 2q’, so $\sqrt 2 = p \div q = p’ \div q’$, so we have a q’ less than the smallest q. Such a q’ cannot exist, therefore p, q didn’t exist, therefore $\sqrt 2$ is irrational.

Why are p, q both even? We said $\sqrt 2 = p\div q$. We multiply both sides by q and square both sides and get $2q^2 = p^2$. Now the square of an odd number is always odd. Since $2q^2$ is even, p must be even, say p = 2p’, so $2q^2 = p^2 = (2p’)^2 = 4p’^2$, so $q^2 = 2p’^2$, and the same argument as before shows that q is even. p and q must both be even, which concludes the proof.

Part 2: We showed $\sqrt a$ is irrational if a = 2. We can show the same if a > 1 is a square free integer (that is it is not divisible by any square > 1): Let b be the smallest prime factor of a, then we can just copy the proof above, except we replace everywhere “x is even / odd” with “x is divisible / not divisible by b”.

Part 3: $\sqrt a$ for an integer a is irrational if a is not a square: We showed this for square free a. If a is not square free but not a square, then $a = b^2 c$ where c > 1 is square free. $\sqrt c$ is irrational, therefore $\sqrt a = \sqrt {b^2 c} = b \sqrt c$ is irrational.

Part 4: $\sqrt {a \div b}$ for integers a, b > 0, is irrational unless $a \cdot b$ is a square. Proof: $\sqrt {a \div b} = \sqrt {a \cdot b} \div b$ is the quotient of an irrational and a rational number.

Part 5: $\sqrt x$ is rational if and only if x is the quotient of two rational numbers $x = a \div b$ where $a\cdot b$ is the square of an integer. Proof: Part 4 plus the fact that the square of a rational number is rational, so the square root of sn irrational number is irrational.