Solutions to $x^2 = x + 1$ are irrational

Question

I'm trying to prove that all the solutions to the equation $x^2 = x + 1$ are irrational. This statement is equivalent to: If $x^2 = x + 1$, then $x$ is irrational.

I want to prove this using contraposition. The contrapositive statement is: If $x$ is rational, then $x^2 \neq x + 1$.

A rational number is one that can be described as a ratio. So, $r = \frac{k_1}{k_2}$, where $k_1$ and $k_2$ are integers and $k_2 \neq 0$.

Let $x$ be a rational number.

Using the above definition of a rational number we simplify $x^2 = x + 1$ to $(\frac{k_1}{k_2})^2 = \frac{k_1}{k_2} + 1$, where $k_1$ and $k_2$ are integers and $k_2 \neq 0$. All is left is to show that this equation cannot be satisfied for any combination of $k_1$ and $k_2$ (considering they are integers).

How can we do this? Can someone point me in the right direction?

Answer 1

The polynomial $x^2-x-1$ is irreducible over $\mathbb{F}_2$, hence it is irreducible over $\mathbb{Q}$ and cannot have any rational root. (Not really) Alternative way: assuming that $\frac{p}{q}$ is a root of $x^2-x-1$, we have $$ p^2 = pq + q^2 \tag{1}$$ but since $p$ and $q$ are not both even (due to $\gcd(p,q)=1$), the RHS and LHS of $(1)$ have opposite parity, contradiction. (Really) Alternative way: since the roots of $x^2-x-1$ add to one by Vieta's theorem, it is enough to prove that one of them (say, the greatest) is irrational to have that both of them are irrational. But if a number $x>1$ fulfills $x^2=x+1$, it also fulfills $x=1+\frac{1}{x}$, hence its ordinary continued fraction is given by $x=[1;1,1,1,1,1,\ldots]$. That is not a finite ordinary continued fraction, hence $x\not\in\mathbb{Q}$.

Answer 2

HINT: if I multiply through by $k_2$ I get $${(k_1)^2\over k_2}=k_1+k_2.$$ This means $(k_1)^2\over k_2$ is an integer. This isn't obviously a contradiction . . .

. . . but what if we assumed at the beginning that $k_1\over k_2$ is in lowest terms?

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Note that to get a contradiction from the assumption that $k_1\over k_2$ was in lowest terms, we need to know that $k_2\not=1$ - that is, that there is no integer satisfying $x^2=x+1$. But this has a nice proof! HINT: is $x$ even or odd?

Answer 3

By the rational roots theorem...

Answer 4

I just want to make a note that this is not the easiest way to do this problem. The best way (in my opinion) would be to actually find the solutions and prove their irrationality.

A hint for your proposed solution method: look at the denominator of each side. What equality would $k_2$ have to satisfy for these rational numbers to be equal?

Answer 5

I'd just rewrite it as $x^2 - x - 1 = 0$, run the quadratic formula, and then see that the roots are irrational, i.e. that $\sqrt{5}$ is irrational.

Answer 6

$4+1=4(x^2-x)+1=(2x-1)^2$ but $5$ isn't a perfect square.