Intereting Posts

Limit of $\left(\frac{2\sqrt{a(a+b/(\sqrt{n}+\epsilon))}}{2a+b/(\sqrt{n}+\epsilon)}\right)^{n/2}$
Flawed proof that all positive integers are equal
Orthogonal Latin Square 6*6
$\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$
Unexpected examples of natural logarithm
Function mapping challange
Can $f\in L^2(\Omega)$ imply $\nabla f\in H^{-1}(\Omega):=(H_0^1(\Omega))^*$?
Calculating Distance of a Point from an Ellipse Border
Sufficient / necessary conditions for $f \circ g$ being injective, surjective or bijective
Gradient operator the adjoint of (minus) divergence operator?
Spline interpolation versus polynomial interpolation
If $f$ is ananlytic in $D_r(z_0)$\ {$z_0$} and $Ref(z)>0$ for all $z\in D_r(z_0)$\ {$z_0$} then $z_0$ is a removable singularity
Show that the right half-open topology on $\mathbb R$ is not metrisable.
Am I wrong in thinking that $e^{i \pi} = -1$ is hardly remarkable?
A lemma about extension of function

Possible Duplicate:

$\sqrt a$ is either an integer or an irrational number.

I have this homework problem that I can’t seem to be able to figure out:

Prove: If $n\in\mathbb{N}$ is not the square of some other $m\in\mathbb{N}$, then $\sqrt{n}$ must be irrational.

- CHECK: Let p be a prime number. Show that $\binom{2p}{p}$ is congruent to $2$ mod $p$.
- suppose $\gcd(a,b)= 1$ and $a$ divides $bc$. Show that $a$ must divide $c$.
- Integer solutions to $2x^2+5x+y^2=19$
- Proof verification: Let $a$ be an irrational number and $r$ be a nonzero rational number. If $s$ is a rational number then $ar$ + $s$ is irrational
- Diophantine equation: $7^x=3^y-2$
- Prove $\gcd(a,b,c)=\gcd(\gcd(a,b),c)$.

I know that a number being irrational means that it cannot be written in the form $\displaystyle\frac{a}{b}: a, b\in\mathbb{N}$ $b\neq0$ (in this case, ordinarily it’d be $a\in\mathbb{Z}$, $b\in\mathbb{Z}\setminus\{0\}$) but how would I go about proving this? Would a proof by contradiction work here?

Thanks!!

- Does $k=9018009$ have a friend?
- How to compute $2^{2475} \bmod 9901$?
- Nested Division in the Ceiling Function
- Is that true that all the prime numbers are of the form $6m \pm 1$?
- Solving $a^2+3b^2=c^2$
- Prove: $\gcd(a,b) = \gcd(a, b + at)$.
- How are the integral parts of $(9 + 4\sqrt{5})^n$ and $(9 − 4\sqrt{5})^n$ related to the parity of $n$?
- When is there an $m$ that divides $u^{an+b}+v^{cn+d}$ for all $n$
- How to find last two digits of $2^{2016}$
- Solve $3x^2-y^2=2$ for Integers

Let $n$ be a positive integer such that there is no $m$ such that $n = m^2$. Suppose $\sqrt{n}$ is rational. Then there exists $p$ and $q$ with no common factor (beside 1) such that

$\sqrt{n} = \frac{p}{q}$

Then

$n = \frac{p^2}{q^2}$.

However, $n$ is an positive *integer* and $p$ and $q$ have no common factors beside $1$. So $q = 1$. This gives that

$n = p^2$

Contradiction since it was assumed that $n \neq m^2$ for any $m$.

Here’s an explanation that I find clearer, and that uses unique factorization explicitly: If a positive number $n$ is not the square of any integer, then when you write it as a product of primes, at least one prime shows up to an odd power. Let one such prime be $p$, and look at the supposed equation $\sqrt{n}=a/b$, with $a$ and $b$ positive integers. This gives $n=a^2/b^2$, hence $nb^2=a^2$. How many times does $p$ show up in the factorization of the left side and of the right? Oddly many times on the left, evenly many on the right. Contradiction to the unique factorization of $nb^2$.

This can also be done with the rational root test: consider the polynomial equation $$x^2 – n = 0$$

and suppose that it has a rational root. Then, this rational root must be an (integer) factor of $n$. So, if $\sqrt{n}$ is rational, then there exists $t\in \mathbb{N}$ (since $x^2 – n$ is an even function of $x$, we may assume, without loss of generality, that $t>0$) with $t \vert n$ such that $$t^2 – n = 0$$ which is to say $$n = t^2$$ and hence $n$ is the square of a natural number.

In fact, this argument generalizes to showing that if $\sqrt[m]{n}$ is rational, then $n$ is an $m^{th}$ power.

- Evaluate $\int_0^{2\pi}\frac{1}{1 + A\sin(\theta) + B\cos(\theta)}\, \mathrm{d}\theta \ \text{ for }\ A,B <<1$
- Is is possible to obtain exactly 16 black cells?
- Problem with 2nd order differential equation
- Is the following set a manifold?
- Prove that $1.49<\sum_{k=1}^{99}\frac{1}{k^2}<1.99$
- combinatorial geometry: covering a square
- Sum of rational numbers
- Square covered with tiles
- Prove that a complex-valued entire function is identically zero.
- Is 2048 the highest power of 2 with all even digits (base ten)?
- Let $z_k = \cos\frac{2k\pi}n + i\sin\frac{2k\pi}n$. Show that $\sum_{k=1}^n|z_k-z_{k-1}|<2\pi$.
- Recurrence relation for words length $n$
- Why does $\lim_{n\to\infty}(1+\frac{1}{n})^n = e$ instead of $1$?
- Proving that $ e^z = z+\lambda$ has exactly $m+n$ solutions $z$ such that $-2\pi m<\Im z<2\pi n$
- A question concerning Borel measurability and monotone functions