Prove that sqrt2 + sqrt6 is irrational

[StephenPrivitera]

Prove that sqrt2 + sqrt6 is irrational.
Where do I start with this?

 

[Hurkyl]

Assume otherwise!

Or alternatively, consider the integer polynomial that has that number as one of its roots.

 

[StephenPrivitera]

I thought about the first one. It doesn't suit me well.
Suppose it is rational. Then,
sqrt2 +sqrt6 = p/q
With some other rational-irrational proofs, I squared both sides. So,
2 +2sqrt12 +6 =p²/q²
4(sqrt3+4)=
(2q)²(sqrt3+4)=p²
Now what?

As for the second part, I'm not sure I know what you mean by "integer" polynomial. Do you mean a polynomial whose domain is the integers?

(x - (sqrt2 + sqrt6))(x - r)=0
x^2 - x(sqrt2 +sqrt6) -xr +r(sqrt2 + sqrt6) = 0
(sqrt2 + sqrt6)(x -r) = x(x-r)
Let's hope x does not = r.
Then sqrt2 + sqrt6 = x
...
This of course makes perfect sense, since x doesn't equal r, it must equal it's other root. What pointless work! Certainly, I don't understand what you mean.

 

[Hurkyl]

(2q)²(sqrt3+4)=p²

Now what?

Solve for sqrt(3)!

As for the second part, I'm not sure I know what you mean by "integer" polynomial. Do you mean a polynomial whose domain is the integers?

I meant one whose coefficients are integers...

x = sqrt(2) + sqrt(6)
x^2 = 8 + 4 sqrt(3)
x^2 - 8 = 4 sqrt(3)
x^4 - 16 x^2 + 64 = 48
x^4 - 16 x^2 + 16 = 0

And depending on how much you remember about solving polynomials, the result is trivial from here.

 

[StephenPrivitera]

Ok, wow, you're going to think I'm really incompetent. I really don't understand what to do for either instance. Sure, I can solve for sqrt3, but then what?
You can square both sides and you get something quite similar to your second method.
3=(p/2q)^4 - 8(p/2q)^2 +16
or 0=(p/2q)^4 - 8(p/2q)^2 +13
I could solve this, but the result is obvious. I'd just be undoing the square I just applied to both sides.
(p/2q)^2=4+sqrt3
So that's not right...

Clearly, you want me to solve the polynomial you wrote. But then I get,
x²=(16+8sqrt3)/2=8+4sqrt3=4(2+sqrt3)
So, x=2sqrt(2+sqrt3)
What nonsense is this!

Am I getting anywhere near a proof of anything besides my ignorance?
Please help.

 

[StephenPrivitera]

Here's another one I can't get.
If a is rational and b is irrational is a+b necessarily irrational? I say yes, but suppose not. Then,
a+b=p/q for some p and q in J
and a=c/d for some c,d in J
then b=p/q-c/d=(pd-cq)/(pd)
Now if we can believe that the integers are closed under * and +, then we have derived a contradiction since pd-cq will be an integer and pd will be an integer. But how can I show they in fact are closed under * and +?

What if a and b are both irrational?
Not necessarily because
pi+(-pi)=0
Does that suffice?
How can I know that the additive inverse of an irrational is also irrational?

 

[Hurkyl]

Ok, wow, you're going to think I'm really incompetent.

Nah, this is one of those things that you usually have to see a few times before the method sinks in.


When you solved for sqrt(3), you got:

sqrt(3) = (p/(2q))^2 - 4

We assumed that sqrt(2) + sqrt(6) is rational, and from that assumption we've proven that there is some integers p and q so the above equation holds... the left hand side of this equation is an irrational number, but what about the right hand side?


As for the integer polynomial, there is a theorem that if p/q is a rational root of an integer polynomial in lowest terms, then p divides the constant term and q divides the leading coefficient. For example, if the equation

2x^3 - 3x + 6 = 0

has any rational roots, they must be among these possibilities:

1/1, -1/1, 2/1, -2/1, 3/1, -3/1, 6/1, -6/1, 1/2, -1/2, 3/2, -3/2

so you just have to exhaust all the possibilities to prove that this polynomial has no rational roots.

So try this theorem with the polynomial

x^4 - 16 x^2 + 16 = 0

for which we know sqrt(2) + sqrt(6) is a root.

 

[Hurkyl]

As for the integers being closed under + and *, you should be able to assume that to be true without proof.

 

[StephenPrivitera]

Guess what? Problem number 18 in my text asks me to prove that very theorem about integer polynomials. It wasn't assigned so I might not get around to it. Sounds tough.

But how about this one:
prove or disprove: If n is a natural number, then
41 + 2 + 4 + 6 + ... + 2n is prime.

My main problem is understanding what the pattern is. To me this is like asking me to prove that any sum of numbers is prime. Can you see a pattern? Maybe the first term is supposed to be one (not 41)? Then the pattern would be 1+2+4+6+8+10+12+...+2n, right?

 

[Hurkyl]

If I had to guess, I think they mean:

41 + Σ2n