Proof that sqrt(6)-sqrt(2)-sqrt(3) is irrational

In summary, the conversation discusses a proof for the irrationality of √6 - √2 - √3 without using a polynomial or the rational root theorem. Various approaches are suggested, including using Galois theory and conjugates, and squaring the expression. Ultimately, it is determined that proving the irrationality of one of the terms is sufficient to prove the irrationality of the entire expression. The concept of infinite probability is also brought up in relation to proving the irrationality of two irrational numbers.
  • #1
abruzzi
4
0
I want to prove that [tex]\sqrt 6 - \sqrt 2- \sqrt 3[/tex] is irrational.

I already know that [tex]\sqrt 2+\sqrt 3[/tex] is irrational (by squaring it). I would like a proof that doesn't use a polynomial and the rational root theorem.

Thanks.
 
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  • #2
abruzzi said:
I want to proof that [tex]\sqrt 6 - \sqrt 2- \sqrt 3[/tex] is irrational.

I already know that [tex]\sqrt 2+\sqrt 3[/tex] is irrational (by squaring it). I would like a proof that doesn't use a polynomial and the rational root theorem.

Thanks.
Is [itex]\mahtbb{Q}(\sqrt{2}, \sqrt{3})[/itex] Galois over Q? If so, something is rational if and only if it is equal to all of its conjugates. (The same is true if the field isn't galois... it's just that in that case, its conjugates would live in other fields)
 
  • #3
Sorry, but I have no idea about groups, rings, fields, etc. I am looking for a basic proof.

For example the proof that I have that [tex]\sqrt 2 + \sqrt 3[/tex] is irrational is that, supposing it is rational, its square should also be rational. But [tex](\sqrt 2 + \sqrt 3)^2 = 5+2\sqrt6[/tex] is irrational because [tex]\sqrt 6[/tex] is irrational.

I would like to come up with a similar proof for [tex]\sqrt 6 - (\sqrt 2 + \sqrt 3)[/tex]
 
  • #4
abruzzi said:
II already know that [tex]\sqrt 2+\sqrt 3[/tex] is irrational (by squaring it).

Hi abruzzi! :smile:

How about squaring [tex]n+\sqrt 2+\sqrt 3[/tex], for some whole number n?
 
  • #5
By taking n=-1 and squaring we get [tex]2\sqrt 6 - 2\sqrt 3 - 2\sqrt 2 + 6 = 6 + 2(\sqrt 6 -\sqrt 3-\sqrt 2)[/tex]

But from this I cannot conclude anything - since knowing that number a is irrational doesn't mean that a^2 is (for example [tex]a=\sqrt 2[/tex]).

Or am I not on the right path?
 
  • #6
oops!

ooh, you're right! :rolleyes:

ok, let's try this:

rational + irrational = irrational.

rational x irrational = irrational.

Suppose √6 - √3 - √2 is rational.

Then (1 +√3)(1 + √2) is irrational, because it is (1 + 2√6) - (√6 - √3 - √2).

But (1 - √3)(1 - √2) is rational, because it is 1 + (√6 - √3 - √2).

So the product (1 +√3)(1 + √2)(1 - √3)(1 - √2) is irrational.

But it isn't - it's 2. :smile:
 
  • #7
sqrt2 + sqrt3 = sqrt 6 + r, with r rational,

implies, by squaring both sides, that 1 + r^2 = (2-2r)sqrt6. which equates a rational and an irrational.
 
  • #8
… mathwonk is cool …

:smile: ooh, mathwonk, that's much better! :smile:
 
  • #9
That was exactly what I was looking for, thanks!
 
  • #10
Much nicer than mine
let x=√6-√3-√2
x³-3x²-15x-3=4√2
->x is irrational
or
let y=√323-√19-√17
y³-19y²-393y+4883=72√17
->y is irrational
mathwonks works for that too
√323=(297+y²)/(2-2y)
->y is irrational
just two was of writing m√ab
one in Z[√a,√b]
one in the field of fractions
the fraction sure give nice numbers though
 
  • #11
Well I think if you can prove that anyone of the terms is irrational, which should be easy, and if the number subtracted by it will be irrational, unless the number you subtract by is the same number or follows the same irrational pattern, which would be quite hard as it is impossible to find a pattern, thus if you were to subtract 2 irrational numbers, there should be an infinite probability that they will be irrational and rational =\.

for example you have an irrational number

.342526524525352325...n where n are the rest of the terms
-.1412413431413...n then the 2 n's would cause repeating zeros, thus being rational.

Just bringing up irrat-irrat

But there is no way to prove that at some point the rest of the digits would be the same (aka n), yet as it goes on for ever it could happen. Very weird little thing infinity =P, I'm in very low level maths, and I just wanted to contribute so this is all I could think of =], I'm sure there is a clever way to prove it though.
 

1. What does it mean for a number to be irrational?

A number is considered irrational if it cannot be expressed as a ratio of two integers (a fraction) and has an infinite number of non-repeating decimal places. In other words, it cannot be written in the form a/b, where a and b are integers.

2. How do you prove that sqrt(6)-sqrt(2)-sqrt(3) is irrational?

To prove that a number is irrational, we must show that it cannot be written as a fraction of two integers. In this case, we can use a proof by contradiction, assuming that sqrt(6)-sqrt(2)-sqrt(3) can be written as a fraction and then showing that this assumption leads to a contradiction.

3. What is a proof by contradiction?

A proof by contradiction is a type of mathematical proof where we assume the opposite of what we are trying to prove and then show that this assumption leads to a contradiction, thus proving that the original statement must be true.

4. Can you explain the steps of the proof for sqrt(6)-sqrt(2)-sqrt(3) being irrational?

First, we assume that sqrt(6)-sqrt(2)-sqrt(3) can be written as a fraction a/b, where a and b are integers with no common factors. Then, we manipulate this expression to get a new expression that also equals a/b. However, this new expression can be simplified further to show that a and b have common factors, which contradicts our initial assumption. Therefore, sqrt(6)-sqrt(2)-sqrt(3) cannot be written as a fraction and is irrational.

5. Why is proving that sqrt(6)-sqrt(2)-sqrt(3) is irrational important?

Proving that a number is irrational is important because it helps us understand the properties and relationships between different types of numbers. It also helps in further mathematical proofs and can have practical applications in fields such as cryptography and number theory.

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