Is the Sum of Square Roots of 2 and 3 Irrational? A Proof by Contradiction

In summary, the conversation discusses proving the irrationality of 6^(1/2) and using that answer to prove the irrationality of 2^(1/2) + 3^(1/2). The method of proof by contradiction is suggested but a better proof using prime decompositions is provided. The conversation ends with gratitude for the help.
  • #1
MathematicalPhysics
40
0
Yeh just having a problem seeing a way to prove that 6^(1/2) is irrational.

Using this answer and proof by contradiction I need to prove that
2^(1/2) + 3^(1/2)is also irrational, however I sould be able to attempt this if I can get the above right.

Any help much appreciated.
 
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  • #2
Here's a better proof than contradiction.

let z be the sqrt of any integer, if it is rational write it as a/b, do the usual squaring thing so that

z.b^2= a^2

look at the prime decompostions on both sides. Every prime mus occur with multiplicity two on the rhs, so it does on the lhs, which tells us in the prime decomposition of z every prime occurs twice, that is z is a perfect square. 6 is not a perfect square.
 
  • #3
Thanks for that, the rest worked out a treat!
 

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. In other words, its decimal representation does not terminate or repeat in a pattern. Examples of irrational numbers include π and √2.

2. How is the irrationality of root 6 proven?

The irrationality of root 6 can be proven by contradiction. We assume that root 6 is rational, meaning it can be expressed as a ratio of two integers, a/b. We then manipulate this expression to show that it leads to a contradiction, thus proving that our initial assumption was incorrect.

3. Can you provide an example of the contradiction used to prove the irrationality of root 6?

Sure, let's assume that root 6 is rational and can be expressed as a/b. This means that (a/b)^2 = 6. We can then rearrange this equation to get a^2 = 6b^2. This shows that a^2 must be even, since it is equal to 6 times an integer. However, we know that the square of an odd number is always odd. This is a contradiction, as a^2 cannot be both even and odd. Therefore, our initial assumption that root 6 is rational must be false.

4. Why is proving the irrationality of root 6 important?

Proving the irrationality of a number is important for the field of mathematics as it helps us understand the properties of numbers. It also has practical applications, such as in cryptography where the security of certain algorithms relies on the use of irrational numbers. Additionally, it showcases the power and elegance of mathematical proofs.

5. Are there any other methods for proving the irrationality of numbers?

Yes, there are other methods such as the continued fraction method and the proof by infinite descent. Each method uses different techniques and can be more suitable for certain numbers. However, the proof by contradiction is one of the most commonly used and versatile methods for proving the irrationality of numbers.

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