Prove sq root of 2 + sq root of 3 is irrational

You need to prove that sqrt(6) is irrational to prove that the addition of a rational and irrational number is irrational. This can be done by contradiction. Assume that sqrt(6) is rational, then it can be written as a fraction p/q where p and q are integers with no common factors. However, this leads to a contradiction when you substitute sqrt(6) in the equation 5 + 2*sqrt(6) = p^2/q^2. Therefore, sqrt(6) must be irrational and the addition of a rational and irrational number is irrational.
  • #1
yanjt
14
0
hi,i read the post of sq root of 2 + sq root of 3.i understand tat i should use contradiction to solve it.yet,i stuck halfway when i tried to solve it.
sq root 2 + sq root 3 = p/q
2 + 2*sq root 6 +3=p^2/q^2
5+2*sq root 6 = p^2/q^2
wat should i do after this?i hav to prove that the addition of a raitonal and irrational number is irrational?
 
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  • #2
yanjt said:
hi,i read the post of sq root of 2 + sq root of 3.i understand tat i should use contradiction to solve it.yet,i stuck halfway when i tried to solve it.
sq root 2 + sq root 3 = p/q
2 + 2*sq root 6 +3=p^2/q^2
5+2*sq root 6 = p^2/q^2


Rearrange, and you see that you have to show that sqrt(6) is rational: rational plus rational is rational


wat should i do after this?i hav to prove that the addition of a raitonal and irrational number is irrational?

See above.
 

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 (i.e. a fraction) and has an infinite number of non-repeating decimal places.

2. How do you prove that the square root of 2 plus the square root of 3 is irrational?

To prove that the square root of 2 plus the square root of 3 is irrational, we assume the opposite - that it is a rational number. Then, using the definition of a rational number, we can show that this assumption leads to a contradiction, thus proving that the original statement is true.

3. Can you provide an example of a proof for this statement?

Yes, one example of a proof is the following: Assume that the square root of 2 plus the square root of 3 is rational, and can be expressed as a/b where a and b are integers with no common factors. Then, we have (sqrt(2) + sqrt(3)) * (sqrt(2) + sqrt(3)) = 2 + 2sqrt(6) + 3 = (a/b)^2. Simplifying this equation, we get 2sqrt(6) = a^2 - 3b^2. This means that 2 divides a^2, and therefore 2 divides a. Substituting a = 2k, we get 2sqrt(6) = 4k^2 - 3b^2. Simplifying again, we get sqrt(6) = 2k^2 - (3b^2)/2. This means that 2 divides 3b^2, and therefore 2 divides b^2. But if 2 divides b^2, then 2 also divides b. This contradicts our initial assumption that a and b have no common factors, thus proving that the square root of 2 plus the square root of 3 is irrational.

4. Why is proving the irrationality of the square root of 2 plus the square root of 3 important?

Proving the irrationality of this statement is important because it helps us better understand the properties of irrational numbers and their relationship with rational numbers. It also serves as a foundation for more complex proofs in mathematics and has various applications in fields such as number theory and cryptography.

5. Are there other ways to prove the irrationality of the square root of 2 plus the square root of 3?

Yes, there are other ways to prove this statement. One common method is using the rational root theorem, which states that if a polynomial with integer coefficients has a rational root, then that root must be a divisor of the constant term. Using this theorem, we can show that the square root of 2 plus the square root of 3 cannot be a rational number, thus proving its irrationality.

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