Prove that sqrt(2) + sqrt(3) is irrational [ILIEKMATHPHYSICS]

This series mainly references the book Daniel Velleman's book "How to Prove It", which introduces how to write proofs in mathematics. The exercise in this video does not appear in this book.

Thanks and enjoy the video!

[MyOneFiftiethOfADollar]

Nice conjugate Idea utilizing the truth that sqrt(2) is irrational.

More generally sqrt(2) + sqrt(K), where K is a nonnegative integer, is irrational by the same conjugate trick you used.

[sharwinkumar]

Keep going bro! You're content is so good you'll hit big one day

[MikeB-qv]

Very nice proof. Another way is to look at (sqrt(2)+sqrt(3))^2 = 5+2*sqrt(6), 2*sqrt(6) is not rational and if a^2 is not rational then a is not rational.

[WhyNotPhysics]

VERY IMPRESSIVE CHANNEL I love your intuition, explanations and overall vibe man! Keep up the hard work you are of great help

[egg]

BRILLIANT! Worked on this problem for 30 mins and still didnt get the answer. The solution is marvelous

[pauselab]

a trick to try would be algebraic conjugates. get the minimal polynomial obviously the product X - (+, - (sqrt(2)+sqrt(3)) and their conjugates). whatever that is, it is monic and thus it is only necessary to check the integer solutions as Z is integrally closed.

[vafkamat]

Here is a proof that is short and sweet.
Now √ 3 + √ 2 and √ 3 - √ 2 are either both rational or both irrational
for their product is 1 which is rational.
If they are both rational then their difference would be rational too.
But their difference is 2√ 2 which we know to be irrational.
Hence √ 3 + √ 2 and √ 3 - √ 2 are both irrational.
This method also shows that if p and q are prime then
√p + √q and √p - √q are both irrational

[Eli_Sanders]

Super cool material, I enjoy learning with you!

[davidgillies]

Or BWOC for a, b in Z assume sqrt(a) + sqrt(b) in Q => (sqrt(a) + sqrt(b))^2 in Q => a + b + 2 sqrt(a b) in Q, but for n in Z sqrt(n) in Z (and thence in Q) iff n a perfect square. 6 is not a perfect square, contra.

[darkmask]

sqrt(2)+sqrt(3) is a root of the polynomial x^4-10x^2+1. Applying the rational roots theorem shows that this polynomial is irreducible over Q. Thus sqrt(2)+sqrt(3) is irrational

[instinx]

This is awesome. I know this isn't rigorous, but I assumed that the sum of an irrational and irrational number is irrational in addition to the sum of a rational and irrational number being irrational. So no matter the irrationality of sqrt3, sqrt3 +sqrt2 is irrational.

[kennnnn]

If the square of a number is irrational, then the number is also irrational.
(sqrt2 + sqrt3)² = 5 + 2sqrt6, which is irrational.

[mathmachine]

Assume √(2)+√(3) is rational, and can be expressed as p/q, with p integer and q positive integer. Since √(2) and √(3) are positive, √(2)+√(3) is positive, thus p is positive.

Since p/q is a nonzero rational, its reciprocal is also rational. Since (√(3)+√(2))*(√(3)-√(2)) = 1, √(3)-√(2) is the reciprocal of p/q, thus √(3)-√(2) is the rational q/p. Since √(2)+√(3) is rational, √(3)-√(2) is rational, and the rationals are closed under addition, that means (√(2)+√(3))+(√(3)-√(2))=2√(3) is rational. Since rationals are also closed under multiplication, (2√(3))*(1/2)=√(3) is also rational. √(3) is not rational, however, thus our original premise is false and √(3)+√(2) is not rational.

[dennissvensson]

And how about sqrt(2) and [1 - sqrt(2)]. Both are irrational, but the sum is rational.

[Ivnkkk]

Where can i find the proof that √2 is irrational

[goknil]

Hey, I proved it like this. Do you think my way is correct? Assume sqrt3 + sqrt2 =a/b, a and b natural numbers. Then (a^2)/(b^2) must be rational as well. But (a^2)/(b^2)= 5 + 2sqrt6. Therefore sqrt3+sqrt2 can't be rational.

[deenthebean]

Can you prove that rational + rational = rational?

[KingGisInDaHouse]

Can't we argue that a numbers that a numbers square cant be irrational unless the original number was then argue that 5+2*sqrt(6) is irrational.

[alphalunamare]

Isn't this true for any integer N and not just 3? You just get (2-N) in the denominator, which is rational and just multiply it out? So isnt Sqrt(2) + Sqrt(N) irrational for all positive integer N? Maybe N*Sqrt(2) is irrational for all integer N? Or in pub terms, 'once you are screwed, no ammount of doubling down is gonna help you' ? :-)

[richardslater]

The square root of any integer that is not a perfect square is irrational.