Proof of irrationality of sqrt2 in Penroses Road to Reality

In summary, the conversation discusses the proof by contradiction of the irrationality of the square root of 2, as outlined in the book "The Road to Reality" by Roger Penrose. The proof involves assuming that the square root of 2 is rational and then showing that this leads to a contradiction. However, the book adds a step where it mentions the existence of an unending sequence of positive integers, which is used to show that the proof should never stop. This connection between finite and infinite sequences is confusing to the person asking the question. They wonder why this step is necessary and how it relates to the overall proof. They also mention an alternative way of proving the irrationality of the square root of 2 without involving unending sequences
  • #1
Feymar
11
2
Hello Everybody, I hope that I've picked the right sub-forum for my question.

The problem is as follows:

I understand perfectly well the proof by contradiction of the irrationality of the square root of 2, where we prove that if we assume that it is rational, then the integers of which the ratio is made of can't be coprime, which is a condratiction.

Well, but recently I came along this amazing book of Roger Penrose "The Road to Reality". I love it and most of it I can understand or manage to find some more elaborative explanation somewhere. But there was this one issue, to which I keep coming back, seemingly one of the simplest ones but that I just fail to grasp.

In the book he mentions the proof by contradiction of the irrationality of the square root of 2, and outlines the proof, but by the end instead of just saying that the contradiction lies in the impossibility of the two integers to be coprimes, he writes as follows:

"But any decreasing sequence of positive integers must come to an end, contradicting the fact that this sequence is unending. This provides us with a contradiction to what has been supposed, namely that there is a rational number which squares to 2."

I know that I did not write the full proof here, up until that point it is the same as outlined in any textbook and any page you come across online, so I believe that there is no need for it. If it is, I will gladly write the full proof as outlined in the book here.

Now, the thing that I do not understand is what has an unending sequence to do with all of this? Where did that come from and how does it make sense that if (hipothetically) there was an unending sequence, there would have been a rational number which squares to 2 , or what?

I just don't see the connection that he made all of a sudden between finite and infinite sequences and the (non)existence of a rational sqrt2. He did not just merely wrote something along the lines of: "And since we get that both integers must be even if we assume that the sqrt of 2 is a rational number, we also get that the integers can't be coprimes which is a contradiction to the assumption that the sqrt of 2 is rational." or something like that.

No, he wrote what he wrote, and I can't quite comprehend the connection between the proof and how he , through the proof concluded what is written in the quote. If somebody could help me with this I would highly appreciate it.

Sorry that I made it so long, I must confess that I am not a man of few words, you'll have to forgive me for that.
 
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  • #2
I don't have the book, but it sounds like an unnecessary complication.

Instead of starting with a/b without common factors, you can start with a/b where you accept common factors, and then show that you can reduce this to (a/2)/(b/2) and so on until you get a contradiction.
 
  • #3
Each step in the proof leads to a ratio of numbers not co-prime and smaller tan the previous pair. Repeating this process leads to a stop since the number of positive integers less than the previous pair is finite. However the proof says it should never stop.
 
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Related to Proof of irrationality of sqrt2 in Penroses Road to Reality

1. What is the proof of irrationality of sqrt2 in Penroses Road to Reality?

The proof of irrationality of sqrt2 in Penroses Road to Reality is a mathematical proof that shows that the square root of 2 (sqrt2) is an irrational number. This means that it cannot be expressed as a fraction of two integers and has an infinite decimal expansion without a repeating pattern.

2. Who first discovered the proof of irrationality of sqrt2 in Penroses Road to Reality?

The proof of irrationality of sqrt2 in Penroses Road to Reality was first discovered by the ancient Greek mathematician Pythagoras, and later formally proved by the Greek mathematician Euclid.

3. How does the proof of irrationality of sqrt2 in Penroses Road to Reality work?

The proof of irrationality of sqrt2 in Penroses Road to Reality is based on a contradiction. It assumes that sqrt2 can be expressed as a fraction of two integers, and then uses logical reasoning to show that this assumption leads to a contradiction. This contradiction proves that sqrt2 is irrational.

4. Why is the proof of irrationality of sqrt2 in Penroses Road to Reality important?

The proof of irrationality of sqrt2 in Penroses Road to Reality is important because it is a fundamental result in mathematics. It not only shows the irrationality of sqrt2, but also provides a general method for proving the irrationality of other numbers. It also has many practical applications in fields such as engineering, physics, and computer science.

5. Is it possible to extend the proof of irrationality of sqrt2 in Penroses Road to Reality to other irrational numbers?

Yes, the proof of irrationality of sqrt2 in Penroses Road to Reality can be extended to other irrational numbers. In fact, this proof is a special case of a more general proof called the proof of irrationality by contradiction. This method can be used to prove the irrationality of any number that cannot be expressed as a fraction of two integers.

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