Square Root Of 2 Irrationality Proof

In summary, the conversation discusses a proof from a textbook where the author assumes that t^2 contains an odd number of 2's. However, it is mentioned that t^2 is equal to s^2/2 and s^2 has an even number of 2's as prime factors. The conversation questions why the author assumes this and points out that it is already a contradiction.
  • #1
moe darklight
409
0
Hi, I'm having trouble understanding some statements in this proof from my textbook:

"Thus, 2 = s^2/t^2 and 2t^2 = s^2. Since s^2 and t^2 are squares, s^2 contains an even number of 2's as prime factors (This is our Q statement), and t^2 contains an even number of 2's. But then t^2 contains an odd number of 2's as factors. Since 2t^2 = s^2, s^2 has an odd number of 2's. (This is the statement ~Q.) This is a contradiction, because s2 cannot have both an even and an odd number of 2's asfactors. We conclude that sqrt(2) is irrational."

Why does he assume that t^2 contains an odd number of 2's all of a sudden? ... and even if it did, s^2 would still be even, because it is equal to 2t^2, not t^2. :confused:
 
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  • #2
moe darklight said:
Why does he assume that t^2 contains an odd number of 2's all of a sudden?
There is something funny in the phrasing; did you copy it exactly? Anyways, he didn't assume, because having an odd number of factors of 2 is a consequence of the previous observations.
 
  • #3
yes I copied it exactly. ok, let me see if I get it:

t^2 must have an odd number of 2's because it equals s^2/2, and s^2 has an even number of twos (because any squared number has an even number of twos). but why not stop at that? isn't that a contradiction already? ... and the second statement still makes no sense to me, because if you multiply a number with an odd number of twos as factors (2t) by two, then you have an even number of twos, so why does he say s^2 is odd after 2t^2 = s^2?.

I assume, as usual, I'm overlooking something very obvious :biggrin:
 
  • #4
moe darklight said:
yes I copied it exactly. ok, let me see if I get it:

t^2 must have an odd number of 2's because it equals s^2/2, and s^2 has an even number of twos (because any squared number has an even number of twos). but why not stop at that? isn't that a contradiction already?
Yes it is.
 
  • #5
Well s^2 is an even number right? (since it equals 2t^2 and t is an integer therefore s^2 = 2k, for k \in Z ) but it's also a perfect square which means s is an even number I believe. Well if s is an even number then s^2 must contain an even number of 2s.
 

1. What is the proof that the square root of 2 is irrational?

The proof that the square root of 2 is irrational was first discovered by the ancient Greek mathematician Pythagoras. It states that if the square root of 2 were rational, then it could be expressed as a fraction of two integers. However, this leads to a contradiction since the square root of 2 is an irrational number, meaning it cannot be expressed as a fraction.

2. Why is the proof of the square root of 2's irrationality significant?

The proof of the square root of 2's irrationality is significant because it was one of the first proofs that showed the existence of irrational numbers. This discovery challenged the ancient Greek's belief in the rationality of numbers and paved the way for more complex mathematical concepts.

3. Can the proof of the square root of 2's irrationality be applied to other square roots?

Yes, the same proof used for the square root of 2 can be applied to other square roots. In general, the square root of any non-perfect square number is irrational. This means that it cannot be expressed as a fraction of two integers.

4. How does the proof of the square root of 2's irrationality relate to the Pythagorean theorem?

The proof of the square root of 2's irrationality is closely related to the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is based on the fact that the square root of 2 is irrational.

5. Are there any real-life applications of the proof of the square root of 2's irrationality?

The proof of the square root of 2's irrationality has many real-life applications, especially in fields such as engineering, physics, and computer science. It is used in calculations involving right triangles and in the development of algorithms for computing square roots.

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