Is the Square Root of 2 an Irrational Number?

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Discussion Overview

The discussion centers around the nature of the square root of 2, specifically whether it is an irrational number. Participants explore various proofs and implications of this concept, touching on aspects of mathematical reasoning and proof techniques.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants affirm that the standard proof demonstrates that the square root of 2 is irrational.
  • One participant questions the logic of assuming that if the square root of 2 can be expressed as a rational number, then it must be a rational number.
  • Another participant emphasizes the importance of showing the contradiction in the proof, noting that this part was missing from the original post.
  • One contributor suggests that expressing m and n as products of prime numbers is necessary to complete the proof, arguing that the squaring process leads to an even number of prime factors, which cannot satisfy the equality when an extra factor is introduced.
  • Another participant claims that the same reasoning applies to the square root of any prime number, asserting that it cannot be rational.
  • A participant expresses surprise at the implications of the discussion, suggesting that their previous understanding of mathematics may have been incorrect.
  • One participant mentions that the square root of any natural number that is not a perfect square is also irrational, and introduces the rational root theorem as an alternative proof method.

Areas of Agreement / Disagreement

Participants generally agree that the square root of 2 is irrational, but there are differing views on the completeness and clarity of the proofs presented. The discussion includes various interpretations and methods of proof, indicating that multiple perspectives remain.

Contextual Notes

Some participants note that the proof's completeness depends on certain assumptions, such as the representation of numbers as products of prime factors. There is also mention of the rational root theorem, which introduces another layer of complexity to the discussion.

Who May Find This Useful

This discussion may be of interest to those studying number theory, mathematical proofs, or anyone curious about the properties of irrational numbers.

ltkach
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SquareRoot 2 is Irrational?

\sqrt{}2 I've attached an image of what I'm talking about. Tell me what you think.
 

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Yes, that is the standard proof.
 
Aren't you basically saying that assuming that ##\sqrt{2}## can be written as a rational number, then it is a rational number?
 
DrClaude said:
Thanks. The last part, namely showing the contradiction, was missing from the OP.

i didn't think that the proof was complete until you express m and n as products of prime numbers. because of the squaring, there will always be an even number of any prime factor in both m2 and n2. but with the extra 2 (or whatever the prime number) on one side, you can show that equality is not possible, thus the proof by contradiction.

it's the same for the square root of any prime number. it cannot be rational.
 
I have no experience with upper level math. I am barely in ODE. Anyways, someone showed me this and I thought it was amazing. Basically everything I learned is wrong.
 
what are it's implications in math?
 
ltkach said:
Basically everything I learned is wrong.

Had you learned that ##\sqrt{2}## was rational?
 
  • #10
rbj said:
it's the same for the square root of any prime number. it cannot be rational.

It's the same for any natural number that is not a perfect square (actually, the n'th root of any natural number that is not a perfect n'th power is irrational).

FYI. An easier proof uses the rational zero theorem. Consider the possible rational roots of the polynomial ## x^2 - 2 = 0 ##.
 
  • #11
wow i cannot believe myself. yeah ignore me.
 

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