Inquiry about the properties of square roots

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Discussion Overview

The discussion revolves around the properties of square roots, specifically focusing on the relationship between natural numbers and their square roots, and whether these roots can be classified as rational or irrational. The scope includes mathematical reasoning and proofs related to number theory.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the proof that establishes that if the square root of a natural number is not another natural number, it must be irrational.
  • Another participant reiterates the same inquiry and suggests using unique factorization into primes as part of the proof.
  • A different participant notes that a perfect square will always have pairs of equal factors, providing examples such as 81 and 100.
  • One participant states that the square root of a prime number is irrational and extends this claim to square-free numbers, mentioning that the proofs for these assertions are not necessarily straightforward.

Areas of Agreement / Disagreement

Participants express various viewpoints and inquiries regarding the properties of square roots, but there is no consensus on a specific proof or resolution of the claims made.

Contextual Notes

The discussion includes references to unique factorization and the nature of perfect squares, but the limitations of these approaches and the assumptions involved are not fully explored.

spherenine
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What is the proof that states that if the square root of a natural number is not another natural number, it must be irrational? In other words, the square root of a natural number must be either natural or irrational.
 
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spherenine said:
What is the proof that states that if the square root of a natural number is not another natural number, it must be irrational? In other words, the square root of a natural number must be either natural or irrational.

Use unique factorization into primes.
 
And continuing with g_edgar's advice, a perfect square will always have pairs of equal factors. E.g., 81 = 9*9, 100 = 10*10 and so on.
 
The critical component is that the square root of a prime number is irrational and by extension so is the square root of any square-free number. The proofs of these facts are not necessarily straight forward.

--Elucidus
 

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