Is √N Always Irrational for Non-Square Integers?

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SUMMARY

The discussion centers on proving that √N is irrational for non-square integers N. The proof utilizes the fundamental theorem of arithmetic, which states that every integer has a unique prime factorization. If √N is assumed to be rational, it can be expressed as A^2/B^2, leading to the equation B^2N = A^2. This ultimately demonstrates that N must be a square, contradicting the initial assumption.

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  • Basic knowledge of rational and irrational numbers
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  • Concept of prime factorization
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bgwyh_88
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I came across this question. How do you show that √N is irrational when N is a nonsquare integer?

Cheers.
 
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It depends on what you're allowed to use.

But the simplest way would be to use the fundamental theorem of arithmetic (that every integer has a unique prime factorization). For any N, if sqrt(N) is rational, you can write that as

N=A^2/B^2

and therefore

B^2 N = A^2

and it's not hard to get from that + the fundamental theorem to the conclusion that N is a square.
 
Nice. Thanks hammster143. Appreciate it.

bgwyh_88
 

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