Thread: Explaination needed for Irrational Proof method

1. Explaination needed for Irrational Proof method

Question proof √3 is irrational and show reason.
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We prove that √3 is irrational. Assume to the contrary that √3 is rational,
that is
√3=p/q,

where p and q are integers and q≠0. Morever, let p and q have no common divisor > 1.Then

3=(p/q)² => 3q²=p² (1)

Since 3q² is odd,it follows that q² is odd. Then p is also odd.
This means that there exists k ∈ Z such that

p=3k. (2)

Substituting (2) into (1), we get

3q = (3k)² => 3q²=9k² => q² = 3k².

Since 3k² is even it follows that q² is even. Then q is also even This is a contradiction.

Note. I'm stuck with this solution and unable to understand the meaning of it. Any more simple alternative solution to reason irrational ?

2. Re: Explaination needed for Irrational Proof method

Yes, this proof is strange.

Originally Posted by vincor
Then

3=(p/q)² => 3q²=p² (1)

Since 3q² is odd
It is not clear immediately why q is odd.
Originally Posted by vincor
it follows that q² is odd. Then p is also odd.
This means that there exists k ∈ Z such that

p=3k. (2)
This is clearly non sequitur. Not every odd number is a multiple of 3.

One proof that $\displaystyle \sqrt{3}$ is irrational parallels a similar proof for $\displaystyle \sqrt{2}$. Suppose 3q^2 = p^2 where GCD(p, q) = 1. Then 3 | p^2, so by Euclid's lemma, 3 | p, i.e., p = 3k for some integer k. Then 3q^2 = 9k^2, so 3 | q^2 and 3 | q, which is a contradiction with GCD(p, q) = 1.

3. Re: Explaination needed for Irrational Proof method

For me, when I look at 3q^2 = p^2 it is obvious this is impossible for integer p,q by fundamental theorem of arithmetic and exponent law (ab)^2 = (a^2)(b^2)

No matter what values I pick for p,q I can never get that 3 'into the square' it doesn't fit.

For non-square integer h, hq^2 = p^2 has no solutions in integers and therefore h is irrational.

Now, if h is a square like 4, 9, 16, etc. then I can put h 'into the square' because it fits.

Examples:hq^2 = p^2

h = 4

4q^2 = p^2, (2^2)(q^2) = p^2, (2q)^2 = p^2 it fits

h = 225

225q^2 = p^2, (15)^2(q^2) = p^2, (15q)^2 = p^2 it fits

If h is not a square, for instance 72 how am i going to make it fit?