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SF/PF

How do we know that √2 is irrational?

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I'm currently reading a book on the history of mathematics, and came across this little proof. I found it kind of neat and thought others might as well!

How do we know that √2 is irrational?

- Suppose that √2 is a rational number. That is, √2 is expressable as a ratio of two integers:

√2 = m/n

- Assume that m/n has been reduced to lowest terms.

- Square both sides of the equation:

√2 = m/n becomes 2 = m2/n2

- Do a little bit of algebra:

2 = m2/n2 becomes 2n2 = m2

- Observe that both sides of this equation must be even, by virtue of multiplication by 2:

2n2 = m2

- Therefore, m2 is even. And if m2 is even, m must also be even. Thus, m may be expressed as:

m=2x (This is what it means for a number to be even!)

- Substituting m=2x back into our equation:

2n2 = m2 becomes 2n2=(2x)2 becomes 2n2=4x2

- Do a little bit of simplyfying:

2n2=4x2 becomes n2 = 2x2

- Observe again that both sides of this equation must be even. Therefore, n2 is even, and n is also even.

- So we know that both m and n are even.

- But wait... we assumed that m/n was reduced to lowest terms. If m and n are both even, then m/n cannot be in lowest terms, as both the numerator and denominator are divisible by 2.

- Therefore, √2 is not rational.

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Yup. Neat. Pretty standard proof.

Here's a tougher one for you: prove that pi is irrational.

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Very interesting!

My education in math has not progressed to the point where I'm proving stuff, so that was very interesting to see. Bonam, if you could give me a clue on where to get started on the Pi thing, I'll give it a try...

I am presently attempting to determine whether SF/PF is rational.

-k

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I'm a history/geography guy....

Math sucks....

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I took a peak at some of the proofs of Pi's irrationality, and they're all beyond my ability in math to understand.

My background is in Philosophy, so I think it was the "proof by contradiction" structure of this proof that really interested me.

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How do we know that √2 is irrational?

- Suppose that √2 is a rational number. That is, √2 is expressable as a ratio of two integers:

√2 = m/n

And let me assume that 42 divided by 6 can be expressed as a ratio.

Or rather, let me assume that I can fly through space at 1000 km/h. (Hey, I just did - but I wouldn't want to do this right now!)

---

SF/PF, here's a better question: Can you express an irrational number using two rational numbers?

For the math challenged, let me lead you in the wrong direction: if two reasonable people marry, does that guarantee that the marriage will last? Or, do celebrity marriages require at least one crazy person?

(Bonus added later: Was it Demi or Ashton who was difficult/crazy? PS: I just finished my grocery shopping!)

Edited by August1991
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And let me assume that 42 divided by 6 can be expressed as a ratio.

Or rather, let me assume that I can fly through space at 1000 km/h. (Hey, I just did - but I wouldn't want to do this right now!)

---

SF/PF, here's a better question: Can you express an irrational number using two rational numbers?

For the math challenged, let me lead you in the wrong direction: if two reasonable people marry, does that guarantee that the marriage will last? Or, do celebrity marriages require at least one crazy person?

(Bonus added later: Was it Demi or Ashton who was difficult/crazy? PS: I just finished my grocery shopping!)

Meanwhile, Voltaire is rolling over in his grave. :)

Edited to add:

SF/PF, here's a better question: Can you express an irrational number using two rational numbers?

Nope.

Edited by SF/PF
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Meanwhile, Voltaire is rolling over in his grave.
Voltaire may roll in his grave but we still don't know whether it is Demi Moore or Ashton Kutcher who was the crazy one.
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Voltaire may roll in his grave but we still don't know whether it is Demi Moore or Ashton Kutcher who was the crazy one.

On what grounds do you assume that only one is crazy?

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Very interesting!

My education in math has not progressed to the point where I'm proving stuff, so that was very interesting to see. Bonam, if you could give me a clue on where to get started on the Pi thing, I'll give it a try...

It's actually quite difficult and not at all elegant. I had to do it as an exercise in a third year math class in university, but don't really remember it at all. Best I can do, sadly, is link the wiki article:

http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational

Edited by Bonam
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