Understanding Irrational Numbers: Is it Possible to Exact Measure?

[nsnayak]

Okay, I was thinking about irrational numbers, and I came to this conclusion: It is impossible exactly measure an irrational number.I am probably wrong, and that's why I posted this thread to check the validity of that statement.
Here is my proof:

If you wanted to cut a piece of paper exactly 1.284736 cm, you would probably measure it to the tenths place (1.3) or the hundreths place (1.28). If you wanted to be even more exact, you could keep on going until the ten millionth place.
Now, suppose you wanted to cut this piece of paper exactly the square root of two.
As we all know, the sqrt(2) is approximately 1.414213562. I say "approximately" since this number goes on forever. Therefore, you can never get an EXACT measurement since you always have another number in the decimal that you haven't taken into account.
As I said before, I am probably very wrong about this statement.

 

[Hurkyl]

All measurements have some error; we can measure 1 no more exactly than we can measure √2.



Let's live in hypothetical land for a moment; suppose there are measurements we can do exactly, and that measuring integer lengths are among them.

Well, it's trivial to construct any number that can be made from integers, +, -, *, /, and square roots, using a straightedge and compass. With origami or a ruler and compass, I think you can do cube roots as well. You can construct a circle, and then pi by rolling the circle.

I would be entirely unsurprised if you could construct a lot more.

 

[matt grime]

Just using ruler and compass you cannot obtain cube roots. The extensions you construct are all degree 2^r over Q, and in particular one cannot square the cube.

If you have a marked ruler, which is how I understand it that we can 'measure' the integers, I'll swing with you being able to obtain cube roots - you can certainly trisect and angle (which is impossible in ordinary ruler and compass construction).


As to the original question, if you can 'measure' any rational accurately, you can 'measure' any irrational to any arbitrary degree of precision, ie you give me ANY e>0, and no matter how small I can produce something no more than e in error from what you want. But as 'measuring' rationals is equally as hard

 

[HallsofIvy]

Matt, what do you mean by "ruler"?

Hurkyl first said that you can get square roots by using a straightedge and compasses. He then referred to getting a cube root by using a ruler and compasses. Certainly, you can get cube roots by using compasses and a marked straightedge (I believe it was Archimedes who showed that) which is what most people mean by "ruler"- we are allowed to mark a length on the straightedge and transfer that length to a different line.

 

[matt grime]

Yeah, it is all a bit vague. The standard in Galois theory is to take a 'ruler' to be an unmarked straightedge, at least this tallies with what the greeks did, it appears (as in what they thought to be constructible or not, eg see Stewart's Galois Theory, 3rd edition). This, it is alleged, is Euclid's strategy becuase it was 'purer' in spirit. In particular the greeks could not trisect the angle using ruler and compasses, which is possible with a marked ruler, and which was known to them.

I didn't notice the switch from straightedge to ruler in Hurkyl's post, to be honest.

There is also the added consideration of what one means by marked (where are the marks, are they uniformly distributed..). You can also argue that the compasses construe a form of measuring device.