A naive question about irrational numbers

[Mark44]

I see.

If space is in fact continuous, can we then talk about exact physical lengths in terms of space?

We can talk about exact physical lengths in the abstract, but as you have been told several times in this thread, there is no way to measure them exactly.

 

[la6ki]

Hmm, sorry if I offended you somehow, didn't mean to be rude.

 

[Diffy]

las6ki,

Take the length of a rod. By rod, I take it you mean some type of long cylinder.

In math we take things to be perfect, because it makes thing nice. So in a math world, the cylinder has a length, because the circle on the top and bottom of the cylinder are perfectly flat, and the cylinder is perfectly straight.

A rod in the real world can never be. As you pointed out before the rod in the real world would be made of atoms. So consider the top and the bottom of your rod. It is akin to a petri dish full of different sized marbles.

Now what is length? Do we measure from the tallest marble? or do we measure from the average height of the marbles?

How do we determine if in fact the rod is perfectly straight? It is not solid in the same sense the mathematical cylinder is? Indeed at an atomic level there is space between the atoms.

Objects in a mathematical world like a cylinder can be kind of like objects in the real world. However they are not.

I hope I have helped.

 

[lordofpi]

I'm sorry if this has already been said, but I'd like to point out that the $\sqrt{2}$ is in fact a real number.

Assuming for a minute that we had the tools of perfect accuracy and precision to be able to actually measure the given idealized distances (the difficulty of which has already been discussed), this number is as much able to found ($\sqrt{2}$) as 3 or 17 using a real-world number line a.k.a a measuring stick.

To maybe express an irrational number as a quantity of some fixed indivisible item is another matter altogether (i.e., $\sqrt{7}$ marbles).

I guess we also have to analyze what we mean by the word "measure." We really want to be able to measure arbitrary distances based on a defined fixed unit of continuous space. Unfortunately, we have no way of doing this without using a reference -- which then leads our measurement to be a multiple of that fixed reference (e.g., the number of atoms in a meter stick to get us to 1 cm), which leads us back to having a hard time with irrationals.

I got to the end of this not feeling as confident in my explanation as I did before I typed it, so I hope it at least helped a -small- bit.

 

[la6ki]

I just saw your responses, guys. Thanks, they both make things clearer for me :) I still feel like there are a few things that need some clarification, but I'll ask when I can formulate them better.

 

[Integral]

Every REAL measurement has associated with it error bounds. Generally you can take your error to be 1/2 of the smallest division on your measuring instrument.

So your 1 m rod by measurement is 1m +/- Δ, where the magnitude of Δ is determined by your measurement instrument. Note that in the real world a device which will measure 1m to +/-.1mm is a specialty device which will cost you a good sum of money.

No matter how you do the measurement there will always be uncertinaty. That is why you cannot have a real rod exactly 1m. It may be, you just cannot prove it.

Your measurement of the √2 length will have the same errors, so the fact that √2 is irrational is immaterial, its length is known only as good as your ability to measure. Measuring any length to more then 3 or 4 decimal places is essentially impossible for the man on the street.