The distance we can't measure

[rktpro]

Lets take the example of trigonometry of heights and distances.

We usually come across something like: the height of a point is 100√2 and say angle of elevation used in the problem was a

Now, √2 is irrational no. and so would be 100√2. This distance can only be approximated. So is it that a point can't be observed at an angle a but at any other angle closer to it or is it that a point can't be at 100√2 units from an observer.

Sorry for my ignorance.

Thanks

[Schrödinger's hat]

Lets take the example of trigonometry of heights and distances.

We usually come across something like: the height of a point is 100√2 and say angle of elevation used in the problem was a

Now, √2 is irrational no. and so would be 100√2. This distance can only be approximated. So is it that a point can't be observed at an angle a but at any other angle closer to it or is it that a point can't be at 100√2 units from an observer.

Sorry for my ignorance.

Thanks

 

If I have a triangle with base and height 100m, then 100√2m is the hypotenuse.

I can just as easily redefine my unit system so that they hypotenuse is 1flam (where flam is a new unit of distance) and 1/√2flam the base and height. Now which one is un-measurable?

 

The point I'm making is that you have ignored the concept of experimental error.

No realy measurement is going to be exactly 100m, even if we measure down further than the planck length (not thought to be possible), it'll be 100.0000000000000000000000000000000000000m

We can easily calculate √2 further than this.

 

You also have to remember that any numbers we apply are only a model. They do not represent the objects we are modelling exactly., only up to a level where any deviation from the model cannot be measured or is not important.

[ajb]

You also have to remember that any numbers we apply are only a model. They do not represent the objects we are modelling exactly., only up to a level where any deviation from the model cannot be measured or is not important.

 

This really is the key point. Mathematically such a triangle exists as a figure in Euclidean space. There is no problem in dealing with sides of length [math]\sqrt{2}[/math] and similar. Everything is well defined.

 

The trouble is you can not exactly draw this. First of all the sides will be of finite width and then the length of the lines cannot be exact. Just physically at some level you have to consider the carbon atoms in the graphite pencil line, then we are into quantum mechanics and the definition of the length becomes obscure.

 

The same would be true of any measurement of any physical object.

[John Cuthber]

Because root 2 is irrational I can't, even in principle, measure out a line 100√2m long.

It's not an issue of experimental error, or the size of the Planck distance. It's impossible.

 

However I can construct that line perfectly easily- it's the diagonal of a square that's 100 m each side.

 

The issue here is one of incomensurability not impossibility.

 

This part of the OP is incorrect

"This distance can only be approximated. "

because, in principle, it can be drawn.

[DrRocket]

This really is the key point. Mathematically such a triangle exists as a figure in Euclidean space. There is no problem in dealing with sides of length [math]\sqrt{2}[/math] and similar. Everything is well defined.

 

The trouble is you can not exactly draw this. First of all the sides will be of finite width and then the length of the lines cannot be exact. Just physically at some level you have to consider the carbon atoms in the graphite pencil line, then we are into quantum mechanics and the definition of the length becomes obscure.

 

The same would be true of any measurement of any physical object.

 

All true.

 

What is also true is that you cannot precisely draw a line of rational length either which is a corollary of your last sentence.

[JohnStu]

In engineering, if they encounter square root of 2, they often just times it by another square root of 2 to solve the problem or they leave 9 digits or more.

 

Rarily in housing construction did I encounter square root of a whole number. It has always been square root of 3.5. square root of 6.64. Thanks to use of inches and feet while overhere they use meter.

[The Observer]

I really don't see how you can say that a measurement of root 2 is impossible to measure out precisely and at the same time believe that you can measure out 1 or 2 exactly. We can calculate the values of root two to any precision. Certainly to as precise as you could ever, even theoretically measure. The infinite number zeroes coming after the number aren't written down but in a sense they are still there.

[rktpro]

I really don't see how you can say that a measurement of root 2 is impossible to measure out precisely and at the same time believe that you can measure out 1 or 2 exactly. We can calculate the values of root two to any precision. Certainly to as precise as you could ever, even theoretically measure. The infinite number zeroes coming after the number aren't written down but in a sense they are still there.

 

4.000000000000000.......upto infinity is same as 4.0

And simply because 4 can be represented in form of p/q but root 2 can't.

[Schrödinger's hat]

Because root 2 is irrational I can't, even in principle, measure out a line 100√2m long.

It's not an issue of experimental error, or the size of the Planck distance. It's impossible.

 

However I can construct that line perfectly easily- it's the diagonal of a square that's 100 m each side.

 

If I can construct one, I can also construct a line of √2m and measure out 100 of them.

You have to remember that our base units are an entirely arbitrary choice (the derived ones, however, follow logically from that choice).

 

4.000000000000000.......upto infinity is same as 4.0

And simply because 4 can be represented in form of p/q but root 2 can't.

 

This is irrelevant. It's just as hard to compare an infinitely precise 4.000.... as an infinitely precise √2 to any real object.

[John Cuthber]

I can, in principle, measure out any rational distance with a ruler.

I cannot, even in principle, do that with an irrational distance.

That's the difference.

You can use any base unit you like, but you still can't find the point on your ruler marked "root two".

[Schrödinger's hat]

I can, in principle, measure out any rational distance with a ruler.

I cannot, even in principle, do that with an irrational distance.

That's the difference.

You can use any base unit you like, but you still can't find the point on your ruler marked "root two".

 

I can construct a line that is root two metres long, and then mark its length on my ruler.

Or I could define a new unit system where one flam is sqrt(2) metres, then anyone can measure any fraction or multiple of sqrt(2) flams.

 

Or rather, I can't. Because I can't measure exactly 1 metre either.

[ajb]

What is also true is that you cannot precisely draw a line of rational length either which is a corollary of your last sentence.

 

Absolutely, details of the physics at some level are always going to mess up your very neat mathematical ideas of geometric figures. Of curse, for the most part we don't really see this in everyday life.

[Schrödinger's hat]

Of curse, for the most part we don't really see this in everyday life.

 

I see what you did there.

[swansont]

Because root 2 is irrational I can't, even in principle, measure out a line 100√2m long.

It's not an issue of experimental error, or the size of the Planck distance. It's impossible.

 

However I can construct that line perfectly easily- it's the diagonal of a square that's 100 m each side.

 

But that triangle is now a length to which another object can be compared. If they are exactly the same length, then can't you say you have measured it? You could even use such devices to put these markings on a standard ruler.

[John Cuthber]

I can construct a line that is root two metres long, and then mark its length on my ruler.

 

Yes, but it wouldn't correspond to any other mark on the ruler- which is the point of ruling marks in the first place.

[swansont]

Yes, but it wouldn't correspond to any other mark on the ruler- which is the point of ruling marks in the first place.

 

Only as a matter of practicality, because you usually want a ruler to be an easily made general use tool. That's not the same as an inability to make such a device.

[John Cuthber]

No It's nothing to do with practicality. It's a matter of principle.

That's the whole point of incomensurability. They cannot be measured on the same scale.

The diagonal of a square is not a rational fraction of the length of the side.

If I take my ruler and try to measure the diagonal of the square I find it's between the 1 and the two.

If I divide that space into ten I find the diagonal is between 1.4 and 1.5 Another division tells me it's between 1.41 and 1.42

If I divide that gap into ten again I find the length of the diagonal is between 1.414 and 1.415

But no matter how finely I divide the scale (and no matter what fraction I divide it into, so it still won't work if I try dividing the scale in thirds or halves) I will never put a mark exactly on root two because it's decimal expansion doesn't terminate.

 

It's trivial to construct the line, but it's impossible to measure it exactly- not for any practical reason, but because the line won't tally with any marking on the ruler.

You can say it's more than ... but less than ... but you can't say it's exactly "here" on the scale.

 

If I measure a line that 1.34 inches long (exactly) then it exactly lines up with the 1.34 inch mark- that's what rulers do.

No ruler calibrated in inches can measure the diagonal of a one inch square.

[rktpro]

So, assuming that in a hypothetical situation, where 100 m can be actually measured:100√2m obtained as a distance. What would you say about my original question. Is it that a point can't be observed at that angle but at any other angle closer to it or is it that a point can't be at 100√2 units from an observer.

[John Cuthber]

The point can be observed at 100 root 2 metres from the observer.