A naive question about irrational numbers

[la6ki]

I've been thinking about this recently and couldn't find the answer to my question (even though I assume it's a really simple one, so forgive me if it's too trivial).

Let's say we have two rods of length 1 meter and we put them at right angles to each other. Then we cut a third rod just long enough to connect the non-touching ends of the first two rods - now we have a right isosceles triangle. Question: what's the length of the hypotenuse? Clearly the answer is √2

Now suppose we physically measured the length of the rod which is the hypotenuse (maybe by multiplying the number of atoms on top of each other times the length of each atom?) Aren't we going to get a rational number as a result?

 

[arildno]

"Now suppose we physically measured the length of the rod which is the hypotenuse (maybe by multiplying the number of atoms on top of each other times the length of each atom?) Aren't we going to get a rational number as a result?"
the relevance being?
For example, a mathematical triangle is characterized by that "points" on two sides of the triangle will get ARBITRARILY closer to each other as they converge to their commmon joint. (Agreed?)

No physical "triangle" will achieve that, so the only insights from this example is that
A) maths do not attempt to describe reality, but elucidates logical structures
and
B) In the studies of reality, maths will be an extremely useful tool for model building and prediction.

 

[la6ki]

A) maths do not attempt to describe reality, but elucidates logical structures
and
B) In the studies of reality, maths will be an extremely useful tool for model building and prediction.

Yeah, I get the overall point. Could you talk about B) a bit more? I completely agree with the statement in general, but I can't see how it follows from my example...

 

[MarneMath]

Simple. Even though the mathematics may not give you the exact answer, it gives you a good idea what the answer should be near. So, let's say you were only told those two rods were both 1 meter and the rod used to make the hypotenuse was 3 meters, then you should be able to say, "hey wait, that sounds like the third rod is to long and that there were will be a lot of excess."

 

[Akshay_Anti]

I've been thinking about this recently and couldn't find the answer to my question (even though I assume it's a really simple one, so forgive me if it's too trivial).

Let's say we have two rods of length 1 meter and we put them at right angles to each other. Then we cut a third rod just long enough to connect the non-touching ends of the first two rods - now we have a right isosceles triangle. Question: what's the length of the hypotenuse? Clearly the answer is √2

Now suppose we physically measured the length of the rod which is the hypotenuse (maybe by multiplying the number of atoms on top of each other times the length of each atom?) Aren't we going to get a rational number as a result?

In the same regard you can always plot an irrational number on a real number line and you can always measure the distance of that point from the zero point even though it has no exact value.

 

[HallsofIvy]

In the same regard you can always plot an irrational number on a real number line and you can always measure the distance of that point from the zero point even though it has no exact value.

Be careful of your wording! Every real number has an "exact value" whether it is rational or irrational.

 

[la6ki]

Is it accurate to say that if we had two rods which are exactly 1 meters long, it would be impossible to connect them with a third rod without a slight residual (of, say, 1 atom)?

 

[arildno]

What does exact mean?

 

[la6ki]

With no residual :)

 

[Akshay_Anti]

Be careful of your wording! Every real number has an "exact value" whether it is rational or irrational.

True, indeed. My mistake! Sorry for that. Here's what i meant to say- Irrational numbers have a non-terminating, non- repeating decimal expansion and taking that definition into account, we can have a value that is approximately close to the exact value and not the exact value can be represented coz we don't know it.

 

[Akshay_Anti]

Is it accurate to say that if we had two rods which are exactly 1 meters long, it would be impossible to connect them with a third rod without a slight residual (of, say, 1 atom)?

this doubt of yours is solved by the concept of unit cells in chemistry. no atom exists of its own but in a lattice and when you calculate the edge length of that lattice, many a times the result is in terms of irrational numbers. So, that would suffice cause, an irrational multiplied by a suitable rational will give a number that will show us the exact number of atoms.

For more, search for Bravis lattice on wiki

 

[coolul007]

The real world does not have an exactly 1 meter anything. The comparison relies on the way we measure and the size of the measuring stick used for reference. Take the coast line of any country. If our measuring stick is a kilometer, we leave out out a few inlets, so shrink the stick and the coast line gets larger. If the measuring stick gets down to an atom, the coastline becomes very large. That is why it is impossible to have a theoretical mathematical value placed on a real world item.

 

[chingel]

I think the answer is simple, if you create an exactly 90 degree triangle with exactly 1 meter legs, the hypotenuse would be exactly the square root of two if you were to measure it (keeping in mind not to round the measurement of the hypotenuse, but measuring as many decimal points as you like with your absolutely exact measurement tool and verifying that all the decimal points are the same as for the square root of two). Any difference from the square root of two is because of measurement error, you just have to measure more exactly.

 

[Mark44]

I think the answer is simple, if you create an exactly 90 degree triangle with exactly 1 meter legs, the hypotenuse would be exactly the square root of two if you were to measure it (keeping in mind not to round the measurement of the hypotenuse, but measuring as many decimal points as you like with your absolutely exact measurement tool and verifying that all the decimal points are the same as for the square root of two). Any difference from the square root of two is because of measurement error, you just have to measure more exactly.

It isn't possible to do any of the things you said. We can't create an angle that measures exactly 90 °, and we can't create legs that measure exactly 1 meter. Furthermore, whatever the length of the hypotenuse, it is not physically possible to measure it exactly. There is no "absolutely exact measurement tool." The triangles that we use in geometry and trigonometry are idealized figures, not things we can create in the real world. When we measure a physical triangle, any such measurement is an approximation. We can get values that are closer by using more precise measurement tools, but we can't get exact values.

verifying that all the decimal points are the same as for the square root of two

How do you propose to do that?

 

[la6ki]

I would like to ask a clarifying question. Is it the case that space is continuous and that is why we can't ever measure exact distances, or we don't know if space is discrete or continuous but we simply don't have the instruments to make the exact measurements (like to pick an object with a length of exactly 1m)?

 

[arildno]

The case is that maths and the study of reality are logically separate areas, and that you shouldn't go about presuming that the conditions and premises set up in maths (and leads to some results there) are present in "reality".
Thus, it is utterly irrelevant how the world behaves, that has no bearing whatsoever on whether a mathematical proposition is true or not.

 

[la6ki]

Excuse my lack of understanding, I'm not a physicist and don't have any related background. I'm not sure how the last post answered my last question.

I understand that math deals with idealized objects and not everything that's true about math is also true about the physical world. But why is it that lengths of exactly 1m don't exist in the physical world? Is it possible they exist but we simply don't have the proper instruments to measure them?

 

[Akshay_Anti]

if you are a physicist, you might already know that we take all units and measurements relative to something. like weight of 1 kg is taken as weight of Ir-Pt cylinder kept in sealed jar and all measures of weight are taken relative to the same. similarly about all other 7 base units.

 

[la6ki]

Hmm, I said I am NOT a physicist :)

 

[Akshay_Anti]

sorry, i read it wrong. the other thing i told of is true. you may refer any college level or class 11 or class 12 level physics books and then you'll know. the seven base units- second, kg, candela, metre, ampere, mole and Kelvin are defined as base units and all other units can be derived using these 7. for more, refer-

http://en.m.wikipedia.org/wiki/Base_units

 

[la6ki]

Actually, I knew about the way units are defined, but I don't immediately see how it answers my question. Are you saying that it is because we lack measurement tools? But still, using the relative measurement, we can still, in principle, measure exact values in those relative terms, no?

 

[coolul007]

The tools are dependent on changes in temperature pressure, etc. If there are minute changes in velocity and gravity the tools only work relative to their surroundings. Hence, curvature of space, etc.

 

[la6ki]

But we can still assume that exact lengths do exist in the universe, we simply can't measure them, right?

 

[mfb]

There is no reasonable way to define an exact length of a real object. And you cannot measure something you cannot even define.
Exact lengths exist in mathematics only.

 

[la6ki]

mfb, I think we're finally coming to what I'm asking. Could you elaborate on why there is no reasonable way to define an exact length of a real object?

 

[mfb]

Real (solid) objects consist of atoms* - the surface is not perfectly flat and looks a bit like that. Even if you manage to remove all hills there, the individual atoms are not flat. And even worse: Most of their volume is occupied by electrons in orbitals, which have no fixed boundary. The "probability" to find an electron at a specific location (if you measure its position) drops with distance, but it does not become zero.

*apart from neutron stars, but that is not relevant here

 

[la6ki]

Yeah, that definitely makes sense, thanks for the explanation.

One last thing that's still a bit puzzling. If we don't consider length in terms of objects (and therefore atoms), but just the length of space, can we then talk about exact lengths? Is space continuous?

 

[mfb]

Is space continuous?

That is unknown. The established theories treat it as continuous, but some other approaches have discrete "units" of space.

 

[la6ki]

I see.

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

 

[Bacle2]

I see.

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

I would say that even if space is not in actuality continuous, that mathematical

theories which assume it is--thinking mostly of Riemann integration in Calculus--

have beeen successful in solving real-life problems. So one may argue that the

assumption of the continuity of space is at least a good approximation to the

actual properties of space. But then there are mathematical theories that

have tried to discretize continuous areas of math, e.g., dicrete differential geometry.

 

[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.