what is so strange about triangle with length of sqrt(2) and circumference of circle?

[PG1995]

Hi

Two of the concepts which at a time really confused me are sqrt(2) and the other related one is circumference of a circle.

If we have a triangle with a base and perpendicular of unit length, then it's hypotenuse would be sqrt(2) long. Obviously, such a length sandwiched between two other definite and measurable lengths should also be definite and measurable. Here, I use the words "definite" and "measurable from mathematics' point of view. If humans can't measure any thing definitely or absolutely, it's their limitations, not mathematics'. As sqrt(2) is irrational number and has never-ending trail of digits after the decimal point, so as a result one might argue that if something never ends then how can it be used to represent something definite (in this case hypotenuse of the triangle)? In calculus we find limits for functions using mathematical methods and we says it's the limit which the function tries to reach without ever actually reaching there. Likewise, the length of the hypotenuse which equals the value of sqrt(2) is the 'pictorial' limit and in this case human methods are the reason which prevent us from really calculating the value of sqrt(2).

The same argument goes for circumference of a circle which is 2*pi*r. Pi is an irrational number just like sqrt(2) with unending trail of digits after the decimal point. We have the 'pictorial' limit before us but we don't have the instrument to reach it.

I understand my reasoning is, perhaps, convoluted and a little erroneous. But I'm sure you can see beyond what I wrote and can extract the correct bits of information, and can fill the gaps between them to make the reasoning understandable for yourself. What is your opinion on length sqrt(2) and circumference? Please let me know. Thank you.

Regards
PG

 

[alec_t]

The world is full of irrational things. I can live with that.
When I become King I will declare √2 to be officially 1.414 and π to be 3.14159. Until then ......

 

[PG1995]

Alex: I wish you become the King soon. I hope you would consider me for a ministerial position and then I won't have to ask so much stupid questions to frustrate others!

 

[squishy36]

I'd say don't worry about it because it's of no practical importance. You can adequately do any problem you'll come across in practical science or engineering with rational numbers. Plus, modern computer tools will let you do your calculations to any number of significant figures you wish (I use the mpmath library when I need such things).

 

[misterT]

So you can divide a stick to 3 parts using a tape measure if the stick is 12 units long, but not if it's 10 units long

 

[Robin2]

Why not measure the 10 unit stick in different units so that its length is 12 of the new units?

 

[ElectroNewby]

Instruments have their limits and that's why we can include the uncertainty in the formulas (physics)
Same goes for the methods.

edit: that was unclear but for instance, say you calculator displays 6 digits (0.0000) you'll always have an uncertainty of .0001
and then there's the rule of Gauss to round the numbers

here' a wiki quote from the gaussian rounding:
"
Round half to even

A tie-breaking rule that is even less biased is round half to even, namely

If the fraction of y is 0.5, then q is the even integer nearest to y.

Thus, for example, +23.5 becomes +24, +22.5 becomes +22, −22.5 becomes −22, and −23.5 becomes −24.

This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. In addition, for most reasonable distributions of y values, the expected (average) value of the rounded numbers is essentially the same as that of the original numbers, even if the latter are all positive (or all negative). However, this rule will still introduce a positive bias for even numbers (including zero), and a negative bias for the odd ones.

This variant of the round-to-nearest method is also called unbiased rounding (ambiguously, and a bit abusively), convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, or bankers' rounding. This is widely used in bookkeeping.

This is the default rounding mode used in IEEE 754 computing functions and operators.

https://en.wikipedia.org/wiki/Rounding
"

 

[carbonzit]

When I become King I will declare √2 to be officially 1.414 and π to be 3.14159. Until then ......

Nah, just declare pi to be 3 and be done with it, like the state of Alabama did.

(Actually, that never happened, contrary to Internet rumors.)

 

[MrAl]

Hi,


It appears that we can not measure a distance of sqrt(2) feet perfectly can we. But then again, we can not measure a distance of 2 feet perfectly either. So what's the difference there? Not a whole lot when it comes to measurement.

Here's an interesting little problem you can look at that might offer more insight into this kind of thing...

Get out a map of the world or maybe a map of the country you live in.
Now you know intuitively that if you look at the pic on that map of your country, you can see that it has a certain area (say in square miles or whatever), and that area has a border of some definite distance.
You can also see that it borders perhaps the ocean or a sea of some sort. If you follow the border with your eyes, you'll see that it twists and turns and is quite a jagged line in most cases. Pick one of those jagged coastal borders and try to figure out just how long that border is say in miles or kilometers.
Ok, so you get out the ruler and measure along the border, and because the edges curve in and out from the bulk area of the country you have to move the ruler around a little and add up all the measurements. Ok, so you've done that and come up with a figure, lets say 1374 miles.
But guess what? That's only a rough approximation. Do you think it is really EXACTLY 1374 miles? It would be a miracle if it was EXACTLY that long, including all the in and out zigs and zags. What if you got a magnifying glass and did it again? Ok, so now we end up with 1374.5 miles. But is it really EXACTLY that? Not likely again.
How about getting a microscope out and doing it that way? Well, that would lead to a BETTER answer, but still not exact, and eventually we would run into the problem of the photo resolution or pixel resolution and that would limit our accuracy, so we're beat there. What's a better way?
Lets get in a boat and visit the shore and measure it with a good ruler, and we'll follow the coast south to north and note the little cracks and stuff and measure where the earth goes in and out zigging and zagging and we'll try to get every little fissure. We come up with a new estimate, 1374.479 miles. Is that EXACTLY correct now?
Well, we didnt get out a magnifying glass this time so we still missed some tiny zigs and zags, so we'd have to go back and do that again to get a better measurement. Then, we'd again realize that we really need a microscope, and then since that cant work well enough either, perhaps an electron microscope. Would that be good enough? Well, it still would not tell us the EXACT total length of the shore line.
We'd have to get down to the atom, and that still isnt good enough, so we'd have to get down to the subatomic particles. But they are moving, so at any given time we'd get a different reading even though we consider even all the moving electrons. So then what 'time' is good enough to take a reading at? 1 o'clock, 2 o'clock, what time? Then, since all of the electrons are moving and we only have a finite time to measure each atom, we find the problem that we can not measure every single electron 'orbit' at the same time. We just cant be everywhere at the same time.

So in the end we just cant do it. It's the same for everything. Try to measure the true length of the border of a 1 foot wooden ruler. The small imperfections in the ruler means the true border length is longer than 1 foot. And we would run into the same problem with the sub atomic particle measurements.

This problem is a little more apparent when it comes to painting a surface. We buy a gallon of paint so we can paint a surface of say 1000 square feet (so it says on the side of the can). But what if the surface is very rough, jagged. Then the paint doesnt seem to cover the same amount of surface area. That's because the surface goes up and down a lot. What we see when we look at the surface is a 2d 'projection' of that surface, we dont see the true 3d view where we take into account all the dips and hills. Once we start to consider those kinds of things, we find that the total surface area could easily be twice the 2d projection surface area. That would mean we would have to buy two gallons of paint instead of one. So you see one common example of this kind of problem.


Fractal geometry might explain this a bit better though, or at least try to address this kind of thing.
You'll have to do some reading on that perhaps and gain some more insight.

 

[PG1995]

Thank you, everyone.

@MrAl: Special thanks to you for the explanation. I do have some questions which I will ask soon.

 

[carbonzit]

This problem is a little more apparent when it comes to painting a surface. We buy a gallon of paint so we can paint a surface of say 1000 square feet (so it says on the side of the can). But what if the surface is very rough, jagged. Then the paint doesnt seem to cover the same amount of surface area. That's because the surface goes up and down a lot. What we see when we look at the surface is a 2d 'projection' of that surface, we dont see the true 3d view where we take into account all the dips and hills. Once we start to consider those kinds of things, we find that the total surface area could easily be twice the 2d projection surface area. That would mean we would have to buy two gallons of paint instead of one. So you see one common example of this kind of problem.

Heh; this remind me of a funny thing I remember my calculus teacher in college showing us. There's a geometric figure called Gabriel's horn, a solid of revolution (y = 1/x, x ≥ 1, revolved around the x-axis). The funny thing? Well, you can fill it with paint, but you can't paint it!

Now why would that be? (This may actually help you with your understanding of math concepts. Or not.)

 

[PG1995]

Thanks, carbonzit.

So, we have a shape, Gabriel's horn, which behaves much like Pi and sqrt(s). Its one end keeps on stretching.

 

[MrAl]

Hello again,

The interesting thing about that mathematical object is that you might find a lump of clay that you can use to build a solid object of that shape, according to the mathematics, but you could never find enough paint to paint it with. But then again, it would be impossible to build an object that is infinitely long, no matter how narrow it was. Thus there are really two impossibilities, not one.

It's again a cross of theory and reality...they sometimes just dont mix. Once you decide to go with theory, you have to go with that until you start to allow the finite realities to set in. For example, in the above we allowed an infinitely long object (Theory) but then decided to use real life paint. If we stuck with theory we could have painted it with infinitely thin paint.

 

[carbonzit]

So, we have a shape, Gabriel's horn, which behaves much like Pi and sqrt(s). Its one end keeps on stretching.

Um, not quite.

Its volume is finite, but its surface area is infinite.

 

[PG1995]

Um, not quite.

Its volume is finite, but its surface area is infinite.

1: Is it really so?! I have been thinking about it. If it has an infinite surface area, then it must also has infinite volume. If it doesn't, then it would be a self-contraction, in my humble opinion which I know is short sighted!

From physical point of view you can say when one of its side is stretched to infinity then at some stage it's bore reaches a limit when no atom etc. can pass through it and any liquid put in it won't leak beyond that point where the bore doesn't allow the tiniest atoms to pass through. But from mathematical point of view, its bore can never ever become zero which implies infinite volume.

2: I was thinking that if there is a way to make numbers such as Pi and sqrt(2) rational. In other words, is this this possible to make rational irrational? I don't think changing number system could affect the ultimate outcome. Pi would still be irrational even if we had binary system. Are such mathematical concepts independent of the systems in which they are studied? If my question is not clear, then please let me know. I would try to rephrase it. Thanks a lot.

Regards
PG

 

[misterT]

2: I was thinking that if there is a way to make numbers such as Pi and sqrt(2) rational. In other words, is this this possible to make rational irrational? I don't think changing number system could affect the ultimate outcome. Pi would still be irrational even if we had binary system. Are such mathematical concepts independent of the systems in which they are studied? If my question is not clear, then please let me know. I would try to rephrase it. Thanks a lot.

If you have a circle with diameter D and circumference C, then Pi is the number C/D. Just draw a circle with rational diameter and rational circumference.

 

[carbonzit]

Um, not quite.

Its volume is finite, but its surface area is infinite.

1: Is it really so?! I have been thinking about it. If it has an infinite surface area, then it must also has infinite volume. If it doesn't, then it would be a self-contraction, in my humble opinion which I know is short sighted!

Short-sighted? I don't know; do you understand calculus?

The surface area and volume of this solid of revolution are calculated as integrals. As I said, the volume is finite, but the surface area is infinite.

Keep in mind that this is an object that can only exist in our minds, not in the physical world.

 

[carbonzit]

2: I was thinking that if there is a way to make numbers such as Pi and sqrt(2) rational. In other words, is this this possible to make rational irrational? I don't think changing number system could affect the ultimate outcome. Pi would still be irrational even if we had binary system. Are such mathematical concepts independent of the systems in which they are studied?

It sounds as if you're asking if the rationality or irrationality of a number would be changed by the system by which the number is represented (decimal, octal, hexadecimal, binary, etc.). The answer is no; all these systems represent the same numeric quantity. (Of course, you would have to set up fractional representations in any of the non-decimal systems--whatever is to the right of the decimal point in base 10--as these other systems are ordinarily only used for integers, but that's a trivial detail.)

 

[MrAl]

1: Is it really so?! I have been thinking about it. If it has an infinite surface area, then it must also has infinite volume. If it doesn't, then it would be a self-contraction, in my humble opinion which I know is short sighted!

From physical point of view you can say when one of its side is stretched to infinity then at some stage it's bore reaches a limit when no atom etc. can pass through it and any liquid put in it won't leak beyond that point where the bore doesn't allow the tiniest atoms to pass through. But from mathematical point of view, its bore can never ever become zero which implies infinite volume.

2: I was thinking that if there is a way to make numbers such as Pi and sqrt(2) rational. In other words, is this this possible to make rational irrational? I don't think changing number system could affect the ultimate outcome. Pi would still be irrational even if we had binary system. Are such mathematical concepts independent of the systems in which they are studied? If my question is not clear, then please let me know. I would try to rephrase it. Thanks a lot.

Regards
PG

Hi there PG,

#1
That would be true if things worked that way, but they dont quite work that way. The problem is a mix of dimensions. We can have something described one way in X dimensions that reaches a limit, while something else related to that in Y dimensions doesnt reach a limit.
We're talking about an 'object' one way in three dimensions, then talking about that same 'object' another way in two dimensions. One is volume (3d) and one is surface area (2d) so the limits dont have to coincide. We cant forget that we're dealing with a fictitious object here, not a real life object. We can only construct an approximation anyway. You have to remember that theory is not real life. The universe was here before theory, or at least Man's theory as it is known today.

#2
You can make pi look rational if you envision a circle as a multi sided object rather than a perfect circle. This is good for practical applications, but i dont see what good it would do to try to make pi perfectly rational. pi converts linear systems to circular systems, which have their very own rationale. Radial lines map perfectly to rectilinear lines in theory only. As soon as we try to build something like this to prove it we find it just doesnt work.
Mathematically it works, and again we might call in the purposed theory that we (as mankind on earth) have too much information, more information than reality, at least for the time being. This means we'll find lots of things that work mathematically but dont work in real life. To put it another way, if math was a solid object it would be bigger than the universe.

 

[KeepItSimpleStupid]

And then the tide went out.