Circles do not exist

The circumference will be Pi ft. Multiply the value of Pi by 1 ft and you will have the measurement in feet. Of course you have to round the answer, but even so there is a limit to the accuracy of current measuring devices, so it would be practical to round to the accuracy of your particular measuring method. After all, this is an engineer's discussion forum, did you expect one to agree that circles don't exist? I once had a physics professor that would never multiply and round when the solution involved Pi because he wanted an exact solution and always left it in terms of Pi. Well, a lot of good that does an engineer, we have to make estimates all the time, it just has to be close enough so that no one dies.

Thank you for your response. However my original submittal required no rounding. If the circle existed it would have a 'known' circumferral length for a 'known' diameter. I did not round the diameter. I did not ask for an estiamted circumference. (is circumferral a word?)

The word you're looking for is "circumferential."

Mevel - What are you talking about??? The Circumference is 1x3.14159 or 3.14159 ft

Thanks for your response, however the circumference would only be 'close' to 3.14159 Close only counts in horseshoes. I would accept your answer if I stated the diameter was 'Close' to 1 foot.

Correction, close only counts in horseshoes and hand grenades, but you don't even have to be close with a small thermonuclear weapon. The circumference of a 1 foot diameter circle would be 3 feet, close enough for me, especially without my hp48GX. Thanks, Mevel123 for getting this started, it has been entertaining.

Just round it to the accuracy of the instrument you used to measure the 1 ft diameter. You can never measure the exact diameter or circumference of the circle. Though you can get close enough for government work.

Thanks for your input. But if a circle existed the length of it's circumference should also be known. At this point I can only concede that circles exist in estimation form only. Try telling the IRS your estimated tax payments are 'close enough' for their purposes :)

(My 1.0ft diamter is 100.0% accurate)

Of course the circle exists. The only thing in question is the true length of the circumference. Your 1.0ft diameter may be 100.0% accurate but is it 100.0000000000000000000000000000000000000000% accurate? I think the circle exists in my mind. I think therefore I am. Don't you think?

I don't really know what your problem is. All you have to do is draw your precise 1 ft diameter circle and then measure it with your precise mesurment. When your are done, give all of us, the precise measurment so we can change Pi.

π is neither an estimate nor unknown. It is no less a number than 14 or 451.96 and can be known to whatever arbitrary precision one has the patience to calculate. A 1-foot diameter circle has a circumference of exactly π feet, and its circumference may be verified as precisely as its diameter.

For a fabricated steel disc, I can verify the circumference to the same precision that you can verify the diameter. You can't define that the diameter is 1.000... if you require that I measure 3.1415926535898... We can both define or we can both measure. The circle's π foot circumference is just as "accurate" (arbitraily precise) as the 1 foot diameter.

==Those wacky IRS folks==

Actually, you're quite free to tell the IRS your estimated payment is close enough. Although a marginal tax rate may be 32.5% (0.325), the IRS won't accept fractions of a penny. If fact, I recall that you're encouraged to round to the nearest dollar, which is quite close enough for their purposes.

Very good sir, you destroyed my analogy. In fact you (and some others) have convinced me to believe. Please see posting below for more details. O

The circle comprises all those points in the chosen plane that are exactly six inches from the centre. Of course it will be very difficult to measure the circumference since it is a line of zero thickness and so a little difficult to see. Yes the circumference is known, to wit Pi feet. There is more than one formula for Pi so you can calculate its length in arabic numerals to whatever degree of precision you require or you can accept that Pi is a precise number for which arabic numerals can only provide an approximation. Over here the Inland Revenue ask us to round all our income to the £ below when filing our tax returns, so my estimates are more precise than theirs.

No! It is not 100% accurate!

You specified 1.0 feet. Your precision is only to one decimal place!!!

Besides, you have proved circles exist! You have been leading us on one!!!

Someone has said "mathematics is a science of inexactitude." I guess you'll just have to live with it!

I wondered what this question was about, now I see what your asking.

Well, I suggest before we consider Pi we first define a circle. There are many ways, but I think the most useful definition in this particular example is a curve with a constant curvature. The reason why I chose that definition is because it is directly related to how many places you take pi out to (3.14 or 3.141 or 3.1415 etc). If you take a circle and look at a particular curved section, it has constant curvature. Take that selected curved section and magnify it 100 times and take a subsection of your magnified curve and it too has constant curvature. You can continue to do so all the way to infinity and each curvature will be the same. There is no straight lines in a circle, and this is why pi is irrational. When you round pi off at a certain decimal place, you are measuring the circumference with a set precision (estimating). Now another way of estimating the circumference of a sphere is to use straight line segments. You take the sphere and enclose it in a triangle. The sum of the lengths of the sides of the triangle are your first approximation for the circumference of the enclosed sphere. Next enclose the circle with a square. The sum of the sides of the square is a closer approximation of the circomference of the sphere than the triangle. By adding a side you have improved your estimate. Next use a Pentagon and you get a better approximation. Next a Hexagon, so on and so forth. What you're doing is using straight line segments (Polyhedrons side length) to estimate a curved surface (just like in calculus). Clearly the smaller the length of the side (more polyhedron sides), the better the estimation.

So you see, Pi is irrational because a circle has constant curvature. You may also find e interesting. e is a curve with constantly changing curvature and is also irrational. So the circumference of a circle of ANY diameter is an irrational number and only approximate circumferences are ever determined, unless left in a form with pi in the answer, which is how mathematicians like to handle your paradox. Of course in the real world there is quantum mechanics to consider, but lets skip that for now.

Thank you for the input. You are right; it is impossible to measure something that has no 'ends' It cannot be constructed from straight lines regardless of how short the segments. However one thing you said confused me; quantum mechanics=real world?

I think you have misunderstood what I was saying. Having no ends has nothing to do with it. It's the fact that it has constant curvature. Constant curvature means that if you were to take the slope of the curve of a circle at any point on the circle, the value would be a constant.

Thanks, I am sure this is elementary to those accustomed to dealing with irrational(or any non real) numbers. I felt that when mathematics makes the leap to non-real it could indicate a problem with the issue, rather I guess it is simply exposing a limitation of real numbers.

The term "real" is a bit arbitrary. Do you know why we use a base ten system (base tem means ten distinct digits to make all numbers: 0,1,2,3,4,5,6,7,8,9)? The reason is because we have ten fingers which is what we used to count. That's it. When new numbers are added to the "Real" number system, it doesn't reflect limitations in math, it reflects our expanding imagination. Irrational, Rational, Imaginary, are just terms we use to categorize numbers we forgot to include when counting was invented.

In truth, we've probably barely scratched the surface.

Why do you insist that the circumference be known in order for the circle to exist? You have assumed a priori that full knowledge of a thing is required for its existence. You've also set up a false dilemma, in that more than one of your proposed conclusions is correct. Regardless of those minor gaps, the information you're looking for is known.

The circumference, of course, is pi feet, which is an exact answer. The fact that you can't write it out precisely in arabic numerals in decimal notation doesn't make it any less exact.

That doesn't mean that circles in general, or your proposed circle in particular, exist, but it does answer your question; thus, by your own statement, you're wrong. Incidentally, circles are also "imaginary," in that they are theoretical geometric constructs. Proof is left as an exercise.

You are right, a lack of precision in our measuring (notation system) does not constitute non-existence. Gaining knowledge is a process of learning your own ignorance.

There are a few ways to find something out. If You make a mistake and write the following: U = ( P * i ) * d You only have a point in the complex plane. You can do: r^2 = x^2 + y^2 to describe the two dimensional Version, but remember from analysis that most of the volume of an N-dimensional sphere is close the surface. From probability try a game: 1. Draw a circle within a square on paper and put it at a wall of Your choice. 2. Throw as many darts as you can with eyes closed in that direction. 3. Do some hit statistics (square vs. circle)and you will have Pi in a precision depending on Your compliance to solve problems scheduled from above, high above mankind. Nature always finds a way to escape the cognition of human beings. Have a nice day Uwe

Moaning All, Can the inacurices of the Pi question reside in the fact the we know not wether we measure the inside of the known diameter coordinates or the out side or indeed the centre line/curve, now I have added a little charcoal to this burning issue "back to work and the real world"

I think he's gone.

Perhaps it was the concept of digital vs analog that had the boy tripping.

Thank you: You win for the best answer-Hope I do not hurt anyone with the experiment:)

Mevel123, You missed an opportunity! You should have made the diameter 1/3 of a foot. (0.333333333333333 ad nauseum)

Do circles exist? Yes. Do circles of any diameter other than 1 foot exist? Yes. By mere inference alone, circles of 1 foot diameter exist. You, yourself wrote: "circumference = Pi * diameter This formula leaves the circumference undefined." I have never seen a purer case of contradiction! By writing (in shorter notation), C = Pi * D, you have EXPLICITLY defined the circumference, and THEN you immediately follow that with the statement: "This formula leaves the circumference undefined!"

Seems as though this is getting more undefined as we go on - division by circle? Multiplication by one (1) is not undefined (identity?), division by zero is! Anyway - Just measure the diameter in inches.

Actually, division by zero is not quite undefined, except for simple arithmetic. The limit of a non-zero number divided by a number as it approaches zero is +/-infinity, thereby giving division by zero a meaning.

TO all: I was wrong, and I will now conceded that circles exist. I will believe in circles without being able to define them accurately. Call it faith. Seriously though, I mis-understood Pi. Is pi unknown solely because of a lack of accuracy of the input variables? Seems kind of silly that people (not me) look for patterns in Pi if the value is wrong to begin with. I always assumed it was calculated based on some other formula describing a circle not requiring physical measurement for input data. I thought the number was indefinite in precision, neither repeating or terminating. Bubble just burst learning that it is not a natural paradox, rather just an example of our physical limitations. Then again, my original belief was that the formula was wrong...requiring the addition of 1 more variable: t (time) Thank you all for playing. O

For an excellent treatise, see Pi at Wikipedia. Pi was defined long, long ago, and its definition has not changed. Methods of making Pi more accurate have advanced, but what the heck, who needs to be more accurate than a gazillionth of a nanometer? I do know that Pi has been calculated to at least 100 decimal places -- I think it's used as an excercise in one-up-man-ship for mathematicians to get more accuracy.

Allow me to interject a little simplicity into all of this: Let's use a real pie.

Cut it into 3 pieces.Each piece is 1/3 or .3333333333333333333333333333333333333333333... ad nauseum of the whole pie.

Each piece is real.The RATIO to the whole will never come out even.

A bubble forms on the beach.It has an absolute diameter, volume, and area.However, the ratio between them is irrational.The bubble does not have to calculate Pi to the last digit, or it could never completely form.

Or did I just muddy the water?

SHP

My response is not an answer to your question, but a question in itself: Does an inch really exist? If I asked you to cut a piece of wood one inch long, could you do it? No you could not. Is a 2x4 block of wood 2 inches by 4 inches? No it's not. There is a fundamental topic which all engineers learn in their freshman year at college. The idea of tolerances must be understood because of this thing you bring up. Everything has a tolerance. Without tolerances, we would not be able to make anything. Some things require tighter tolerances than others, but even the most fundamental parts have limits which cannot be exceeded during manufacturing. So, it may be true that a circle does not exist, but neither does an inch or a foot or anything else for that matter. The only thing that can be done is to define one particular object as representative of that unit of measurement. This is known as a standard. Every thing is made as close to that standard as possible, but there is nothing that can truely reproduce that standard. I guess it comes down to the theory vs. the practical. Theoretically, you are correct, but for practical purposes, a circle can in fact exist if a tolerance is given.

i suppose we are reinventing the circle which is natural to every human in every walk of life thats called Karma!

integer * integer = integer integer * real = real real * real = real real * irrational = irrational If the diameter specified is real then the circumference is irrational Surprised how many people defend the irrational as real :)

( EASIER TO READ )
integer * integer = integer
integer * real = real
real * real = real
real * irrational = irrational

If the diameter specified is real
then the circumference is irrational
Surprised how many people defend the irrational as real :)

note to self...Use PREVIEW BUTTON!

Can a straight line be defined as a segment of a circle with an infinite radius?

If so, then all straight lines have a slight curve to them.Therefore, even a "Straight" line going thru the center of a circle is slightly curved, similar to scribing a circle onto the surface of a sphere,with the "North" pole as the center.The apparant radius is not the "Real"radius. Therefore, straight lines do not exist.

(Just thought I'd stir things up a bit)

S P Idass

You wrote:

"Can a straight line be defined as a segment of a circle with an infinite radius?"

NO, a circle with and infinite radius would have a curvature that APPROACHES a straight line (but still has a very small curvature). This is not the same thing as being a straight line, which is why your argument doesn't work.

Then the curvature would be infinitely small.Can we measure something that is infinitely small? Could we detect an infinitely small curvature?If all of our "straight lines are influenced by the local space-time curvature, where is our reference point.We would have to go to a place with in space time with no matter, which means we could not be there.Therefore, I speculate that "true" straight lines and circles can only exist as an idea, not as reality.

"A segment of a circle with infinite radius APPROACHES a straight line."

Pi approches a rational number the further we calculate it.....but 3 or 4 places is good enough for me.

I agree that a circle is an ideal that can never be reached in the real world. So what. That has nothing to do with pi being irrational. It has nothing to do with measurement. Here are some other things that don't exist in the real world:

Parallel lines, Perpendicular lines, True Ellipses, right triangles, squares, regular hexagon, regular octagon, regular polygons, points, lines, planes, cubes, spheres, cylinders, blah, blah, blah.

Can we measure something that's infinitely small? No. Can we do calculations with infinitely small things, yes, it's called Calculus.

All you are saying is "Can you have perfection in the real world?"

The answer is no. wow!

Pi is not rational to our menial mentality as humans.

But Pi is a real quantity, as evidenced by the very existence of spheres in our universe.

Spheres seem to be the universal shape assumed when all outside forces are eliminated.

We cannot express the ratio of diameter to circumference with our number system,

but it does not matter, bubbles form anyway.

"Not everything that counts can be counted, and not everything that can be counted counts." (Sign hanging in Einstein's office at Princeton)

You don't understand.

Irrational Numbers are Real, by definition they are a subset of the real number system. That doesn't mean real in the way you're talking about. These are just classifications of number systems, not descriptions. From largest to smallest, there are Complex, Real, Irrational, Rational, Integer, and Natural number systems.

If you're interested in understanding all of this, please read this article on Number Systems in Wikipedia.

In a layman's view, which I am, and having no extended education beyond high school, it would seem to me to be an inherent flaw in a number system that cannot express exactly the relationship between diameter and circumference.

The measurement is exact, as witnessed by the fact that bubbles do exist.The ratio cannot be expressed in 10,s or fractions of 10.s.

What would happen if we used the ratio of Diameter and Circumference as a unit of measurement, or use Pi as we use the number 1, 10, etc? it would start out Pi, Pi squared, Pi cubed, etc.?

Instead of Binary(2) Octal(8) Hexadecimal(16) or Decimal(10)

I realize that the binary system is nescessary for our computers, and that Octal and Hex are just shorthand methods of expressing long strings of ones and zeroes.

There are certain decimal numbers, less than 1, that cannot be expressed to a high degree of acccuracy when converted to any of theses systems.

Maybe the same is true with Pi and the decimal system?

Pi seems to be a universal constant, why not use it as a base for a number system?

(Just a curious redneck.)

Circles do not exist. When you draw a line on top of another line does it become "one line" with the other or does it stay "one line on another"? The human eye cannot tell, to us it is one line. Just the same, the human eye cannot tell that a circle is truly a spiral. For a circle to exist, it would have to end exactly where it started, which makes it a dot, which is then technically a sphere.

Spheres exist, circles do not. But try to implement that into the mathematics being taught in school. You won't get away with it. Math likes to pretend is solid, untouchable to philosophy. Math has more to do with practicality than factuality.

No circle (and when i say circle i mean a perfect circle, but is there really another kind?) can truly exist. The definition of a circle is the set of ALL points equidistant from a central point. The problem is with the word all meaning an infinite number of points that are equidistant from the central point make up a circle. My argument would be that if you can make a "perfect circle" i could add in another point to make it a better circle because it contains an infinite number of points so there is no way that you could ever include all of them. Really what people are mean when they are referring to a circle is actually a polygon with many many sides.

Just my 2 cents. Circles dont exist, because you cant calculate Y given any X.

Lot of people say R² = X² + Y² ,but, you then get Y = Sqrt(R²-X²)

No all real numbers have real sqrt's. But, then what about Isosceles triangles?

sqrt(2) is irrational

therefore Isosceles triangles do not exist.

.......

Some ratios cannot be expressed precisely in decimal form.For instance,1/16th.

We can easily divide a circle, or a square, into 16 segments,no problem.But if we convert it to decimal form, the remainders go on forever.

This is because our system bases everything on base 10.

Consider Pi to be a number, in and of itself, a ratio that cannot be converted exactly into decimal.

You can divide a pie into 3 exactly equal pieces, can't you?

Just for argument and amusement, check this out:

Pi is dead!

http://tauday.com/