Problems with Pi

Sure it can, at least close enough for engineers. About two decimal places is usually all we need.

If you design bridges or skyscrapers I'm staying the hell out of Kuala Lumpur

If you are using more than two decimal places for pi in a bridge or a skyscraper then you are not really understanding the significance of significant figures....

(as the engineer holds six or eight decimal places on his calculator, because he can, and the welder or fitter on the job uses a measuring tape with 1 mm resolution to build the bridge, while being supported by a bamboo scaffold in the wind as he strikes the arc.)

Design and manufacturing are two very different animals. The designers of the Verrazano Narrows bridge had to take into account the curvature of the earth to make it work. The builders had to worry about losing their lunch. See what I'm saying?

Now you are being irrational.

Yes it can, just never resolved to an infinite number of decimal places (the number pi is after all 'irrational' and 'transcendental', much like the arguments of free energy and over-unity tinkerers).

http://en.wikipedia.org/wiki/Pi

http://en.wikipedia.org/wiki/Circle

Don't get the concept of 'immeasurable' measurement mixed up with 'non-absolute' measurement. If you have a practical application that requires an accurate value of pi then it has been calculated to 1 trillion decimal places.

http://www.super-computing.org/pi_current.html

Jack you are truly "immeasurable" The area of a square is always accurate to the trillionth decimal place by executing the simple formula A=s2, agreed ? what of the humble circle Pi(r2) does not yield a "concrete" value or does it. How many Rand = a Euro?

"How many Rand = a Euro?"

There, you just answered your own question. Why are currency conversion rates NOT taken out to a trillion digits? Because after (say) 12 it simply doesn't matter, not even if Bill Gates cared about that extra penny.

Again have a close look at the links I posted before. They should help you understand the difference between calculating the area of a circle and square (they are both shapes but the theory and calculation behind the circle shape is much more complicated by the shapes very nature!).

http://en.wikipedia.org/wiki/Pi

http://en.wikipedia.org/wiki/Circle

How many Rand = a Euro?

Do you need to know to 1000 decimal places, no because it isn't practical. Standard practice is currency conversion to 4 decimal places and as for the end result, rounded up or down to the nearest practical unit (ie- a single Rand).

Your next CR4 thread question is going to be on fraction division causing an infinite answer (ie- 1/3 = 0.3333333333....) isn't it. Your questions are all along the lines of trying to understand the concept of infinity (be it practical, natural or theoretical).

Jack you are an oasis in a dry and desolate place! Please take a nano second to review my premise! I am simply taking the position of the skeptic, Is it possible to state the area of a circle with the same certainty as that of a rectangle or sq. or triangle or trapezoid? And if not Why Not? Hows the winter?

I understand that you are posting this and other questions as "esoteric" challenges, but the answer is rather obvious, it is not an apples to apples direct comparison as the fundamentals behind the area of a circle involve non-absolute values (by its very nature). The area of any other shape with flat edges is easy to calculate because the area formulas and area involve definite values )and even complex shapes can be broken down into a sum of squares and triangles whose area can be summed together to find the area of the complex shape).

Elementary math + suitable (and practical) rounding = acceptable answer. There is not really any more that I can say that will answer your question (which isn't really esoteric due to is well understood fundamentals and extremely broad distribution and dissemination).

Hows the winter?

Hot and wet (but that bizarre combination is because of the local geography).

Kay, you are confused. Pi is not an 'infinite quantity' and can be measured by anyone with some string and a ruler.

Pi is irrational, that is its numerical representation can be extended indefinitely. The Japanese hold the current record at 2.5 trillion digits.

For engineers it's not how many digits there are that matters but how many digits are "enough" to get a particular job done correctly. There might be some esoteric physics or astronomical experiments out there that require extreme precision (look that word up) but my experience has been that 6 digits of pi is sufficient in electrical engineering.

Anyone have an experience that needed more?

Steve , Thank you for your thoughtful response, but doesn't it support my premise? Your statement seems to suggest "close enough" while the area of a sq.(A=s2) would always be exact.(not merely close enough)

Not merely close enough for what exactly? (This is an engineering forum and we tend to have a practical bent which I suspect is the crux of the biscuit, so to speak.)

Steve, I must apologize as I am not an engineer,I am a carpenter. I enjoy placing "esoteric" challenges on this "wonderful"site These are merely cranial gymnastics and no disrespect to the engineering community is intended. I lived in NY and have no clue as to where your town is located BEST REGARDS K

No apology necessary, we live and breathe esoteric challenges and cranial gymnastics here.

Actually a the area of a square is only as precise as the function of the square of the error in precision of the measurements of the side, which will be far greater than the error in precision of Pi.

the SQ.ROOT of 2 But when would you need an area???????

Alright, so please measure me a square with a side of √2.

And you need areas all the time, for example:

• solar panels for certain output
• cross sectional areas of structural members
• ...
• ..
• .
  

Thank you for your thoughtful response, but I cant help but believe your statement is supporting my premise solar panels for what output exactly? cross sectional areas of structural members? I'll LEAVE THESE FOR YOU TO ELABORATE ON

eg.

• a standard solar panel produces 1800 W/m²; you require 4500 W; how many solar panels do you need?
• you have steel that has a strength of 20,000 psi; you need to support a load of 15,000 lbs; how much cross sectional area of a square bar do you need.

Besides this whole discussion/argument is a moot point.

How accurately can you measure your square (or rectangle, or ....)

• 2"
• 2.0001"
• 2.000000001"
• ...
• ..
• .

And will guess that you will come back with a hypothetical, theoretical situation.

I suggest you find and read up on the topic of "Squaring the circle" - a geometrical puzzle that has been around for quite some time.

You should start with "Squaring the circle": a history of the problem

And then advance into items such as:

http://zakuski.math.utsa.edu/~jagy/papers/Intelligencer_1995.pdf

Thank You for your response,you have certainly given it a strong dose of mental effort! But please allow me to direct you you the basis of the discussion. Is Pi(r2) the Area of a circle an ascertainable value

Yes it is.

Use the Leibniz formula

Now I have to ask you: do you want to keep the discussion theoretical or practical?

Ahhh ........... I give up on waiting for your response.

I was setting you up for this: quantum mechanics.

You only need to use the value of pi on the order of 10³⁴ digits inline with Planck's constant. Any more accuracy of pi would be meaningless

Not a particularly clever comment this but what's to say we can measure the radius of the circle anyway? Even with a "perfect" measurement system the unit of measure itself is arbitrary. If you look at the history of the definition of 1 metre you'll see what I mean, each is based either on some other arbitrary quantity or on a quantity we can only measure to a certain accuracy. Pi is a ratio I know so not subject to unit of measure uncertainty, hence the area of the circle is entirely dependent on the initial accuracy of the radius!

You could also achieve this by a rectangle that measures 1 x 2 ??????

Well, I also have a circle with a radius of 1/(√∏), and voila - I have an area of 1

Riddle me this, then - what would the length of one side of a square equal if you wanted an area of 2?

You could also achieve this by a rectangle that measures 1 x 2 ??????

Imagine this argument if applied to a real world situation - eg - A customer talking to a builder (who just happens to have a cheesy Yugoslav accent for some reason, don't worry he still speaks English). Customer begins the conversation..........

"I asked for square with an area of exactly 2"

"Right, a rectangle with area of exactly 2"

"No, a SQUARE with an area of exactly 2"

"Rectangle with area of exactly 2 is what you get, don't argue is perfect area of 2"

"But it is a RECTANGLE, it won't physically fit my square hole"

"Is rectangle, is better than square"

"Gaaah"

"Tell you what, I give you discount. Better?"

"Your solving the wrong problem"

LOL

Jack, Your intelligence seems to have been usurped by your creativity! Please take a moment to revisit my premise and lose the Yugoslavian Builder Best Regards K

A square is a rectangle, but not all rectangles are squares.

Or was that what you were getting at?

Chas

Kay, the basis of your 'esoteric exploration' assumes Pi to be a quantity, whereas it is a ratio.

Other 'ratios'; Sine, Cos, Tan and hypotenuse run into resolution 'problems' if inputs and/or outputs are not 'rounded' to an appropriate tolerance against scale.

I guess, therefore only squares of finite dimensions (see post 11), not constructed using Trig, are 'truly quantifiable'.

Though small squares are useful, given a perfect plane, finite dimensions and perfect demarcation, big ones are not too good at giving answers outright. For instance, will A = s 2 give the area of a liquid or that of the imaginary tangential plane?

Gotta get in on this

The area of a square, for example is "exact" only if 's' is exact and 's' can never be exact (except in theory) since it is always possible to improve the precision of the measurement. Think, for example, about the length of the coast line of an island. If you measure with a stick the length of a mm you'll measure around every pebble, but if the stick is the length of a meter you'll pass over the stones and find a very different answer.

Concerning money - some time ago a banker type person transfered all fractional cents less than 1/2 to an account in his name. After a while he had thousands of $. You see even fractional cents can matter.

yes you are correct in your analysis if the area of a square = A=s2 and the area of a circle = Pi(r2) And both r & s are refutable quantities (or if you wish indescerinible quantities) than the areas of both polygons are essentially inascertainable Please refer to my original premise Thank You K

Now as I understand it using our common base ten numerical system and mathematics is where ∏ runs out forever. However I have read in several odd publications over the years that other numerical base systems they don't actually have ∏ calculate out as a irrational number.

Is that true?

The reason I ask is I had a college professor with an unhealthy obsession with different base systems who also had brought up examples of how different base systems change how certain ratios have their values represented.

The area of a circle can be calculated without using pi.
I am on a baby laptop at present and did not read all the other posts.

The equation of a circle is (x-h)² + (y-k)² = R²

A GOOZ card for the first solution to calculate a 45degree wedge.

Would that be a circular GOOZ card with area Π in² or a square one with sides equal to √2in?

Pi is not infinite; it is irrational, like the square root of 2.

It can be calculated from an infinite series, though, which is a different thing.

"pi is an infinite quantity"

Pi is a a ratio between is diameter and circumference. We know it lays between say the numbers 3.14159 and 3.14160. Just because we have not been able to compute the decimal part out to a final digit. This does mean the it has an infinite quantity its there.

"therefore the area of a circle can not be ascertained"

A circle has a definite area what pi lets you do is compute it with in reason. If the loss of area in what you are dealing with is significant to the problem we just use a more accurate pi.

Your argument appears sound Let r = Any positive non integer and tell me what the area is? You are a structural engineer I presume Thank you for this well thought out argument

Area/Area = pi/4pi = 1/4

Very clever Roger, but if you substitute anything for "Pi", say "whale" it also comes out ¼. It just proves you used the same whale, of in-determinant value, although a transcendental one.

GOOD FORM, WELL SAID!!

You Wrote:"Very clever Roger, but if you substitute anything for "Pi", say "whale" it also comes out ¼. It just proves you used the same whale, of in-determinant value, although a transcendental one."

Forgive me, but your argument makes no sense. Even you'll acknowledge that Pi is a number. A whale is not. There is no such thing as dividing a whale by a whale, that's basically gibberish.

You are confusing non-terminating with indeterminate. We certainly have exact values of pi. Just because we can't calculate pi's value doesn't change the fact that it has one. That value may have an infinite number of digits, but that value doesn't change. You are mistaking our own limitations for pi's.

If you express an area in terms of pi, such as 4pi or 6pi etc. you have expressed that area exactly.

Roger, my admiration at your cleverness in simply 'cancelling out' Pi algebraically is quite genuine. No offence inferred or intended.

The smiling whilst typing part was at the thought of a transcendental whale.

Naturally I was a bit leery of saying "non-terminating" ref whale fearing an immortality interpretation.

However; Pi is a RATIO.

"no such thing as dividing a whale by a whale" – and why we do it by "x" = "whale". After all, math is a modeling system for just such "gibberish". I could have used "W", but it represents another word/concept, as does "C" (for cetacean), but it's taken too.

"If you express an area in terms of pi, such as 4pi or 6pi etc. you have expressed that area exactly" – (you just don't know the exact answer – in our number system, that is); is nevertheless a difficult very philosophical proposition to refute.

"It has a name – therefore it is (defined)" – I like it, it's up there with that French guy's "Je pense donc je suis".

Your argument, is that pi is a never ending number and therefore can't have an exact value.

.9999999999999...... also is a never ending number, so following your logic, we can't know it's exact value. Since the digits continue on to infinity, it never is fully realized, right?

Except it is. It turns out .999999999...... is exactly equal to one. Not approximately equal to 1, exactly equal to one. Here's the link:

http://en.wikipedia.org/wiki/0.999...

So your premise that a nonterminating number can't ever be fully realized, and thus the area can't ever be completely known, is wrong.

Please, feel free to respond by telling me how clever I am and than disproving the above with what you personally feel makes sense.

I think Pi is a beautiful expression of an infinite sided polygon. It has a definite value. It's the same argument that 1/3+1/3+1/3 <1 rather than =1. Of course .9999....=1. Pi=Pi, no doubt, as an expression of a ratio... carrying it into the nth decimal place is a practice for philosophers(and computers), not mathematicians or engineers.

Good answer.

Thank you Roger for your kind invitation, but my argument is NOT !!!!!; "that pi is a never ending number and therefore can't have an exact value."

If follows that your premise of my premise is also in error.

My "argument", as repeatedly stated; "Pi is a RATIO"

Perhaps I should have added; A (geometrical) ratio is inherently "exact" and A ratio is non-dimensional, but assumed these self evident

The 'problem' here is resolution using a base 10 system, results in a decimal equivalent with a 'yet to be discovered end'. Therefore, this system does not provide a literally exact 'area', merely a more than adequately correct one.

And thanks for the wiki link, and yes I agree; should we ever encounter "99999…" at 'the end' of that longish Pi number we could say the first of those '9's' = one.

we could say the first of those '9's' = one.

Is it so?

I would say, the digid just before first 9 has to be increamented by 1.

Please people, read the link. You are not understanding:

http://en.wikipedia.org/wiki/0.999...

Here's a proof from the link, there is no rounding involved:

You guys need to learn how to listen, please, I know you guys can understand this if you give it a chance.

I like the digit manipulation proof much better Roger, I think it both plays on and resolves the misconception quite well. Let me use 1.99999... (Remember, a.mmm.... is a short hand notation for an infinite series summation. Hmm, did I add one too many periods. )

Let x=1.999..... therefore,

10x=19.999....

10x-x=9x=19.999....-1.999....=18.000...

9x=18.000... → x=2.000... → 1.999....=2.000...

That seems like a good proof* too.

*note to any mathmaticians. I'm sorry I'm throwing around "proof" like this. I know "illustrative example" is probably a better term for what I've done in earlier posts. It just sounds awkward in a post. To be honest, I'm not entirely sure where an illustrative example ends and a proof begins.

your reasoning throughout this discussion has been SOUND!!!! Sound Reasoning is much preferred to Questionable word choice I made an "off the cuff" statement to see what responses i would get. This site is peopled with high quality "THINKERS" I thought a somewhat provocative premise would stimulate discussion... it seems as though i was "almost" right Thank You all for you input K

You wrote:"and yes I agree; should we ever encounter "99999…" at 'the end' of that longish Pi number we could say the first of those '9's' = one"

No, you don't understand, the number .9999.... is equal to one. There is no "end" to the number .999...., however, it is equal to one. The same way 3/3 is equal to one. It's equivalent. Please try actually reading the link this time:

http://en.wikipedia.org/wiki/0.999...

You don't seem to understand that not all numbers end. In fact, most numbers don't. Terminating decimals are the minority, not the majority of numbers. Read the following link. Just the first paragraph where they explain that most numbers are irrational.

http://en.wikipedia.org/wiki/Irrational_number

Pi is an irrational number.

If you read, you will learn, otherwise you are going to be hopelessly encumbered by your misconceptions. You need to educate yourself regarding irrational numbers.

Just a bit off topic, but not too much I hope .

I regularly take some training programs for the induction level engineers. Mostly on different aspects of SQC (Statistical Quality control).

In that aspect one of my questions are "Do arithmetic Exist ?"

The Obviously no, since existence is related to real life and the real life only deals with statistics, ie a value with a probable range distribution (equal or unequal).

This range distribution is what we take as wide or narrow depending upon the characteristics it affects.

On the same logic we ma take the value of pi as lying between 3.14 or 3.15 or 3.141 and 3.142 or 3.14159 and 3.14160 depending on what we want to do with it and which of the final CTQ (Critical to Quality) it affects and the degree to what it does.

The pi is an nonrecurring irrational number (ie a number that can not be expressed as m/n and it doesn't have a pattern in its numbers). But then like any other universal constants G ? that too is irrational number.

Any of these constants are computed as a ration (including powers) of several known value variables and hence each of these do have a margin of error since the variables themselves have (What about modulus of elasticity?)

We as engineers are habituated of working within these uncertainties and hence don't bother about it, bother about the uncertainty that is resulted by it in the final parameter (and that will have much more stronger ones from the measured variables in the equation)

Limitation in human being is the guard of your ability to work. A man's ability to enter into the Universe is maximum up to Moon so far. He is try to go further up. May be he can reach to Mars or Jupiter. But If some one is trying to cross the Universe, he would live life and would not be able to reach till end. The word "Infinite" is used for Universe and is surely immeasurable. The nature has created us to live on earth and not on Milky-Way, hence "He" instilled human qualities and its limitation in us according to the requirement of earthly life. This mean do things only what you can do. Going beyond your limitation is FAILURE. God do not want to see us failed with the given human qualities . Therefore; when finding out area of circle, go up to the maximum decimal figure, which is workable and logically possible . There is no end to Universe or infinite.

After reading all the posts, it appears that much of the argumeent is semantic.

You are right to the extent that it is not possible to give an EXACT value to an irrational number like ∏, which inexactitude will be carried over into any calculation that does not cancel out (as in ∏/∏) such irrational entities.

Bioramani

No, you're wrong.

Since when did math become subjective?

Its astonishing how shameless the ignorance is here sometimes.

Roger-

MATHEMATICS is exact. Unfortunately, reality is not. Therefore, mathematics is a poor model for the real world (but a lot more fun to work with than reality!).

Obviously, but if you read the original post, he's saying that the area of a circle can't be known even in the abstract.

I think there is just a bit of confusion.

The values of constants (pi, G or any other) are exact, there is no doubt about it.

So pi/pi=G/G = 1

(though pi<>G)

The point here is these can not be calculated to exact, since these are the constants that are calculated based on some concepts and ratios, that are thereselves getting affected by the uncertainties. So for the applied science professionals, it is not very critical to calculate these to millionth place of decimal (at least currently) since the uncertainties on tha parameters on which these constants are applied upon will have its uncertainty much more than this.

So if you do calculate these, thein it is only for fun, or to see your name appear in the International Journal of Mathematics or things like that.

Does it have any significance in the real life? and real life here means the applied science.

Say i do calculate the area = πr²,

The r can mbe measured upto ? 7th place of decimal? of course at A⁰ level you are already in the uncertain area.

So what is the uncertainty in the r²? Now do you really want the pi upto 10th degree of decimal?

But still a Pi is a pie . Pi is determinate and not indeterminate.

so pi/pi=1

Is it so confusing?

Look at the other constants

c= ? Wiki says 299792458 m/s but then it says the meter is based to have this value of c.

m itself is havinig now a variability.

The same thing goes about G.

Now look at this link

http://en.wikipedia.org/wiki/Physical_constant

do you see the universal constants having uncertainties (except where we have forced them to have a value- defined ones- and sacrificed the basic unit itself instead). Anyway the basic units themselves have their uncertainties in these cases like m,s,Kg,...

also you may look at these

http://physics.nist.gov/cuu/Constants/Table/allascii.txt

but still for these, since they are determinate, the value divided by itself is 1.

I agree. good answer.

Isn't the exact value of pi 22/7? It only becomes indeterminate if you try to express it in decimal form.

No. PI is not 22/7.

22/7 becomes 3.14285714

and PI in decimal is 3.14159265.

^{Thus 22/7 is very rough approximate figure used for simplicity.}

You can ascertain the area of a circle without consideration of Pi by using the measurement of the circumference times the radius divide by 2. So it could be ascertained exactly as long as you can measure it.

Ok, my faith in cr4 is restored. Many of you have put it much better than I did. Good answer.

Redfred

GA from me also.

Kay, please note: Since Pi is defined in terms of the radius (or dia./2) and the circumference and has an infinite (as so far calculated by computers) number of digits describing it, the calculation of a circle's area can only approach the absolute exact value. This is just like .333333... ad infinitum can only approach the exact value of 1/3. Make sense?

Red This is an excellent response

RED so my statement concerning area of a square A=s2 yields a Qauntity that is inaccurate Due to our inability to measure s

Great reply, Red! Especially for someone from Lon GIsland...

Okay, okay, so Richard Feynman was from Far Rockaway...

After all folks, I managed to note that besides all arguments are valid, you are messing significant digits with decimal digits. No matter how big or small a number is, in engineering, what matters is the number of significant digits.

For instance, if you are dealing with a number like 1E6, no matter what it is, one or two or three decimals make really no meaning in engineering. But, 1,1E6 has some meaning. Its not a decimal place. Its 1E5, or ten percent of the value. The significant digits, from left to right, is what matters.

Remember the time there was no computers and calculations were done by sliding rules? How it was possible? Did things fail? No. Just do the calculations using significative digits and forget what is not worth, Or you will not get your work to n end.

Numerical calculus, I think this was the name of the course in the university.

One can know the area of a circle as accurately as one can know the area of a square or any other geometric construction, because the determination is going to be how accurately one can determine the accuracy of the appropriate measure. Pi is an absolute ratio between the diameter of a circle and its circumference, and does not change. One can measure the radius of a circle as accurately as one can measure the dimension of a square, whether it be to 2 or 5 or 10 decimal places. One can select the accuracy of pi such that a small deviation in the value used is going to have no effect on the accuracy of the calculation of the area.

Listen carefully 'I shall say zees onlee wernce'
You are allowing yourself to get confused by maths.
Maths is a tool which we use to help us...
It is not reality, it is just a convenient way of explaing reality or doing calculations. Why do calculations? To help us understand or make stuff.
Thus if you want to cut something in 3, use a suitable method, don't use decimals which will give 0.333333333333333333333 type answers. Use a different measuring system (A good old fashioned imperial rule marked in 1/12 ths, or fold the paper over into 3 equal lengths then cut it.
How many time do you get a situation where the 'maths' gives a result and then the practical approach shows there was a mistake in the maths ?

It's a tool...if you don't understand it don't use it. If you want to understand it, then read suitable books, search the net etc. I'm not sure this forum is the best place for a coherent concise explanation of the concepts.
Mind, if you enjoy the banter, be my guest, fill you boots.
And if you can understand some of the mad humour then tally ho, chocks away...rein ne va plus. (That's French for 'it's stopped raining!')
Del

"None of this effects the reality of the area enclosed by a circle."

Exceeds GA Del.

When did the philospher wander in to have their esoteric discourses on reality versus mathematics.