The area of a circle...

[daepp]

If you must use an irrational number (pi) to compute the area of a circle, does that mean that the area can never be a whole number? Isn't the area finite?

[Bill Verburg]

Originally Posted by DAEpperson

If you must use an irrational number (pi) to compute the area of a circle, does that mean that the area can never be a whole number? Isn't the area finite?

no, arithmetic w/ irrationals can lead to rational results, here one that I have always been intrigued by it has both rational and imaginary #s yet has an integral result
e^(pi*i) = -1 aka Eulers Identity

the thing about irrationals is that they can never be written down explicitly(completely and exactly), they are always symbolized or rounded. But that has nothing to do w/ the results obtained from using them except to the extent that the answer obtained from using a rounded input will itself be rounded

[RWebb]

it is even easier to get a 'rational' result if you live in (KY or Tenn. IIRC) where the legislature voted to set the value of pi equal to exactly 3.0

- that was in the 1920s tho...

[masraum]

Originally Posted by Bill Verburg

no, arithmetic w/ irrationals can lead to rational results, here one that I have always been intrigued by it has both rational and imaginary #s yet has an integral result
e^(pi*i) = -1 aka Eulers Identity

the thing about irrationals is that they can never be written down explicitly(completely and exactly), they are always symbolized or rounded. But that has nothing to do w/ the results obtained from using them except to the extent that the answer obtained from using a rounded input will itself be rounded

I thought the original question was quite interesting, and I was disappointed that I didn't have a good answer. I had an idea, but didn't have any idea of the reason/proof.

Your answer is as interesting as the question.

[jyl]

Interesting, irrational numbers.

Irrational number - Wikipedia, the free encyclopedia

On the OQ, I believe most circles have area that is an irrational number. But not all.

You may read that "all" circles have area that is an irrational number, e.g. here is a blogger at
Plato's Heaven: Can't square a circle
who states "all circles have a circumference and an area which is irrational."

But that seems evidently false to me. Suppose radius r = sqrt ( X / pi ) where X is any rational number. Then area is X = pi * r^2 and is, almost definitionally, a rational number.

Seems like an example of not believing everything you read on the Internet.

Does anyone see something wrong in my logic?

[Bill Verburg]

Originally Posted by jyl

Interesting, irrational numbers.

Irrational number - Wikipedia, the free encyclopedia

On the OQ, I believe most circles have area that is an irrational number. But not all.

You may read that "all" circles have area that is an irrational number, e.g. here is a blogger at
Plato's Heaven: Can't square a circle
who states "all circles have a circumference and an area which is irrational."

But that seems evidently false to me. Suppose radius r = sqrt ( X / pi ) where X is any rational number. Then area is X = pi * r^2 and is, almost definitionally, a rational number.

Seems like an example of not believing everything you read on the Internet.

Does anyone see something wrong in my logic?

you are venturing into an area of number theory that is fascinating
One usually thinks of infinity as being a static thing, but in actuality in the heirarchy of counting the infinity represented by the 3# of irrationals is greater than the infinity that represents the # of rationals

the heirarchy of infinities is represented by the Hebrew letter aleph


counting the # of things is called cardinality if you count all the rational #s the result is . How do you count the # of rationals? easy, just expand on this and add the negatives



interestingly this # is identical to the # of whole #s {..., -3, -2, -1, 0, 1, 2, 3, ...0}

a similar technique can be used to find the cardinality of the irrational #s, First, to simplify things just concentrate on all the irrationals between 0 and 1. Now lets assume that the irrationals are denumberable; this would then mean that there would be some way to list all of them such that we would not leave any number out.


Part of one such example list appears on the diagram above. Since this is clearly an infinte set and since the decimal representation of every irrational must contain an infinte number of no repatitous digits, the diagram really extends indefinitely in both directions. Now take each digit along some diagonal of this list; in this example 0.13497.... This then gives us one of the irrational numbers. But if we now alter every single digit, say by adding 1 (wrapping 9's back to 0's), then we get another irrational number 0.24508.... But this new number can not possibly appear anywhere in our list since it must always have at least one digit with the wrong value no matter which row we may compare it against. We have just produced an irrational number which is not in our list, and therefore our original assumption that the irrational numbers are denumerable must be wrong.

At this point we have discovered some new level of infinity, and infinity which is somehow greater than the infinity of all the integers. For now lets call this new transfinite cardinal number by the symbol . We also have the relation < .


As Cantor proved there is a series of alephs each larger than then previous. This leads one to question whether this series is itself complete or if there may be other levels of infinity which lie between the alephs. The conjecture that there are no intermediate infinities is most famously stated by Cantor's Continuum Hypothesis, expressed by the equation:

[Bill Verburg]

Any set which can be put in a one-to-one correspondence with the natural numbers (or integers) so that a prescription can be given for identifying its members one at a time is called a countably infinite (or denumerably infinite) set. Once one countable set is given, any other set which can be put into a one-to-one correspondence with is also countable. Countably infinite sets have cardinal number aleph-0.

Examples of countable sets include the integers, algebraic numbers, and rational numbers. Georg Cantor showed that the number of real numbers is rigorously larger than a countably infinite set, and the postulate that this number, the so-called "continuum," is equal to aleph-1 is called the continuum hypothesis. Examples of nondenumerable sets include the real, complex, irrational, and transcendental numbers.

[daepp]

Okay Brainiacs - dumb it down just a bit.

Can a circle's area be a finite, rational number?
If the answer is yes, how is this possible if pi is irrational???

[sammyg2]

[sammyg2]

[sammyg2]

Of course it can, but the diameter may be an irrational number.

Just solve backwards. A= pi x R x R
Come up with a rational number for area, just make up a simple easy number like 3.
3= pi x R x R
Divide both sides by pi.

3/pi = r x r

diameter = approx. (.995/r) x 2

I think .........

[TimT]

Its been 25 years or so since I last took any sort of advanced mathematics. Did the whole calculus based engineering program .....

I would say NO, since π (Pi) is irrational...

Occams Razor!

You will have to decide what value of π (Pi) is going to be used.... You could be scratching something out on a scrap of paper and use 3.14 or.. who knows how many places a computer take Pi too??

Interesting question

Also you would think the area is finite...

Say you had to measure the diameter of the circle, then use Pi in the calculation...... You would be using two irrational numbers then, since you can ever really measure to zero tolerance.... (picking nits)

[Icemaster]

Bill, you have obviously forgotten more on this than I will ever know.

[RWebb]

ok, Bill, you did NOT learn that in engineering school!


BTW, Georg Cantor went insane from thinking about the real line...

[Bill Verburg]

Originally Posted by DAEpperson

Okay Brainiacs - dumb it down just a bit.

Can a circle's area be a finite, rational number?
If the answer is yes, how is this possible if pi is irrational???

by definition all rational #s are finite ie they can be expressed as a fraction.

there is no rational # that is not also finite

here's something else to think about, between any 2 rationals #s is an infinite # of irrational #s, the reverse is not true,

[Bill Verburg]

Originally Posted by RWebb

ok, Bill, you did NOT learn that in engineering school!


BTW, Georg Cantor went insane from thinking about the real line...

one of my Masters Degrees is in finite mathematics

[Steve Viegas]

Perhaps I don't understand what has been said up to this point (highly likely) but, if the area of a circle is pi*r^2 then it would seem to me that:

if r= sqrt (1/pi) then the area would be 1...I think.

[Amail]

A circle with radius =1/sqrt(pi), then rē = 1/pi. Area would then be pi/pi, or 1.

So, can you have a circle with radius = 1/sqrt(pi)? You can have a circle bigger than that, and you can have a circle smaller than that.

I remember a paradox about an arrow fired at a target. Before the shot it is distance D from the target. At some point, it will be D/2 from the target. At another point, it is (D/2)/2 from the target. You can continue to cut the remaining distance in half, but you never get to the target. Of course, unless you're me, you will hit the target, so you get past this never ending sequence of halves. Maybe the circle is the same thing - you can't put a real number on it, but you sure as heck can cut a stick to that length.

[sammyg2]

3.141592653589793238462643383279502884197169399375 10582097494459230781640628620899862803482534211706 79821480865132823066470938446095505822317253594081 28481117450284102701938521105559644622948954930381 96442881097566593344612847564823378678316527120190 91456485669234603486104543266482133936072602491412 73724587006606315588174881520920962829254091715364 36789259036001133053054882046652138414695194151160 94330572703657595919530921861173819326117931051185 48074462379962749567351885752724891227938183011949 12983367336244065664308602139494639522473719070217 98609437027705392171762931767523846748184676694051 32000568127145263560827785771342757789609173637178 72146844090122495343014654958537105079227968925892 35420199561121290219608640344181598136297747713099 60518707211349999998372978049951059731732816096318 59502445945534690830264252230825334468503526193118 81710100031378387528865875332083814206171776691473 03598253490428755468731159562863882353787593751957 78185778053217122680661300192787661119590921642019 89380952572010654858632788659361533818279682303019 52035301852968995773622599413891249721775283479131 51557485724245415069595082953311686172785588907509 83817546374649393192550604009277016711390098488240 12858361603563707660104710181942955596198946767837 44944825537977472684710404753464620804668425906949 12933136770289891521047521620569660240580381501935 11253382430035587640247496473263914199272604269922 79678235478163600934172164121992458631503028618297 45557067498385054945885869269956909272107975093029 55321165344987202755960236480665499119881834797753 56636980742654252786255181841757467289097777279380 00816470600161452491921732172147723501414419735685 48161361157352552133475741849468438523323907394143 33454776241686251898356948556209921922218427255025 42568876717904946016534668049886272327917860857843 83827967976681454100953883786360950680064225125205 11739298489608412848862694560424196528502221066118 63067442786220391949450471237137869609563643719172 87467764657573962413890865832645995813390478027590 10

[sammyg2]

Originally Posted by Bill Verburg

one of my Masters Degrees is in finite mathematics

No WONDER he's a moderator.
Although I'd think a masters in child psychology might be more useful .........