Do I take this to mean that there is no way of calculating the area of a rectangle to be mathematically equal to the area of a circle? can this be done?
to square the circle means to construct, using a straightedge and compass, a square with the same area as a circle. you can only construct, with straightedge and compass, rational numbers and their square roots (and square roots of square roots, etc). pi is a transcendental number, therefore it is not constructible, therefore you cannot square the circle.
Umm... Isn't it possible to use a straight edge and a compass to make a line sqrt(2) units long? As far as I can visualize, yes.
phik scalcione, Haven't heard this one since Highschool. A bit dumb now but I will never forget the surprise that we all had when our math teacher told us there was only one way to square a circle: (I was raised in farm country): "Drive a 2 x 2 up a hogs as_".
mathematically it can be figured by a rectangle with on side equal to pi and the other being 1 thus pi R squared.
As I once said in another thread, English is not my mother tongue, please can you explain me the joke?
But if you have a side with size 1, there is no possibility to make the other side with size pi by means of a straight edge and a compass only
I take it that no areaof any circle can be a rational number? then not all areas of rectangles can be equalled by a circular area?
Julixa, ANS: I am not a mathematician but what you say seems to be correct. If deriving the area of a circle uses multiplication by an irrational (non ending number) then the area of a circle is not actually defined and could not be duplicated by a square since the actual number is never derived. Would that be correct Lethe?
Lethe, ANS: Damn. Not being a mathematician I hate having to say I disagree with a mathematician but I feel I must. Your solution is based on an impossible circle. You have no way of knowing what the value of a radius of pi^.5 is. Therefore you can't conclude a known area. What am I missing here? Would it not just become A = pi * ((pi^.5)^2) or pi * pi = pi^2 (still an irrational number)
JAmes R., ANS: Ah, you are correct. But how would that change the ultimate conclusion? A = pi * ((1/pi^.5)^2) = pi/pi = 1. But it is still based on an unknown circle diamater. That is you can state it mathematically but you cannot make a known circle and square it. No circular area it seems can ever be known. You are only speaking hypothetically by making the radius a hypothetical but unknown value. All you are saying is if you have a circle that actually has an area = 1 then you could square that circle. But you have no way of producing such a circle. See my problem?
Why unknown? If you are talking about actually, physically, measuring a circle of radius exactly 1/sqrt(pi), then, yes, that's impossible. But then it is also impossible to measure a distance of exactly 1/3 or, for that matter, 1. We aren't talking about measurements, we are talking about mathematics. One can always posit a length of 1, or 1/3, or pi, or 1/sqrt(pi). "You have no way of knowing what the value of a radius of pi^.5 is." I know exactly what sqrt(pi) is! No, I can't write it out in decimal form but that's not "knowing" the number, that's just knowing a particular representation. Julixa's post asked whether it was impossible to have a circle with rational area. The answer to that is clearly no, one can have a circle of any area- for example, the circle of radius 1/sqrt(pi) has area 1. It is, of course, impossible to construct a line segment of length pi (or sqrt(pi) or, for that matter, cube root of 2). As Lethe said the only numbers that can be constructed with straightedge and compass are "rational numbers and their square roots (and square roots of square roots, etc)". Technically those are called "algebraic of order a power of two".
A lot of people actually miss the point with mathematics. Maths is all about abstraction. In the real world, there are no perfect circles, straight lines extending to infinity, or equilateral triangles. But mathematics doesn't refer directly to the real world; it refers to its own world. It is actually a bit of a mystery as to why it is so damn useful when it comes to solving real-world problems.
