Irrational Pi Defrocked

[C1ay]

OK....employing the given formula requiring the radius for an area of 16 units to the circular plane. I'll employ several different pi values, starting with the irrational pi and followed by that of 355/113 and that followed by the rational value of 256/81. I myself use the finite pi value of 3.1640625.

 

OK, so if we're looking for the area of Earth's orbit and we use the value 93,000,000 for the radius your value of π (3.1640625) yields 27,365,976,562,500,000 sq. miles where as using the real value of π yields 27,171,634,860,898,121 sq. miles. That's a difference of 194,341,701,601,878 sq. miles. Do you think that's some negligible amount?

 

Why would any real mathematician use such a figure that produces errors this large? They would certainly realize that anyone would question the rest of their work if they found even one example such as this. One things for sure, I wouldn't want you balancing my bank statement.

[Qfwfq]

if you live in a manifold that isn't flat, defining the circumference as the ususl locus at distance r from point p, the length c of the circumference won't be exactly 2pir. However, the ratio c/r will depend on r and it will have the lim(r-->0) as being pi. It couldn't very well have only rational values.

 

You might find some wierd rational-valued manifold in which it would be different, I'm not sure, in which the ratio would be rational for every r strictly greater than zero despite the irrational limit. :)

[Robust]

Clay, The area of erath's orbit about what?

 

The questio here simply pertains to the arae of a circualar plane - a closed continuum. The given formula applies equally to all known pi values and giving the same exact results - the irrational pi giving no different result than any other known pi value

 

N o need to make it more than what it is....

[C1ay]

Clay, The area of erath's orbit about what?

 

The questio here simply pertains to the arae of a circualar plane - a closed continuum. The given formula applies equally to all known pi values and giving the same exact results - the irrational pi giving no different result than any other known pi value

 

N o need to make it more than what it is....

 

The Earth's orbit about the sun is a closed circular plane. Please show us how to find this area regerdless of what value of PI is used. I don't think it is possible.

[Qfwfq]

The questio here simply pertains to the arae of a circualar plane - a closed continuum. The given formula applies equally to all known pi values and giving the same exact results - the irrational pi giving no different result than any other known pi value

The exact same results? Exactly the same in which sense?

 

How can different values of radius be the same circle? If you talk about a 'circular plane' then you mean in a flat manifold, so you really can't pick and choose about the value of pi. A trivial algebraic trick will never prove it to be possible. A modified value of pi will only give you something which isn't the radius.

 

The problem originally stated on the Australian forum considered the turning of a circular metal disc to give an area of precisely 16 units. What radius would the mechanical engineer give to his machinist?

Give your machinist the radii that you have calculated using the different pi values and then lay the metallic disks you will get, one on top of the other. You will easily see they have different areas, unless the difference is simply too slight. It's obvious that, if you calculate their areas, using for each one the pi value that you had used for giving that radius, you will get the same area you had started with. But what is the area of each metallic disk?

 

For each one, calculate the circumference, again using that disk's very own pi value, you will not get the same value for all disks. Even if the difference is too small to tell by measuring, calculation with give you different values of circumference although it gives you the same values of area. Try it.

[Robust]

For my sake I was hoping we might keep this topic simple as possible - my not being a maths person persay.

 

We're not dealing here, at least as yet, with the parallax of lines as might be associated with cosmology or quatum physics, but with the simple closed continuum of a circular plane of known area. There is nothing sacrosanct about pi, being merely the ratio of line to arc; i.e., radius Vs radian. Accordingly, dividing radius by radian gives the distance between each angular degree on circumference of the circle. As clearly shown, it is the radius of the circle that is the determinant of its given area - the pi value is inconsequential.

 

Let's put it in closer perspective as relevant to radius/radian relationship (which is all we're really dealing with here) and say that the mechanical engineer requires the turning of a circular disc giving the same area as that of its circumference in degrees (360-degrees). There is but one radius fulfilling the requirement, and it should not be surprising to find it to be twice that of the radian. Check it out - using whatever known pi value you wish.

 

The formula again: sqrt area/sqrt pi ^ = radius. (It is not pi that rules but root 2).

[MortenS]

sqrt area/sqrt pi ^ = radius

 

Is it (sqrt area/sqrt pi)^2 = radius you mean?

or is it

sqrt area/(sqrt pi)^2 = radius you mean?

 

or something completely different...the formulae you wrote makes no sense to me.

 

For your idea to have any practical value, it needs to be able to solve a practical problem, like finding out the area of circle when only the radius is known. How you do this with whatever value of pi, and still get the same area, is just not possible. The value of pi is the ratio of the circumference of the circle, divided on its diameter....you can do this for any circle you make...measure the circumference as precise as you can, then measure the diameter as precise as you can, and you will get pi, for whatever size of the circle. This prove that pi is not any number. It is a constant, and its value is known. Any differences you get, is due to measurement errors.

[maddog]

For my sake I was hoping we might keep this topic simple as possible - my not being a maths person persay.

Difficult, since your challange about PI depends on formulae and mathematics.

 

In fact the Area of Circle A = PI * R^2 where R = Radius of circle

Circumference C = 2 * PI * R

There is nothing sacrosanct about pi, being merely the ratio of line to arc; i.e., radius Vs radian. Accordingly, dividing radius by radian gives the distance between each angular degree on circumference of the circle.

Inaccurate. See above formulae.

Let's put it in closer perspective as relevant to radius/radian relationship (which is all we're really dealing with here) and say that the mechanical engineer requires the turning of a circular disc giving the same area as that of its circumference in degrees (360-degrees). There is but one radius fulfilling the requirement, and it should not be surprising to find it to be twice that of the radian. Check it out - using whatever known pi value you wish.

 

The formula again: sqrt area/sqrt pi ^ = radius. (It is not pi that rules but root 2).

Your math is quite off. From the formulae above:

 

2 * PI * R = PI * R^2

 

PI on both sides cancel, along with one R to equal

 

R = 2 which would be the one answer to your above problem. You just gotta' do the

math to see it. :o

 

Conclusion: This says nothing about whether the state of PI being Irrational or Rational

or otherwise. Specifically, the constant PI is a trancendental number and is neither

Rational or Irrational. This would I think debunk your defrocking of PI.... :o

 

Maddog

[Jay-qu]

it seems like that but when you look at an equation it can have two solutions

 

for example 2^2 = 4 but also (-2)^2 = 4 [coz a -ve times a -ve = +ve]

 

so √4 = + or - 2

 

 

EDIT:: sorry, was replying to something on first page didnt see the the next 3 pages...

[C1ay]

For my sake I was hoping we might keep this topic simple as possible - my not being a maths person persay.

It is simple, pi is an irrational, transcendental value. You are the one that has tried to make it complicated by claiming that it is not.

 

The formula again: sqrt area/sqrt pi ^ = radius. (It is not pi that rules but root 2).

Now back to your equation. Perhaps you should write his out long hand so that we may identify exactly what you are trying to say. It looks as if you are saying, "the square root of the area divided by the square root of pi raised to (some omitted) value equals the radius". That is an invalid or incomplete equation. I think you are simply trying to say that the radius is equal to the square root of the area divided by the square root of pi. Why have you appended an exponentiation operator to this?

 

Perhaps you could illustrate the solution of the radius for an area equal to 10. As a cross check this value should be approximately 1.7841241161527711145389663725651 if your claim is valid.

[Qfwfq]

Not raised to anything C1ay, omit the '^' character and you have it. Robust's argument is a simple tautology and this was cleared up days ago.

 

I have since challenged him to see if the different values of "pi", as well as giving back the same initial area, also give the same circumference. Obviously they don't, the circles can't be the same. Robust replied in a way that makes no sense and that I won't discuss. Who knows what his notions of circle and related things are.

[C1ay]

Not raised to anything C1ay, omit the '^' character and you have it. Robust's argument is a simple tautology and this was cleared up days ago.

 

Yeah, I know. He should be noticing by now that it's his claim that's getting defrocked.

[Robust]

MortenS, I can't believe that my formula is so untoward as not to be quite clear. Let me try to be more succinct about it then - the area to the circular disc still being 16 units.

 

Sqrt 16 = 4; sqrt pi = 1.7724....; 4/sqrt pi = 2.2567; 2.2567....squared = 5.0929 radius; r^*pi = 16 area.

 

Use any other known pi value in place of the irrational pi and you will derive the same answer. The irrational pi is not sacrosanct.

[Robust]

Thanks, Clay. Here it is then:

 

Sqrt 10/ = 3.1622....; sqrt irrational pi = 1.77245....,; sqrt 10/1.77245.... = 1.784124; 1.78124 ^= 3.1830....radius; r^pi = 10 area.

 

The value you give as 1.784124 is correct, the result of dividing sqrt 10 by sqrt of irrational pi. Accordingly, the squaring of that number gives value of the radius squared as 3.1830....; thus r^*pi = 10 area.

 

I have applied the formula to all known pi values (including the finite pi), all of which giving the same exact figure. It is not pi that is the hero of these maths, people, but that of the radius/radian relationship, the pi value being entirely arbitrary.

 

Considering the technological challenges of the times, particularly in the field of astrophysics, there should be no doubts as to the exacting relationship of line to arc. The late Prof. of astronomy, Chas. A Young, remarked that it is not our powers of observation that is at fault but our mathematics.

 

"All things number and harmony." - Pythagoras

[MortenS]

MortenS, I can't believe that my formula is so untoward as not to be quite clear. Let me try to be more succinct about it then - the area to the circular disc still being 16 units.

 

Sqrt 16 = 4; sqrt pi = 1.7724....; 4/sqrt pi = 2.2567; 2.2567....squared = 5.0929 radius; r^*pi = 16 area.

 

Use any other known pi value in place of the irrational pi and you will derive the same answer. The irrational pi is not sacrosanct.

 

I understand every operation you do until:

 

r^*pi

 

I understand r^*pi as radius exponated in pi (or the value of pi you used earlier)

 

since you used the irrational pi, it would be 5.0929^3.1415... = 166.339

 

clearly not the same number you have used...

 

you would have had to use

1.7032 as *pi to get 16 as answer in r^*pi

 

working through your formulae mathematically:

 

Sqrt(Area)/sqrt(pi) = sqrt(radius)

Sqrt(area) = sqrt(radius)*sqrt(pi)

Sqrt(area) = sqrt (radius*pi)

Area = radius * pi

 

This is clearly wrong.....Area is measured in square units (cm2, m2, etc), radius is measured in units (cm, m, etc)

[Robust]

Morten S, clearly I am not the maths person one might desire, but do not see a reason for confusion over the formula. It results in giving a radius to the circle of 2.256758....That radius squared = 5.092958....Multiplied by pi gives the area of 16.

 

The point of it all simply is that one arrives at the the same result regardless of the pi value. Please show me where I might be mistaken in this. I think it to be quite relevant.

[tom]

Your formula is

(( sqrt area / sqrt a ) ^ 2 ) * a = area

 

And this is true for any value of a except 0 :o. So your affirmation is wrong. Here's a demonstration.

 

((sqrt area / sqrt a ) ^2 )*a = area

(sqrt area ) ^2 * a = area * (sqrt a ) ^ 2

area * a = area * a

area = area