Irrational Pi Defrocked

[Robust]

Introduced on an Australian science forum is a formula showing Euler's highly touted irrational pi to have no greater authority than any of the other numerous pi values of historical reference. The new formula was given as solving a mechanical engineering problem requiring a given radius for the turning of a circular disc of any specified area - given as follows:

 

FORMULA: sqrt area/ sqrt pi ^ = radius; thus, radius * pi = area.

 

Please note change in formula from that originally given.

The formula gives the radius to any circle of a prescribed area - regardless of any and all known pi values as might be applied. What do you think?

 

 

"All things number and harmony." - Pythagoras

[Turtle]

Robust's formula "FORMULA: sqrt area/ sqrt pi ^ *pi = radius."

___the asterisk(presumably indicating multiplication) following the caret (presumably indicating exponentiation) relates pi (presumably a multilicand) to no multiplier. Something seems missing. :cup:

 

___I add, that if you use 22/7 as the approximation for pi & re-write it in base twelve as 1A/7 then do the division, you arrive at a repeating base twelve decimal & not a transcendental base ten decimal.

___Very interesting. :cup:

[Robust]

Turtle, thanks for the input - tho' I don't understand your use of an approximation for pi. In any event it apparently makes no difference what pi value is used, for,if I have it straight, it's root 2 that rules, not pi, the given radius being the same in every individual instance. The example gave an area to the circle of 16 units; thus 4/sqrt pi (any pi value) squared = 2.25 radius, which is the figure looked for, and r*pi = 16 area.

 

"All things number and harmony." - Pythagoras

[Qfwfq]

The formula seems to be the inversion of the traditional one, if only '^ *pi' is left out, but things don't match up and I fail to see the point. It might be helpful to have a link to the Australian page before any further judgement.

 

As far as I know it wasn't Euler that proved pi is irrational, Pythagoras had realized it must be but he didn't exactly like to admit there could be such a thing as an irrational number.

[Bo]

i dont really understand the formula...

it seems to contradict the traditional, and by definition right, A=pir^2...

 

Bo

[Qfwfq]

A=pir^2

The square of pir? :cup:

 

I think a part of the trouble is that it's tricky to write formulae in ASCII. :cup:

[C1ay]

The square of pir? :cup:

 

I think a part of the trouble is that it's tricky to write formulae in ASCII.

 

It looks to me like they trying to say:

 

radius = sqrt(area) / sqrt(pi^2) or IOW

 

The radius is equal to the square root of the area over the square root of pi squared.

 

Of course the square root of any number squared is just the absolute value of that number so I don't see how that's supposed to reduce pi to a rational entity, the square of pi squared is still pi. Then again, sinc this makes no sense at all maybe they're trying to say something else. :cup:

[Qfwfq]

I add, that if you use 22/7 as the approximation for pi & re-write it in base twelve as 1A/7 then do the division, you arrive at a repeating base twelve decimal & not a transcendental base ten decimal.

 

base twelve decimal: I would suggest to call that a twelfthal... :cup:

 

One can certainly find rational values, arbitrarily near pi and periodic, in whichever base one chooses.

 

edit: try for example 3490/1111

[Qfwfq]

Of course the square root of any number squared is just the absolute value of that number so I don't see how...

Strictly, the square root of a number squared can be either + or - the same number but this is a detail.

 

According to my interpretation the formula would be equivalent to the usual one. Your interpretation would give the radius as pi times the normal one apart from sign.

[C1ay]

Strictly, the square root of a number squared can be either + or - the same number but this is a detail.

 

According to my interpretation the formula would be equivalent to the usual one. Your interpretation would give the radius as pi times the normal one apart from sign.

 

How can the square root of any number squared be negative? Isn't the square of any number always positive? Isn't the square root of any positive number positive itself? I was taught that this was exactly what the absolute value function meant. Does the absolute value function actually mean something else?

[Qfwfq]

Isn't the square of any number always positive?

Yes

 

Isn't the square root of any positive number positive itself?

Yes and no:

 

Any number has two square roots, and three cubic roots, and four fourth roots etc...

 

I was taught that this was exactly what the absolute value function meant. Does the absolute value function actually mean something else?

It's not a good definition of absolute value, which means what you mean by it.

[Qfwfq]

Sorry, the first answer, strictly, should be: Yes, if the number is pure real or pure imaginary.

 

edit: it was a moment of delirium, :cup: while typing I switched thinking of a positive square to a real one. Pure imaginary has a real and negative square. :cup:

[Robust]

The point of it all being that the irrational pi is not sacrosanct as claimed to be.The formula gives the correct area to any circular plane regardless of the pi value employed. It furthermore validates that any closed continuum must describe an area by a whole number or ending decimal.

 

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?

 

 

"All things number and harmony." - Pythagoras

[C1ay]

The formula gives the correct area to any circular plane regardless of the pi value employed.

 

No it won't. Are you really trying to say that someone could use 4, 10 or 100 as the value of pi?

[Qfwfq]

The formula gives the correct area to any circular plane regardless of the pi value employed.

Show us how. Like St. Thomas, I shall believe when I shall have passed my fingers through the holes in his hands. :friday:

 

We haven't even been totally sure of what the formula is exactly.

[Robust]

Show us how. Like St. Thomas, I shall believe when I shall have passed my fingers through the holes in his hands. :)

 

We haven't even been totally sure of what the formula is exactly.

 

 

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.

 

sqrt area/sqrt 3.14159265359 pi = 2.25675833419 radius; r^ pi = 16 area;

sqrt area/sqrt 3.14159292035 pi = 2.25675823838 radius; r^ pi = 16 area;

sqrt area/sqrt 3.16049382716 pi = 2.25 radius; r^ pi = 16 area;

sqrt area/sqrt 3.1640625 pi = 2.24873078056 radius; r^ pi = 16 area.

 

"All things number and harmony." - Pythagoras

[Qfwfq]

Now I can see what you are doing, Robust. You are dividing 4 by an arbitrary rational number a, with the square of a being near to pi, and calling the result r. It is quite obvious that r times a will be exactly 4 and nearly as obvious that r-square times a-sqare will be just as exactly 16 (4-square).

 

But what's that got do do with the radius and the circumference of a circle? It will work with any rational value of a, even if a-square is quite different from pi. How about 0.0723?

 

pi = 0.0000006103515625, area = 16, radius = 5120

 

How's that, for a perfect circle? Almost worthy of Giotto!!! :)