PI IS EQUAL TO THREE YOU HOMOPROMOTING DEVIL EVOLUTIONIST

[Sally]

Before I reveal the solution.

Anybody else want to give try at the resolution of the paradox:
Pi has an irrational number cannot be expressed as a ratio
But the definition of Pi is the ratio of circumference/diameter.
Both statements cannot be correct.

[roller24]

isnt 22/7 an expression of a ratio?

[Sally]

Yes. But that is only a rational approximation of Pi.

[Sally]

Solution: An error on my part. The definition should read with the Bold being new

 π is the symbol used in mathematics to represent a constant — the ratio of the circumference of a circle to its diameter.
π is irrational. That means it can't be expressed as a ratio of two integers.

Thus In
π = Circumference/diameter.
If diameter is a rational number then the circumference is irrational.
If circumference is rational then the diameter is irrational.*

So π=c/d. is indeed a ratio but with at least one irrational non integer number.
My bad. I failed to properly define an irrational number.


*The circumference of a circle = diameter × pi. (2πr)

[roller24]

Well a circle will always be relative to any other circle in the ratios contained within each circle. That is pi, and that is the rationale.
I'm guessing you dabble in spirals and the wonders of numbers. Fractals and musical scales. ==Sacred Geometry

[roller24]

Then there's the probability and statistics crowd
You should ask GPTchat some of your riddles. Maybe you can break it.

[Prawn Connery]

Solution: An error on my part. The definition should read with the Bold being new

 π is the symbol used in mathematics to represent a constant — the ratio of the circumference of a circle to its diameter.
π is irrational. That means it can't be expressed as a ratio of two integers.

Thus In
π = Circumference/diameter.
If diameter is a rational number then the circumference is irrational.
If circumference is rational then the diameter is irrational.*

So π=c/d. is indeed a ratio but with at least one irrational non integer number.
My bad. I failed to properly define an irrational number.


*The circumference of a circle = diameter × pi. (2πr)

I disagree. If the rational numbers are infinite, then we simply haven't found the two rational numbers that can be expressed as a ratio of Pi as we still don't know what Pi is in decimal form – it doesn't mean the ratio doesn't exist.

No-one has ever calculated Pi to its last decimal. Perhaps no-one ever will. But that doesn't mean it doesn't end . . . or go on for infinity. If it does go on for infinity, we will never know – so there is no way of proving it doesn't.

An "irrational number" is therefore a number for which a ratio exists that we have yet to discover. It is not irrational at all. I just used a rational argument to define it.

[Sally]

I disagree. If the rational numbers are infinite, then we simply haven't found the two rational numbers that can be expressed as a ratio of Pi as we still don't know what Pi is in decimal form – it doesn't mean the ratio doesn't exist.

No-one has ever calculated Pi to its last decimal. Perhaps no-one ever will. But that doesn't mean it doesn't end . . . or go on for infinity. If it does go on for infinity, we will never know – so there is no way of proving it doesn't.

An "irrational number" is therefore a number for which a ratio exists that we have yet to discover. It is not irrational at all. I just used a rational argument to define it.

Your understanding of infinities is abysmal, I understand now why you cheat on your wife with me for infinities, she'd probably leave you.


While there are infinite number of rationals and lrrationals on the number line, pi itself is not infinite, pi has a discrete place on the number line. pi is not an infinite string of non repeating digits. No matter how far we calculate pi out in decimal form, we are only calculating an approximation and always will. Even if we could represent Pi as in infinite string of digits, it would still only be an approximation of pi, since pi is transcendental. That is we cannot experience it but we can have knowledge of it,...and infinite polynomial sums.. I digress..

Note: decimal representation for other viewers.
3.14285714 Is a notation for a ratio of two integers.
314285714/100000000
Which reduces to 22/7

I'm having trouble writing a mathematical proof out with ascii text, I'm going to have to do it on paper and take a picture. You'll need to have a simple understanding calculus and polynomials. And uses our good friend Proof by Contradiction. Because pi is also transcendental, no proof is 'simple'.

...
There is a saying that the best mathematician is a lazy mathematician. So rather then recreate Niven's proof by hand here is a link to several mathematical proofs that pi is irrational.
https://proofwiki.org/wiki/Pi_is_Irrational

[Prawn Connery]

My understanding of semantics is excellent. Infinity is infinity. Mathematics may define infinities as it pleases, but anything other infinity is not infinity. If one infinity is defined as larger than another (as in Cantor's theory), then the original infinity was never infinite to begin with. Logically, it may appear that some infinities are larger than others. For example:

The list of whole numbers from 1 can be infinite.

The list of decimal numbers between 1 and 2 can be infinite.

So if all the numbers between 1 and 2 are infinite, surely that means there are more of them than the whole numbers from one to infinity. After all, there are infinity decimal numbers for every whole number you can think of. So that's infinity x infinity.

But the beauty of the word "infinity" – indeed, the true (human) concept of "infinity" – means there is no end. For every decimal between 1 and 2, there is another whole number. And on it goes. If you reach the end of one before the other, it either wasn't the end, or it wasn't infinity.

Now you say Pi is an approximation . . . but that's only because we haven't found an exact way to measure it, no? To do so, we would need an infinite number of straight lines to intersect to form a circle (a polygon with infinite sides). Or invent a device that could measure a piece of string to its enth decimal place (how long is a piece of string? No-one knows!).

So in fact, if we could measure infinity, we could also measure Pi to the last decimal place. And once we've done that, then we could find a number short of infinity divided by another number short of infinity that could be expressed as 3 + said decimal places.

Viola! The possibility (if not the reality) of being able to measure Pi surely means there is the same possibility of proving it is a rational number.

As for experiencing Pi (or infinity), perhaps you cannot experience it and I cannot experience it, but does that mean it can't be experienced?

If there are infinite possibilities, does that not mean there are infinite experiences? Transcendental really only refers to transcending the human condition, which is pretty limited to begin with.

[Sally]

As for experiencing Pi (or infinity), perhaps you cannot experience it and I cannot experience it, but does that mean it can't be experienced?

If there are infinite possibilities, does that not mean there are infinite experiences? Transcendental really only refers to transcending the human condition, which is pretty limited to begin with.

Yes, the human condition, it was easier than describing the infinite polynomial, that's what the mathematical definition of it came up with, so that's only why the transcendental handle was used. If we try to quantify it with integers we come up with a contradiction. A particular infinite sums of polynomials doesn't fit anywhere on the number line. If we say it does we get a contradiction.

But we know there is an exact value to the length a circle's circumference. We roll out the circle and there's a length in the real world. Now that we have several proofs that pi is irrational we know why we have been unable measure or mathematically quantify it in the real world because it's also Transcendental number, that is a number that is not algebraic, that is not the root of a non zero polynomial with rational coefficients.

pi is wierd. Or maybe our understanding of math or the real world. Just from trying to exactly measure the circumference of a wheel.

Comes back to do we say a unit circle is one unit diameter or one unit circumference. . Ahh the beautiful poetry of math. Can A perfect Circle exist in the real world?

Anywho. I hope that's not too rambling, I'm interested in discussing the rest of your post later.♫♪ I got to yell at some people on the phone now.