Can pi be reduced to a rational number?

[Dennisg]

Just a very wild idea. It starts by defining a "point" as being a Plank length (there is some logic behind this idea) and then using this measur the circumference and to calculate pi by dividing the circle into triangles with the plank length on the circumference. A circle with a radius of one plank length has a pi = 3.0. Pi increases as the radius grows –but will it end with a rational number?

[Kyrisch]

In a circle on a flat plane with radius r and circumference c, [math]\pi=\frac{c}{2r}=3.14159...[/math] The Planck Length is just another number. Substituting the Planck Length for r does nothing to make [math]\pi=3.0[/math].

[big314mp]

His point was that c=planck length, which he approximates to zero (a point).

 

Therefore pi*d=c, becomes pi*d=0, which doesn't make any sense.

 

But make c=3*planck length, then you have: pi*d = 3

 

d must be in integer units of the planck length, therefore, pi=3.

 

At least, I think that is the premise. Correct me if I'm wrong.

 

And as c goes to infinity, the calculated (approximate in this case) value of pi will approach the actual value of pi.

 

Basically this is "quantizing" the circle. As the circumference of the circle goes to infinity, the circle "unquantizes" (Its the same thing as if the units you are dividing the circle into go to zero), which takes you back to the unrounded/quantized/truncated form of pi.

[Dennisg]

His point was that c=planck length, which he approximates to zero (a point).

 

Therefore pi*d=c, becomes pi*d=0, which doesn't make any sense.

 

But make c=3*planck length, then you have: pi*d = 3

 

d must be in integer units of the planck length, therefore, pi=3.

 

At least, I think that is the premise. Correct me if I'm wrong.

 

And as c goes to infinity, the calculated (approximate in this case) value of pi will approach the actual value of pi.

 

Basically this is "quantizing" the circle. As the circumference of the circle goes to infinity, the circle "unquantizes" (Its the same thing as if the units you are dividing the circle into go to zero), which takes you back to the unrounded/quantized/truncated form of pi.

Today 07:27 PM

 

Yes thanks. For a point to actually occupy a location then it must have a dimension - the minimum being a plank length. Anything less than a plank length is meaningless and can not logically be used to define a point. This length then can be used to calculate pi.

[D H]

While the concept of planck scale as a physical limit might well result in interesting mathematics, it will not change mathematics itself. Whether the planck length is a true physical limit is completely irrelevant to mathematics itself because mathematics per se has nothing to do with reality. Show me a one (not one apple, or something that masses one kilogram) in the real world.

 

Closer to this problem, show me a circle in the real world. There are many things in the real world that have a shape that can be described mathematically as circular, but none of these things truly is a circle. The fact that there is no such thing as a true circle in the real world does not stop mathematicians from talking about circles, nor does it stop scientists from modeling things in the real world as circles.

 

Mathematics is not science. The core operating principles of mathematics and science are quite different. The core mathematical principle is proof. Once some mathematical hypothesis is proven true in mathematics it remains true forever. It becomes a mathematical theorem. The core scientific principle is experimentation. While a scientific experiment can discredit a scientific hypothesis, no experiment in the world can prove a scientific hypothesis to be true. Given enough confirming evidence and a strong theoretical basis, a scientific hypothesis becomes a theory. Scientific theories are always provisional in nature: They can be proven false. All it takes is one lousy experiment to do so.

 

For example, suppose some future refinement of the Michelson-Morley experiment shows that the measured speed of light does indeed exhibit some variation as the speed of the observer changes and suppose furthermore that the experimental results are confirmed. This experiment would invalidate much of modern physics; physicists would have to either modify relativity theory or replace it with something else. On the other hand, this experiment would not invalidate the mathematics behind relativity theory. The underlying mathematics of the Lorentz-Fitzgerald contraction would remain valid. They just would no longer be a physically valid mathematical model.

 

Pi is 3.14159265..., period.

Edited August 27, 2008 by D H

[Dennisg]

While the concept of planck scale as a physical limit might well result in interesting mathematics, it will not change mathematics itself. Whether the planck length is a true physical limit is completely irrelevant to mathematics itself because mathematics per se has nothing to do with reality. Show me a one (not one apple, or something that masses one kilogram) in the real world.

 

Closer to this problem, show me a circle in the real world. There are many things in the real world that have a shape that can be described mathematically as circular, but none of these things truly is a circle. The fact that there is no such thing as a true circle in the real world does not stop mathematicians from talking about circles, nor does it stop scientists from modeling things in the real world as circles.

 

Mathematics is not science. The core operating principles of mathematics and science are quite different. The core mathematical principle is proof. Once some mathematical hypothesis is proven true in mathematics it remains true forever. It becomes a mathematical theorem. The core scientific principle is experimentation. While a scientific experiment can discredit a scientific hypothesis, no experiment in the world can prove a scientific hypothesis to be true. Given enough confirming evidence and a strong theoretical basis, a scientific hypothesis becomes a theory. Scientific theories are always provisional in nature: They can be proven false. All it takes is one lousy experiment to do so.

 

For example, suppose some future refinement of the Michelson-Morley experiment shows that the measured speed of light does indeed exhibit some variation as the speed of the observer changes and suppose furthermore that the experimental results are confirmed. This experiment would invalidate much of modern physics; physicists would have to either modify relativity theory or replace it with something else. On the other hand, this experiment would not invalidate the mathematics behind relativity theory. The underlying mathematics of the Lorentz-Fitzgerald contraction would remain valid. They just would no longer be a physically valid mathematical model.

 

Pi is 3.14159265..., period.

 

I agree that shapes such as squares and circles do not naturally occur. What interests me is why pi is an irrational number. This means that the exact area of a circle can never be known. Why is this? We can know the exact area of a square etc but not a circle. Maybe an irrational number is nature’s way of telling us that we have got something basic wrong - in this case the definition of what a "point" is. I think for a point to occupy a location it must have a dimension.

[Sisyphus]

Of course we can know the exact area of a circle. It's pi(r^2). Just because pi is irrational doesn't mean it's indefinite. Nature is not obligated to make things simple, and mathematics is not dependent on nature, anyway. You're mixing up math and physics. Less than a plank length is not mathematically "meaningless," it's just the maximum observable resolution of physical objects, hence physical descriptions on a smaller scale are pointless.

 

Also, a definition can't be "wrong." If you change the definition, you're just defining something else.

 

I think one of my favorite Einstein quotes applies here: "Everything should be made as simple as possible, but not simpler."

[the tree]

Pick a number, any number, it is infinitely more likely that your number is irrational than rational. Rational numbers are everywhere but still less abundant than the space between pages on a tightly packed bookshelf. In short, there is nothing special or interesting about the fact that π is irrational.

 

If you like, Wikipedia has a reasonably neat proof that π is irrational.

[Dennisg]

Of course we can know the exact area of a circle. It's pi(r^2).

 

Yes, but pi is not known exactly thus neither is the area. The boundary of a circle is made up by a set of points of which no line can be formed that includes more than two points. The boundary of a square is made up by points that can be defined by just 4 lines. When the definition of a point is dimensionless then the number of lines that can be derived from a circle in infinite and thus pi become irrational. But if the definition of a point has a finite dimension I am guessing that pi will become rational

[the tree]

Yes, but pi is not known exactly thus neither is the area.

But π is known exactly, it's not rational but it is as exact as any other number.

But if the definition of a point has a finite dimension I am guessing that pi will become rational

If you change the definition of a fork to be a spoon' date=' then all forks wont suddenly become rounded, nor will the colour of the sky change if you switch around the words 'pink' and 'blue'.

What your suggesting is that you approximate a circle with a polygon of a very high degree, and whilst that would be close, it would certainly not be a circle since the radius wouldn't be consistent.

 

Oh and because I really love repeating myself so much, [b']π is exact[/b].

[Sisyphus]

Dennisg, you might want to read a little bit about irrational numbers, since - and I don't mean this in an insulting way - you obviously don't really grasp what they are yet. Check out the Wikipedia article, for one thing. It's not the best, but it's a pretty good place to start.

[Willa]

I agree that shapes such as squares and circles do not naturally occur. What interests me is why pi is an irrational number. This means that the exact area of a circle can never be known. Why is this? We can know the exact area of a square etc but not a circle.

 

 

The side length of a square may very well an irrational number too... then the area of that square might be irrational. At the same time there are circles with rational areas--say the radius is [math]\frac{3}{\sqrt{\pi}}[/math], then the area would just be 9. So, there's no real difference between the rationality of areas of squares vs. circles.

 

 

If you have a square with area 2, then the side length must be [math]\sqrt{2}[/math], which is an irrational number. Would you suggest that this should be a rational number as well?

[Dennisg]

Oh and because I really love repeating myself so much, π is exact.

 

Unless you mean n = n then you have lost me. And if that is what you mean then God help you.

 

Dennisg, you might want to read a little bit about irrational numbers, since - and I don't mean this in an insulting way - you obviously don't really grasp what they are yet. Check out the Wikipedia article, for one thing. It's not the best, but it's a pretty good place to start.

 

Thanks for letting me read this after you were done. I have read it before.

 

If pi is an infinite series of non repeating integers then I don’t think that you can say its value is exactly known – please correct me if I am wrong.

 

There are a couple of points here that I think are interesting and worth discussing.

1. The area of an object in a plane is determined by lines and not points. This being the case then the notion of a “round” circle is fantasy unless the lines are dimensionless objects.

2. The practical out come of pi being irrational is that there are many different values for pi in use. It could be that pi is not a constant but its value varies with the radius of a “circle”.

[Sisyphus]

Unless you mean n = n then you have lost me. And if that is what you mean then God help you.

 

Not n. It's the symbol for pi.

 

If pi is an infinite series of non repeating integers then I don’t think that you can say its value is exactly known – please correct me if I am wrong.

 

Well, yes, you are wrong. The value of pi is pi. Exactly. Just like the value of 1 is 1, or the value of sqrt2 is sqrt2. It's an exact value. There's nothing else to "know." It just happens to be the case that you can't write it as decimal, because it's irrational, but that doesn't make it imprecise.

 

1. The area of an object in a plane is determined by lines and not points.

 

I don't know what this means.

 

This being the case then the notion of a “round” circle is fantasy unless the lines are dimensionless objects.

 

Hence I don't know how you get here. However, you are correct in a way, as circles are indeed "fantasy" in the sense that they are mathematical concepts, and don't correspond to any physical object. I have no idea what you're talking about with lines being "dimensionless objects," though. They're not "objects" at all.

 

The practical out come of pi being irrational is that there are many different values for pi in use.

 

Yes and no. Sometimes it is necessary to approximate pi in decimal form, in which case the "different values" arise from the degree of precision required. When dealing with pure mathematics, however, there is no need to approximate. Pi is simply pi. The area of a circle with radius 2 is not 6.28 (4*3.14), it is simply 4pi.

 

It could be that pi is not a constant but its value varies with the radius of a “circle”.

 

Not in Euclidian geometry it can't.

[Bignose]

If pi is an infinite series of non repeating integers then I don’t think that you can say its value is exactly known – please correct me if I am wrong.

 

OK, consider yourself corrected, because you are wrong.

 

The value of pi is known -- just because it is irrational in no way whatsoever means is cannot be known. It just has an infinite decimal representation.

 

(As much as I am loathe to bring this topic back up)

 

Do you consider 0.999999999..... infinitely repeating 9's, unknown?

 

There are an infinite number of digits after the decimal. It just turns out that the infinite number of digits after the decimal fir pi have no nice patten like 0.99999 or 2/7 or any other rational number.

 

Just because there isn't a pattern doesn't mean we can't find out what those numbers are... because we can. We literally know millions of the digits after the decimal point. There are just an infinite number of them, so it is impossible to find every single number. But, we know the methods to find the digits -- that's completely equivalent to knowing the digits anyway.

 

The big thing is that just because there are an infinite number of digits, doesn't mean that the number is "unknowable".

[ydoaPs]

Not in Euclidian geometry it can't.

 

Unless it's spinning.

[Dennisg]

The big thing is that just because there are an infinite number of digits, doesn't mean that the number is "unknowable

 

I am trying really to understand your logic but you are talking in circles (no pun intended). Your replies sound like an exercise in medieval scholasticism mixed with 20th century psychological “living in denial”. Really. By defining pi = pi you have simply closed your minds to any original and fun thinking on the topic. It sounds absurd for you to say that an infinite series of non repeating integers is knowable.

 

But, we know the methods to find the digits -- that's completely equivalent to knowing the digits anyway.

 

Completely equivalent? Just need to fill in a few intergers and she's sweet:eyebrow: Sloppy thinking at best and self deception at worst.

[Bignose]

Dennis,

 

I think a good analogy would be knowing the definition a word: how to use it properly, how to use it in the right context and all that, but you don't know how to spell it.

 

Does not knowing how to spell it take away from your knowing the word? No. You know what it means and how to use it correctly.

 

To take the analogy further, say you were trying to look up how to spell it, but the dictionary you are using has infinite pages. You know the procedure for looking it up, but you can't ever find the exact word you know because it would take an infinite amount of time.

 

There are algorithms out there than can compute any of the digits of pi you want. You specify "I want the 765,813rd digit" and the algorithm gives it to you. That's completely knowing how to get every single digit. It doesn't matter what digit you ask for, you can get an answer. But, it would take an infinite amount of time to computer an infinite number of digits.

 

Nevertheless, you are missing the bigger point that just because it has an infinite number of digits, doesn't mean we don't know it. It is a number that has many definitions, all equivalent.

 

Again, going to my dictionary/word analogy, if you know someone (say like me and some of my loooong posts) who is unduly prolonged or drawn out, who uses an excess of words, you know exactly what that means. That means, you also know what prolix means. You may have never used/known about the word prolix until just now, but you knew what it meant -- you had a definition for the word prolix, though you didn't know exactly what that word was. In the same way, we have many definitions of pi. We know exactly what it does, what it's properties are, etc. etc., just because we don't know every single digit doesn't mean we don't know it.

[big314mp]

Pi is a ratio. It is no more unknowable than a fraction, such as 22/7. It is a mathematical symbol for circumference divided by diameter. It's actual numerical value, represented in decimal form is irrelevant. For mathematicians, they will use the pi symbol, as that is the most accurate representation. For scientists/engineers, an approximate value of pi is more relevant.

 

It all sounds like semantics, but in a real sense, that's all it is. Pi is a ratio of c to d. That's its definition. It can be approximated in various ways (3.14..., 22/7, etc.), but these are just numerical approximations to a mathematical CONCEPT. They cannot be compared.

 

It is meaningless to say that "we don't know pi out to the last digit, so we don't know it at all." The actual string of numbers comes secondary to the concept of what pi is.

[Dennisg]

There are algorithms out there than can compute any of the digits of pi you want. You specify "I want the 765,813rd digit" and the algorithm gives it to you. That's completely knowing how to get every single digit. It doesn't matter what digit you ask for, you can get an answer. But, it would take an infinite amount of time to computer an infinite number of digits.

 

Thanks for your post. I'm still processing it - this part is the most interesting. It may help - how exactly is pi calculated?

 

Pi is a ratio.

 

Interesting that a ratio is used to calculate area etc. I'm interested in how the ratio is calculated.

[Vts]

Interesting that a ratio is used to calculate area etc. I'm interested in how the ratio is calculated.

 

Actually, if I remember it correctly, 22/7 ratio is the ratio of perimeter of the polygon with 90 straight sides to the maximum distance between the sides.

 

That is, if polygon with 90 sides is circumscribed around a circle with diameter equal to d=7, its perimeter would be equal to 22.00008478.

[Sisyphus]

Actually, if I remember it correctly, 22/7 ratio is the ratio of perimeter of the polygon with 90 straight sides to the maximum distance between the sides.

 

That is, if polygon with 90 sides is circumscribed around a circle with diameter equal to d=7, its perimeter would be equal to 22.00008478.

 

What's your point?

[the tree]

Thanks for your post. I'm still processing it - this part is the most interesting. It may help - how exactly is pi calculated?

There's a few gazillion ways, WP to the rescue again, personally I think [math]2 \sum_{k=0}^{\infty} \frac{k!}{(2k+1)!!}[/math] is the least horrific looking.

[Dennisg]

There's a few gazillion ways, WP to the rescue again, personally I think [math]2 \sum_{k=0}^{\infty} \frac{k!}{(2k+1)!!}[/math] is the least horrific looking.

 

EEK!!!! get away from me!

Can you explain the logic behind such equations. Thanks.

[the tree]

Any of them in particular? None are that easy to explain or prove. Or is it simply that you'd like to know more precisely what the notation means?