Is PI (##\pi##) really a number?

  • #1
dom_quixote
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We know that ##\pi## originates from the L/D relationship of a circumference, where "L" represents the perimeter of a circumference and "D" represents its diameter. The size of a circumference does not matter, as both the perimeter and the diameter of any circumferecence always maintain the same L/D proportionality relationship.


Normally, in engineering problems related to the circumference, ##\pi## is kept until the final result of the calculations, that is: until the moment when a project will be executed. At this time, depending on the precision required, ##\pi## will be reduced to a number with a greater or lesser number of places after the decimal point.


##\pi## is related qualitatively like the L/D ratio of a circumference, but when it needs to be represented quantitatively, it falls into an inevitable trap: it falls into the category of irrational numbers, that is: ##\pi## is never represented numerically in its entirety.


The two best-known counting systems, that is: decimal and binary, are unable to represent ##\pi## in its entirety.


Finally, the question:


Is there a counting system other than the binary or decimal system capable of representing ##\pi## in its entirety?
 
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  • #2
The strict answer to your question is yes, although I don't know what "counting system" implies. For instance the repesentation of pi in radian measure is 1 or in degrees it is 180.
But if you mean typical integer systems then an irratioal number remains irrational no matter which integer one chooses. (Probably true for any rational number choice of base but I don't know offhand.....might be easy to show)
 
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  • #3
dom_quixote said:
Normally, in engineering problems
What use would a full precision number, irrational or not, be to an engineering problem?

For instance, if they're building a bridge and 1/10 mm precision is required, then saying it is a 2 km bridge (exactly) is insufficient for engineering purposes. If it really is that, then it must be stated as a total length of 2.0000000 km, and more zeros is too much.


As for a system that represents Pi exactly, you used in in your OP. The symbol π represents it exactly. This doesn't work for most irrational numbers, and the Greek alphabet is admittedly not a counting system, let alone a number system. Even a value of a third cannot be represented exactly in a binary number on a computer.


hutchphd said:
Probably true for any rational number choice of base but I don't know offhand.....might be easy to show
Indeed, no number represented as a finite number of digits of a rational base can represent any irrational number since any such number can be reduced to a ration of two integers, using nothing but addition, multiplication, finding of least common denominators, and such.
 
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  • #4
dom_quixote said:
The two best-known counting systems, that is: decimal and binary, are unable to represent ##\pi## in its entirety.
You're getting worked up over nothing. How do you represent 1/3 as a decimal number? How about 3/11? ETC.
 
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  • #5
„Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.“ (L. Kronecker, 1823-1891)
(“The integers were made by God, everything else is man-made.”)
 
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  • #6
fresh_42 said:
„Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.“ (L. Kronecker, 1823-1891)
(“The integers were made by God, everything else is man-made.”)
I guess that was before Peano.
 
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  • #7
A relationship of quantities can be represented by a rational number or not. However, only when the relationship is made with a single quantity, the number makes complete sense.

For example:

a) Length A / Length B;

b) Volume A / Volume B;

c) etc.

Note that PI is obtained from a relationship between different things. Although they can be expressed as a number, the quantity "perimeter of a circle" and the quantity "diameter of the circle" are different entities.
 
  • #8
dom_quixote said:
A relationship of quantities can be represented by a rational number or not. However, only when the relationship is made with a single quantity, the number makes complete sense.

For example:

a) Length A / Length B;

b) Volume A / Volume B;

c) etc.

Note that PI is obtained from a relationship between different things. Although they can be expressed as a number, the quantity "perimeter of a circle" and the quantity "diameter of the circle" are different entities.
Why are they different entities? They are both lengths, are they not?
 
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  • #9
martinbn said:
Why are they different entities? They are both lengths, are they not?
Diameter length is a straight line; the circumference perimeter length is a curved line.
 
  • #10
dom_quixote said:
Diameter length is a straight line; the circumference perimeter length is a curved line.
The length of anything is a number.
 
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  • #11
dom_quixote said:
Diameter length is a straight line; the circumference perimeter length is a curved line.
Exactly. They are both lengths. You are way out if left field on this. You are creating a problem in your mind where there is none in reality.
 
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  • #12
martinbn said:
I guess that was before Peano.
1886 in a speech to the Berlin Natural Scientists’ Assembly.

Peano lived from 1858 to 1932 and was 28 at the time. He received his doctorate in 1880 and habilitated in 1884. Means: He could have been in the audience but it is unlikely. It is also unlikely that Kronecker knew much about Peano's work before 1886. So, yes, your guess is probably true. However, I doubt that Kronecker would have changed his polemic if Peano's work had been known to him. Kronecker apparently wasn't a Platonist. But what he definitely was a harsh critic. I once read one and I still feel pity for the author.
 
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  • #13
dom_quixote said:
Although they can be expressed as a number, the quantity "perimeter of a circle" and the quantity "diameter of the circle" are different entities.
No.
A length is a length, whether it is along a straight line or along a curve. Note that when you drive a car, the odometer measures the distance traveled and makes no distinction between driving a straight line or on a road with curves. The house where I grew up was on a street that was a circle three miles in circumference. When you drove around it the odometer worked as you would expect.

In reply to your original question, yes ##\pi## is a number.
 
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  • #14
Given that circles don’t exist in the real world you can stop worrying whether pi exists
 
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  • #15
dom_quixote said:
A relationship of quantities can be represented by a rational number or not. However, only when the relationship is made with a single quantity, the number makes complete sense.

For example:

a) Length A / Length B;

b) Volume A / Volume B;

c) etc.

Note that PI is obtained from a relationship between different things. Although they can be expressed as a number, the quantity "perimeter of a circle" and the quantity "diameter of the circle" are different entities.
This is an ancient view. The Greeks considered numbers, rational numbers, as quotients of lengths they measured as multiples of a given length 1. A lot has been achieved ever since e.g.
https://www.physicsforums.com/insights/counting-to-p-adic-calculus-all-number-systems-that-we-have/
https://www.physicsforums.com/insights/yardsticks-to-metric-tensor-fields/

And as far as I am concerned, I'd say the Greeks' restriction to straightedge and compass constructions is as artificial or as natural as considering limits of Cauchy sequences as literally real numbers.

Really is a philosophical topic. Any scientific view leads us to Cauchy sequences or Dedekind cuts and to yes as the only legitimate answer to your title question. Any other view is historically interesting but in the end philosophy again.
 
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  • #18
dom_quixote said:
##\pi## is never represented numerically in its entirety
Since we're talking engineering here, I would point out that no physical object has a property represented numerically in its entirety.

That bolt? Approximately 3/4 inch thread.
That foot long ruler? Approximately a foot long.
That micron-caliper diameter? Approximately 0.0000001 inches.
 
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  • #19
I've read this thread twice or more, and finally decided to post: the fact is that this discussion makes me think about some quote by L. E. J. Brouwer, and also to ask if it's him's (I mean the cite): "Mathematics are too human".
 
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  • #20
mcastillo356 said:
I've read this thread twice or more, and finally decided to post: the fact is that this discussion makes me think about some quote by L. E. J. Brouwer, and also to ask if it's him's (I mean the cite): "Mathematics are too human".
I haven't found that particular one, but a similar one:
Brouwer said:
Mathematics is nothing more, nothing less, than the exact part of our thinking.
https://www.azquotes.com/author/44864-L_E_J_Brouwer
https://www.goodreads.com/author/quotes/3513841.L_E_J_Brouwer

He said a lot of metaphysical stuff for a topologist who placed great emphasis on mathematical precision.
 
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  • #21
dom_quixote said:
Is there a counting system other than the binary or decimal system capable of representing ∏ in its entirety?
I mean, in a base-pi number system, pi is just 1. :wink:
 
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  • #22
Drakkith said:
I mean, in a base-pi number system, pi is just 1. :wink:
How would that work?
 
  • #23
bob012345 said:
How would that work?
It won't.
 
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  • #24
bob012345 said:
How would that work?
Honestly, I'm not sure. Now that I think about it, I'm not even sure what I wrote is correct. It's something I thought I remembered from a video on non-integer bases. I'd recommend ignoring what I wrote.
 
  • #25
Drakkith said:
Honestly, I'm not sure. Now that I think about it, I'm not even sure what I wrote is correct. It's something I thought I remembered from a video on non-integer bases. I'd recommend ignoring what I wrote.
One can write every real number as ##\sum_{n\in \mathbb{Z}} a_n\pi^n## but the problem is, that the ##a_n## aren't unique. There is no natural system of digits. Classical geometry isn't of help either because you cannot "flatten" ##\pi. ## If we define a unit length ##\pi## then we cannot construct ##1## by straightedge and compass. We need a spiral or something. Things would get complicated quickly.

I would say that ##\pi## is real because of Buffon's needle problem. ##\pi## appears as a result of a real-world experiment.
 
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  • #26
Drakkith said:
Honestly, I'm not sure. Now that I think about it, I'm not even sure what I wrote is correct. It's something I thought I remembered from a video on non-integer bases. I'd recommend ignoring what I wrote.
The system would not work in general but I believe you could at least represent pi which would be 10.
 
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  • #27
fresh_42 said:
I would say that ##\pi## is real because of Buffon's needle problem. ##\pi## appears as a result of a real-world experiment.
Here is another nice appearance:





 
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  • #28
How about just expressing it as the area of a circle with radius ##1##?
 
  • #29
dom_quixote said:
Is there a counting system other than the binary or decimal system capable of representing ##\pi## in its entirety?

##\pi ## is a transcendental number (*), i.e. there is no integer polynomial ##p(x)## such that ##p(\pi)=0.## If on the other hand, there was a number system to the basis ##b\in \mathbb{N}## that could represent ##\pi## in its entirety, then we would have ##\pi= \sum_{k=-n}^m c_k b^k## with ##c_k\in \{0,1,\ldots , b-1\}\subseteq \mathbb{Z}.## We would then have an integer polynomial ##\displaystyle{ p(x)=b^n\cdot \left( \sum_{k=-n}^m c_k b^k \right) -b^n x}## with ##p(\pi)=0## but this does not exists. This means our assumption about such a number system was wrong and ##b\not\in \mathbb{N}.## But what would be the digits if we allow any real number as basis? You would have to invent an entirely new concept and I'm not sure if this can be done without contradictions (see post #25) regarding the fact that nobody ever came up with such a concept. However, PF wouldn't be the place to discuss this either. The only consequence left is that there is no lower bound ##-n## for the ##c_k,## i.e. the sum has infinitely many digits on the right of the point of the representation.

I have now given a formal proof why there is no way to write ##\pi ## completely down in any number system, i.e. I answered the question. The title of this thread asks for a thing that is "really". Such a discussion would be philosophical since "really" has no mathematical meaning. ##\pi \in \mathbb{R}## is literally real from a mathematical point of view, and the philosophical part cannot be discussed at PF.

All questions have been answered so this thread is closed now.



(*) https://people.math.sc.edu/filaseta/gradcourses/Math785/Math785Notes6.pdf
 
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