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Is pi a normal number?

by 7777777
Tags: normal, number
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7777777
#19
Dec19-13, 12:48 PM
P: 18
Quote Quote by jbriggs444 View Post
.... "pi" is a representation of the ideal ratio of the circumference of a circle to its diameter. It is exact.

So possibly what you are saying is that "pi" is not a valid representation and that pi is inexact since all valid representations [using some unspecified notion of what constitutes a valid representation] are inexact.

So you really need to start by defining what it means for a representation to validly represent a number.
I admit that the definition of [itex]\pi[/itex] is exact. But [itex]\pi[/itex] is different than its value, which is always approximate, unless you write down all its infinite digits. Only then you can say that this is
an exact value of [itex]\pi[/itex]. I have never seen exact value of [itex]\pi[/itex], but it does not mean it does not
exist. I told already that I am just making a hypothesis concerning [itex]\pi[/itex] and I may be wrong.
Either [itex]\pi[/itex] has an exact value or it does not. It cannot at the same time both have an
exact value and have it not.

[itex]\pi[/itex]=[itex]\pi[/itex] and this is certainly exact. But [itex]\pi[/itex] is not the value
of [itex]\pi[/itex].
gopher_p
#20
Dec19-13, 01:22 PM
P: 423
Quote Quote by 7777777 View Post
... [itex]\pi[/itex] is different than its value ...
Only then you can say that this is an exact value of [itex]\pi[/itex].
I have never seen exact value of [itex]\pi[/itex] ...
Either [itex]\pi[/itex] has an exact value or it does not.
It cannot at the same time both have an exact value and have it not.
[itex]\pi[/itex] is not the value of [itex]\pi[/itex].
You really need to state precisely what you mean when you use this word.
R136a1
#21
Dec19-13, 01:25 PM
Newcomer
P: 341
Quote Quote by 7777777 View Post
I admit that the definition of [itex]\pi[/itex] is exact. But [itex]\pi[/itex] is different than its value, which is always approximate, unless you write down all its infinite digits. Only then you can say that this is
an exact value of [itex]\pi[/itex]. I have never seen exact value of [itex]\pi[/itex], but it does not mean it does not
exist
True. But note that the following is also true:

I admit that the definition of [itex]1/3[/itex] is exact. But [itex]1/3[/itex] is different than its value, which is always approximate, unless you write down all its infinite digits. Only then you can say that this is
an exact value of [itex]1/3[/itex]. I have never seen exact value of [itex]1/3[/itex], but it does not mean it does not
exist
So ##\pi## is no less exact than ##1/3##. But nobody seems to make a deal out of that...
7777777
#22
Dec21-13, 12:05 AM
P: 18
Quote Quote by gopher_p View Post
You really need to state precisely what you mean when you use this word.
I began to wonder if I did not already state precisely in my above post what did I mean by the value of [itex]\pi[/itex]. So I did some google search with the words "value of pi" and "exact
value of pi" to see what is the reason of the confusion I made. The consensus online seems to be
that "[itex]\pi[/itex] is a transcendental number. It can be proven mathematically that its
exact value can never be written down with digits", direct wiki.answers quote.
I adhere to this consensus and this is what I mean when I use the the words "value of [itex]\pi[/itex]"

Only very few people tend to say or define that the exact value of [itex]\pi[/itex] is [itex]\pi[/itex],
and I think this is reason for the confusion.

There are even those who have arrived at various "proofs" and calculated the exact value of [itex]\pi[/itex]. For example here is a "proof" that
[itex]\pi[/itex] = 17 - 8[itex]\sqrt{3}[/itex] <crackpot link deleted>.
I think there must be some error, wikipedia states that "[itex]\pi[/itex] cannot be expressed
using any combination of rational numbers and square roots or n-th roots".
Mentallic
#23
Dec21-13, 12:48 AM
HW Helper
P: 3,507
Quote Quote by R136a1 View Post
So ##\pi## is no less exact than ##1/3##. But nobody seems to make a deal out of that...
Oh but they have...

Quote Quote by 7777777 View Post
There are even those who have arrived at various "proofs" and calculated the exact value of [itex]\pi[/itex]. For example here is a "proof" that
[itex]\pi[/itex] = 17 - 8[itex]\sqrt{3}[/itex]
<crackpot link deleted>.
I think there must be some error, wikipedia states that "[itex]\pi[/itex] cannot be expressed
using any combination of rational numbers and square roots or n-th roots".
That isn't a proof because the author has made at least one mistake. On page 20, the diagram labelled "1st modification in basic figure" is wrong, hence any conclusions he draws from it are invalid.
gopher_p
#24
Dec21-13, 01:23 AM
P: 423
Quote Quote by 7777777 View Post
I began to wonder if I did not already state precisely in my above post what did I mean by the value of [itex]\pi[/itex].
You'll have to forgive me for not accepting "Exact value is the opposite of an approximate value" as a proper definition.

So I did some google search with the words "value of pi" and "exact value of pi" to see what is the reason of the confusion I made.
I apologize in advance if this comes off as rude or belligerent. That is not my intent. But you really should have some idea about what the words that you use mean. It's fine if your intended meaning turns out to be "wrong" in the sense that it's not the same as the consensus meaning. But if you really mean what you say, then what you say has to have meaning to you.

In other words, I'm asking you what you mean when you use the word value. That's not something that you need Google for. Again, I'm not trying be belligerent here. I'm just trying to understand you.

The consensus online seems to be that "[itex]\pi[/itex] is a transcendental number. It can be proven mathematically that its exact value can never be written down with digits", direct wiki.answers quote. I adhere to this consensus and this is what I mean when I use the the words "value of [itex]\pi[/itex]"
A. ##\pi## is transcendental. That is a fact.
B. No transcendental number has a finite decimal representation. That is a fact.

Neither of these facts help me understand what you mean by value other than (possibly) (1) a number has a value and therefore is not itself a value, (2) a number's value is something different than its decimal expansion, (3) a number's value is something that can be (sometimes/usually/always) written down. Also, this particular quote seems to imply that ##\pi## has an exact value and leaves open the possibility that it can be written down, just not with digits.

Key questions that I still have are (1) do all numbers have a value, (2) how does a number's value differ from the number and its representations (if at all), (3) is a number's value always something that can be written down either in reality or in theory.

Only very few people tend to say or define that the exact value of [itex]\pi[/itex] is [itex]\pi[/itex],
and I think this is reason for the confusion.
What is the exact value of 2? If the exact value of 2 is 2, then why can't the exact value of ##\pi## be ##\pi##?

There are even those who have arrived at various "proofs" and calculated the exact value of [itex]\pi[/itex]. For example here is a "proof" that
[itex]\pi[/itex] = 17 - 8[itex]\sqrt{3}[/itex]
<crackpot link deleted>
I think there must be some error, wikipedia states that "[itex]\pi[/itex] cannot be expressed
using any combination of rational numbers and square roots or n-th roots".
That proof is most definitely bogus. A cursory glance at the front page of that "journal" demonstrates that it is less than legitimate.
CompuChip
#25
Dec21-13, 03:19 AM
Sci Advisor
HW Helper
P: 4,300
Just skimmed the thread, if I get it correctly your definition of an "exact value" is a real number with a terminating expansion in base 10?
lostcauses10x
#26
Dec21-13, 08:36 AM
P: 94
Key questions that I still have are (1) do all numbers have a value, (2) how does a number's value differ from the number and its representations (if at all), (3) is a number's value always something that can be written down either in reality or in theory
Seems again the idea of numbers and what they are is in play here.
7777777
#27
Dec21-13, 10:57 AM
P: 18
Quote Quote by CompuChip View Post
Just skimmed the thread, if I get it correctly your definition of an "exact value" is a real number with a terminating expansion in base 10?
I began this thread by thinking about the possibility that the value of [itex]\pi[/itex] could terminate if it ended up at infinite string of 9's,
and these could be rounded up, or cut. But then I began to suspect that in this case [itex]\pi[/itex]
would no longer be normal. In this way, I made a hypothesis that [itex]\pi[/itex] does not
have an exact value.

It is possible that my hypothesis is wrong, and [itex]\pi[/itex] has an exact value, because it seems
an exact value can also be non-terminating.
7777777
#28
Dec21-13, 11:40 AM
P: 18
Quote Quote by gopher_p View Post
Neither of these facts help me understand what you mean by value other than (possibly) (1) a number has a value and therefore is not itself a value, (2) a number's value is something different than its decimal expansion, (3) a number's value is something that can be (sometimes/usually/always) written down. Also, this particular quote seems to imply that ##\pi## has an exact value and leaves open the possibility that it can be written down, just not with digits.
It is very hard for me to think how can you write down an infinite number of digits. You said that this leaves open the possibility that it can be done just not with digits. Let's think this is done with with the symbol [itex]\pi[/itex]. Does this tell anything about normality of [itex]\pi[/itex]?
To find out if [itex]\pi[/itex] is normal, we should try to write down all its digits and calculate
if all digits appear as frequently. Maybe this is also not possible, I think I am making a mistake,
I am saying that we should try to do something that can't be done.

Quote Quote by gopher_p View Post
Key questions that I still have are (1) do all numbers have a value, (2) how does a number's value differ from the number and its representations (if at all), (3) is a number's value always something that can be written down either in reality or in theory.

What is the exact value of 2? If the exact value of 2 is 2, then why can't the exact value of ##\pi## be ##\pi##?
The exact value of 2 is 2, but there could be room for error. A physicist, for example, could
think of these numbers as approximate, especially because there don't appear any decimals.
2 is exactly the same as 2.0000000000000000000000000000000000000............ and that
is more accurate representation.
So why can't the value of [itex]\pi[/itex] be [itex]\pi[/itex]? Maybe it can, this reminds
me of a joke :
What is a pi?
Physicist: Pi is 3.1415927 plus or minus 0.000000005
Engineer: Pi is about 3.
Mathematician: Pi is Pi.
Mentallic
#29
Dec21-13, 03:30 PM
HW Helper
P: 3,507
I feel like we're going around in circles here.

A rational number can have an infinite number of digits just like 1/3 can, although they would be repeating. Any number such as 0.999... or even 0.12345678999... with an infinite string of 9's at the end does not round up to a rational number. It IS a rational number. For proof of this, see the relevant threads in our FAQ

http://www.physicsforums.com/forumdisplay.php?f=207

In particular, what CompuChip was saying is that while a rational number like 1/3 might not terminate in base 10, it will terminate in other bases such as base 3 and base 9. Every rational number will terminate in an infinite number of bases.

All irrational numbers have an infinite number of digits with no repeating pattern, but they can be both normal and not normal. All normal numbers are irrational.
You were given an example of a non-normal number earlier, and here are examples of normal numbers

http://en.wikipedia.org/wiki/Normal_number#Propertieshttp://en.wikipedia.org/wiki/Normal_number#Properties

These numbers were essentially constructed, as a proof for normality of well known irrationals such as [itex]\sqrt{2}, \pi[/itex] etc. has not been found to date. Being irrational is not enough to ensure the numbers are normal.

As for "exact values". The value [itex]1/\pi[/itex] is irrational, so you'd consider it to be "inexact", correct? Well if you multiply that by [itex]\pi[/itex] which is another "inexact" value, then you get an exact value of 1. Remember that 0.999... = 1 precisely (no rounding whatsoever). So I don't see how it would be possible to multiply two inexact numbers together to get an exact number.
By the way, there is no such thing as an exact value. In Mathematics, all values are exact, even the infinite sums that are representations of [itex]\pi[/itex].

In physics and the real world, our measurements always have an error. We cannot construct a length of exactly [itex]\pi[/itex] units (this has nothing to do with the infinite expansion of [itex]\pi[/itex]) in the same way that we cannot construct a length of exactly 1 unit.
mfb
#30
Dec21-13, 05:40 PM
Mentor
P: 11,576
Quote Quote by Mentallic View Post
I feel like we're going around in circles here.
I agree, and there is no value in keeping the thread open.

Quote Quote by 7777777
To find out if π is normal, we should try to write down all its digits and calculate
if all digits appear as frequently. Maybe this is also not possible, I think I am making a mistake,
I am saying that we should try to do something that can't be done.
As there is no "last digit", it is impossible to write down "all" digits. The first few trillions have been calculated (and they are a strong hint for normality), but there is no way to calculate "all".

If you use mathematical words like "value", please make sure you use them in the same way everyone else does, otherwise it leads to unnecessary confusion.


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