Is Pi a Normal Number?

  • Thread starter 7777777
  • Start date
  • Tags
    Normal Pi
In summary, a normal number is a real number whose infinite sequence of digits is uniformly distributed and all digits have the same frequency. It is widely believed that pi is a normal number, but a proof has not yet been found. The conversation also discussed the difference between normal and irrational numbers, and how irrational numbers can also be normal. One person also shared a "proof" that pi is normal based on the concept of an exact value of pi, but this reasoning is not correct. Pi does have an exact value, but it is an infinitely long and non-repeating decimal.
  • #1
7777777
27
0
In mathematics, a normal number is a real number whose infinite sequence of digits is distributed uniformly in the sense that each of the digit values has the same frequency, also all digits are equally likely. Wikipedia says that it is widely believed that pi is normal, but a proof remains elusive.

What about my "proof"? It seems impossible to me that by counting the endless digits of pi we could end up at the following result:
pi=3.1415926......9999999999999999999999999...endless string of 9's

if this would be possible then pi could have an exact value, because we could round up the endless
string of 9's into 100000... And pi would have an exact value, containing a finite number
of decimals, because we could ignore the last 0's, in other words we could cut the decimals of pi
without making an error.

It is rather more likely that pi really is a normal number, and the digits of pi are "random":
http://info.sjc.ox.ac.uk/users/gualtieri/Is Pi normal.htm
 
Mathematics news on Phys.org
  • #2
Pi is a type of number called an irrational number.(In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.)The wikipedia topic about Irrational numbers exactly states that:
The famous mathematical constant pi (π) is among the most well-known irrational numbers and is much-represented in popular culture
 
  • #3
What's the need of poof for pi being a normal number?Pi is already an irrational number.It can't be a normal number.Finding a proof that Pi is not irrational seems absurd
 
  • #4
7777777 said:
What about my "proof"? It seems impossible to me that by counting the endless digits of pi we could end up at the following result:
pi=3.1415926......9999999999999999999999999...endless string of 9's

7777777, you appear to be conflating non-normal with rational. Just because a number is not normal does not mean it is rational. Consider the following base 10 number: 1.01001000100001000001 … This is not rational, but it also is not normal.


adjacent said:
What's the need of poof for pi being a normal number?Pi is already an irrational number.It can't be a normal number.Finding a proof that Pi is not irrational seems absurd
adjacent, you have it exactly backwards. All normal numbers are not rational.
 
  • #5
It is unknown whether ##\pi##, ##e##, or even ##\sqrt{2}## are normal numbers. Almost all real numbers are normal numbers (in the sense that the set of real numbers that are not normal has Lebesgue measure zero), but as far as I know, the only known specific examples are numbers that were specially constructed to be normal.
 
  • #6
D H said:
adjacent, you have it exactly backwards. All normal numbers are not rational.

Err.Sorry I thought normal numbers are something else.
 
  • #7
This number seems to be rational and normal : 0.17586934201758693420...?
 
  • #8
jk22 said:
This number seems to be rational and normal : 0.17586934201758693420...?
That number is not normal. All ten digits do appear in that number with equal probabilities. However, only ten two digit strings appear: 01, 17, 20, 34, 42, 58, 69, 75, 86, and 93. Where are the other ninety two digit strings? For any string length n, a number that is normal in base 10 has all 10n possible n-digit strings appear with equal probabilities.

To be truly normal, a number has to satisfy these properties in all bases. A rational cannot be normal.
 
  • #9
jk22 said:
This number seems to be rational and normal : 0.17586934201758693420...?

What about two-digit combinations? You need every two-digit combinations to occur with equal probability. And then three-digit combinations and so on.
 
  • #10
You've shown that pi does not have a sequence of recurring 9's (ignoring your use of the phrase "round up") which means exactly nothing for your goal.

Furthermore, pi already has an exact value.
 
  • #11
jk22 said:
This number seems to be rational and normal : 0.17586934201758693420...?

The OP only quoted part of the definition of "normal number" from the web page. Maybe that has some connection with his/her other wrong ideas.

Your example would be "normal" using the OP's incomplete definition of course, but that would make "normal numbers" rather uninteresting.
 
  • #12
pwsnafu said:
What about two-digit combinations? You need every two-digit combinations to occur with equal probability. And then three-digit combinations and so on.
Also, a number such as 0.17586934201758693420... will generally lose the property that every digit is equally represented if you express it in some other base. So it fails the one-digit criterion, too.
 
  • #13
jbunniii said:
Also, a number such as 0.17586934201758693420... will generally lose the property that every digit is equally represented if you express it in some other base. So it fails the one-digit criterion, too.

In particular, if you have a pattern that repeats with a n digit cycle in base 10 then in base 10n it will have a one digit cycle. In base 10n - 1 the representation will terminate.

Of course this means that the number is rational. The same manipulation works in bases other than ten.

http://www.wolframalpha.com/input/?i=0.175869342/9999999999
 
  • #14
D H said:
7777777, you appear to be conflating non-normal with rational...

I did not mean to confuse pi with rational numbers. I found an interesting article with several
formulas showing how pi is approximated by rational numbers, for example:
3.14159 < pi < 3.14160
It is an interesting question if there is a limit of how accurately pi can be approximated
in this way.

http://educ.jmu.edu/~lucassk/Papers/more on pi.pdf

What I was trying to say was that my "proof" is based on the silly idea of an exact value of pi, and why it does not exist, thus meaning that pi is a normal number. I am not sure if my reasoning
is correct, though.
 
  • #15
7777777 said:
I did not mean to confuse pi with rational numbers. I found an interesting article with several formulas showing how pi is approximated by rational numbers, for example: 3.14159 < pi < 3.14160
It is an interesting question if there is a limit of how accurately pi can be approximated
in this way.

http://educ.jmu.edu/~lucassk/Papers/more on pi.pdf

What I was trying to say was that my "proof" is based on the silly idea of an exact value of pi, and why it does not exist, thus meaning that pi is a normal number. I am not sure if my reasoning
is correct, though.
What do you mean by "the silly idea of an exact value of pi"? It's a very specific number, so of course it has an exact value. Here's a smallish subset of the infinitely many different ways to represent pi: http://www.wolframalpha.com/input/?i=Representations+of+Pi.

You appear to have two key misconceptions:
  1. That irrational numbers don't have an exact value.
    This is not the case. All that irrational means is that they cannot be represented as the ratio of two integers. It does not means that they don't have an exact value.
  2. That all irrational numbers are normal.
    This too is not the case. I gave a specific example of a non-normal irrational number in post #4.
 
  • #16
7777777 said:
I did not mean to confuse pi with rational numbers. I found an interesting article with several
formulas showing how pi is approximated by rational numbers, for example:
3.14159 < pi < 3.14160
It is an interesting question if there is a limit of how accurately pi can be approximated
in this way.

Pi is a real number. All real numbers are exact in the sense that for any non-zero error bound, there is an approximation which can be written as a decimal fraction and which is accurate to within that error bound.

[Providing that one does not quibble about how many decimal places can be computed within the lifetime of the universe or written down on the contents of the universe. Or quibble about whether the definition for a particular number allows its decimals to be computed at all]

What I was trying to say was that my "proof" is based on the silly idea of an exact value of pi, and why it does not exist, thus meaning that pi is a normal number. I am not sure if my reasoning
is correct, though.

If you want to reason about whether pi has an exact value, you should probably start by defining "exact value".
 
  • #17
jbriggs444 said:
If you want to reason about whether pi has an exact value, you should probably start by defining "exact value".

Exact value is the opposite of an approximate value. You can of course represent pi in many ways, as D H said, but these are just formulas for calculating the approximate values of pi. And using these formulas you will always end up at an approximate value. Because no-one cannot write down an infinite numbers of digits which the exact value of pi has.

Doubtless there are exact formulas for calculating the value of pi. But it is another thing to use
these formulas and arrive at the exact value. I think you will always obtain only an approximate
value. In some sense, these formulas create pi and its value. Maybe these formulas could
also create an exact value, but it takes forever to calculate and write down its value.
The infinite number of digits in pi makes me wonder if it really exists at all. Maybe pi is
just invented by humans. I don't even know if pi is a very useful concept.

My argument was that because pi is a normal number, it cannot have an exact value. I may
be also wrong.
 
  • #18
7777777 said:
Exact value is the opposite of an approximate value. You can of course represent pi in many ways, as D H said, but these are just formulas for calculating the approximate values of pi. And using these formulas you will always end up at an approximate value. Because no-one cannot write down an infinite numbers of digits which the exact value of pi has.

To rephrase what you are saying...

A representation can be inexact if the value that it represents does not match the value that it is supposed to approximate.

A representation can be exact if the value that it represents is equal to the value that it is supposed to approximate.

But in this sense, "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.
 
  • #19
jbriggs444 said:
... "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].
 
  • #20
7777777 said:
... [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.
 
  • #21
7777777 said:
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...
 
  • #22
gopher_p said:
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".
 
Last edited by a moderator:
  • #23
R136a1 said:
So ##\pi## is no less exact than ##1/3##. But nobody seems to make a deal out of that...
Oh but they have...

7777777 said:
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.
 
Last edited by a moderator:
  • #24
7777777 said:
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.
 
Last edited by a moderator:
  • #25
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?
 
  • #26
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.
 
  • #27
CompuChip said:
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.
 
  • #28
gopher_p said:
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.

gopher_p said:
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.
 
  • #29
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

https://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.
 
  • #30
Mentallic said:
I feel like we're going around in circles here.
I agree, and there is no value in keeping the thread open.

7777777 said:
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.
 

1. What is a normal number?

A normal number is a mathematical concept that describes a number in which all digits appear with equal frequency in its decimal representation. In other words, all digits have an equal chance of appearing in any position in the number.

2. Is Pi a normal number?

The answer to this question is currently unknown. While Pi has been calculated to trillions of digits, it has not been proven to be a normal number. However, it is believed to be a normal number based on statistical analysis of its digits.

3. Why is it important to know if Pi is a normal number?

Knowing if Pi is a normal number would provide insight into the distribution of its digits and could potentially have applications in fields such as cryptography and random number generation.

4. Has anyone tried to prove if Pi is a normal number?

Yes, mathematicians have been working on proving the normality of Pi for centuries. However, due to the complexity of the concept and the vast number of digits in Pi, it has not yet been proven.

5. Are there any other numbers that have been proven to be normal?

Yes, there are a few numbers that have been proven to be normal, such as Champernowne's constant and Copeland-Erdős constant. However, these numbers have much simpler decimal representations compared to Pi, making it easier to prove their normality.

Similar threads

Replies
4
Views
253
Replies
4
Views
972
  • General Math
Replies
14
Views
1K
Replies
3
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
687
  • Introductory Physics Homework Help
Replies
8
Views
559
Replies
5
Views
2K
  • General Discussion
Replies
10
Views
1K
Replies
3
Views
1K
  • General Math
Replies
4
Views
1K
Back
Top