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Is pi a normal number? 
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#19
Dec1913, 12:48 PM

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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
Dec1913, 01:22 PM

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#21
Dec1913, 01:25 PM

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#22
Dec2113, 12:05 AM

P: 18

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 nth roots". 


#23
Dec2113, 12:48 AM

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#24
Dec2113, 01:23 AM

P: 435

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. 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. 


#25
Dec2113, 03:19 AM

Sci Advisor
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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
Dec2113, 08:36 AM

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#27
Dec2113, 10:57 AM

P: 18

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 nonterminating. 


#28
Dec2113, 11:40 AM

P: 18

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. 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
Dec2113, 03:30 PM

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P: 3,516

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 nonnormal 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
Dec2113, 05:40 PM

Mentor
P: 11,631

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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