Multiplying pi to get whole numbers

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Multiplying pi to get whole numbers

Postby Twasbrillig » Thu Nov 15, 2007 1:01 am UTC

So I had a math test today and finished early, so I started playing with my calculator.

I started typing in a few random digits, then multiplied them by pi (just tried the one time), and divided by 2.

I got a whole number.

Needless to say, I was shocked. But is that a reasonable response? How likely is it? Are the odds millions to one against, or what?

(The number was 94,613,256. Using pi to 9 decimal places, my calculator gives 148,618,155 when divided by two)
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Re: Multiplying pi to get whole numbers

Postby Mathmagic » Thu Nov 15, 2007 1:24 am UTC

That... is really odd. Theoretically, that shouldn't happen since pi is irrational (ie. can't be expressed as a fraction), so saying that:

(94613256)*pi = 297236310

Is implying that:

pi = 297236310/94613256

Which is obviously not true. It just must be a very, very, *very* good approximation of pi; such a good approximation that both Google AND my TI-83 both just round up to 1 when you divide that fraction by pi. I'm guessing it's accurate enough to pi to have the same value as the amount of digits that the calculator/Google can store.
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Re: Multiplying pi to get whole numbers

Postby Lamil_Lerran » Thu Nov 15, 2007 1:27 am UTC

Since it is impossible to get exactly a whole number by multiplying pi by a rational number, the probability depends entirely on how many extra significant digits your calculator had that it would have shown you if they were non-zero, but didn't because they were zero. (For instance, if your calculator is a scientific calculator that shows 10 digits but cuts off any trailing zeros after a decimal, the probability would be 1/10 (since your answer was 9 digits).)

I am making a few assumptions here (your calculator needs to store more digits than it shows (otherwise the division by 2 can affect things), and I'm assuming a uniform distribution of the decimal component of multiples of pi in the interval [0,1], and I'm making an assumption about how your calculator displays results (for instance, a calculator could theoretically show ".0" at the end of every number it rounded to an integer to display, but internally wasn't an integer.)). However, these assumptions are probably valid or nearly valid.

[Edit: Google and TI-83s round it? That is impressive.]
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Re: Multiplying pi to get whole numbers

Postby shill » Thu Nov 15, 2007 1:29 am UTC

Well, it's 3.1415926537820450867899525622498..., so it's good to, what, 8 digits?
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Re: Multiplying pi to get whole numbers

Postby btilly » Thu Nov 15, 2007 2:05 am UTC

mathmagic wrote:That... is really odd. Theoretically, that shouldn't happen since pi is irrational (ie. can't be expressed as a fraction), so saying that:

(94613256)*pi = 297236310

Is implying that:

pi = 297236310/94613256

Which is obviously not true. It just must be a very, very, *very* good approximation of pi; such a good approximation that both Google AND my TI-83 both just round up to 1 when you divide that fraction by pi. I'm guessing it's accurate enough to pi to have the same value as the amount of digits that the calculator/Google can store.


Given floating point roundoff, it doesn't have to be that good an approximation. You're near the maximum display size anyways.
Code: Select all
$ perl
use Math::BigFloat;
$pi = new Math::BigFloat "3.141592653589793238462643383279502884197169399";
print $pi*94613256/2, "\n";
__END__
148618154.9909052133288675624294648629676
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Re: Multiplying pi to get whole numbers

Postby Mathmagic » Thu Nov 15, 2007 2:10 am UTC

Given floating point roundoff, it doesn't have to be that good an approximation.

I just have no idea how many digits Google or my calculator actually hold. I figured Google would hold enough to not have to round...
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Re: Multiplying pi to get whole numbers

Postby quintopia » Thu Nov 15, 2007 2:17 am UTC

I think the question wasn't answered and I want to know too. What fraction of integers x have pi*x with at least 9 zeros after the decimal?
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Re: Multiplying pi to get whole numbers

Postby btilly » Thu Nov 15, 2007 2:25 am UTC

quintopia wrote:I think the question wasn't answered and I want to know too. What fraction of integers x have pi*x with at least 9 zeros after the decimal?


It didn't have 9 zeros. It had 2 9's after the decimal. Given the 9 digits in front of the decimal, that caused it to display as an integer on his calculator.

The answer with 9 zeros after the decimal, though, should be 1 in a billion. If you want to say that rounded off it should have 9 zeros, it still should be 1 in a billion, though only half of the first set is in the second set.

I say "should" be because I'm too lazy to try to prove that the portion in question is actually well-defined. But if it is, then that's the obvious answer.
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Re: Multiplying pi to get whole numbers

Postby Mouffles » Thu Nov 15, 2007 3:26 am UTC

If you ask google to multiply any 8 digit number greater than about 70,000,000 by pi and divide it by 2, you always get an integer. It only displays 9 significant figures.
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Re: Multiplying pi to get whole numbers

Postby Cosmologicon » Thu Nov 15, 2007 3:26 am UTC

There's about a 2% chance that if you do what the OP did you'll get as close to a whole number. If your calculator only has 9 display digits, though, then it's impossible for it to display a number in the billions as anything but a whole number. If it shows 11 digits, it should not have displayed it as a whole number. So, I don't really get what happened.... Does the OP's calculator display exactly 10 digits?

mathmagic wrote:pi = 297236310/94613256

Which is obviously not true. It just must be a very, very, *very* good approximation of pi; such a good approximation that both Google AND my TI-83 both just round up to 1 when you divide that fraction by pi.

It might be a very very *very* good approximation in absolute terms, but compared with some others it's not so hot. 245850922/78256779 is about 2.5 million times better, and it's shorter.
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Re: Multiplying pi to get whole numbers

Postby btilly » Thu Nov 15, 2007 3:51 am UTC

Cosmologicon wrote:There's about a 2% chance that if you do what the OP did you'll get as close to a whole number. If your calculator only has 9 display digits, though, then it's impossible for it to display a number in the billions as anything but a whole number. If it shows 11 digits, it should not have displayed it as a whole number. So, I don't really get what happened.... Does the OP's calculator display exactly 10 digits?


It could also show 11 digits, but the decimal point takes up a digit on the display. In that case it might still choose to display an integer.
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Re: Multiplying pi to get whole numbers

Postby Twasbrillig » Thu Nov 15, 2007 4:55 am UTC

Cosmologicon wrote:Does the OP's calculator display exactly 10 digits?



Yes, and does my new avatar throw you off by THAT much, Cosmo?
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Re: Multiplying pi to get whole numbers

Postby Mathmagic » Thu Nov 15, 2007 5:02 am UTC

It might be a very very *very* good approximation in absolute terms, but compared with some others it's not so hot. 245850922/78256779 is about 2.5 million times better, and it's shorter.

Didn't say it was the best. :razz:

And like I said, I didn't really take the time to investigate how accurate Google calculator actually was. In hindsight, it's not all *that* spectacular a feat. :)
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Re: Multiplying pi to get whole numbers

Postby Cosmologicon » Thu Nov 15, 2007 5:09 am UTC

Twasbrillig wrote:
Cosmologicon wrote:Does the OP's calculator display exactly 10 digits?

Yes, and does my new avatar throw you off by THAT much, Cosmo?

What? Anyway, if your calculator only displays 10 digits, then any random number with 9 digits left of the decimal that you choose to display has a 1-in-10 chance of appearing as an integer. So, I wouldn't go buying lottery tickets thinking it's your lucky day.
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Re: Multiplying pi to get whole numbers

Postby Twasbrillig » Thu Nov 15, 2007 7:02 am UTC

Cosmologicon wrote:
Twasbrillig wrote:
Cosmologicon wrote:Does the OP's calculator display exactly 10 digits?

Yes, and does my new avatar throw you off by THAT much, Cosmo?

What? Anyway, if your calculator only displays 10 digits, then any random number with 9 digits left of the decimal that you choose to display has a 1-in-10 chance of appearing as an integer. So, I wouldn't go buying lottery tickets thinking it's your lucky day.


I figured there'd be a simple explanation.

Also, I was previously referring to how you called me "OP", even though we've had so many discussions on the fora in the past...
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Re: Multiplying pi to get whole numbers

Postby SimonM » Thu Nov 15, 2007 7:32 am UTC

Most calculators have a display shorter than their rounding length (my one displays 10 remembers 15)

But if you type in more it starts rounding
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Re: Multiplying pi to get whole numbers

Postby btilly » Thu Nov 15, 2007 7:38 am UTC

Twasbrillig wrote:Also, I was previously referring to how you called me "OP", even though we've had so many discussions on the fora in the past...


OP means Original Poster. As in the person who started the thread.

In this discussion you're the OP.
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Re: Multiplying pi to get whole numbers

Postby zenten » Thu Nov 15, 2007 7:09 pm UTC

According to google: (pi * 100 000 000) / 2 = 157 079 633
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Re: Multiplying pi to get whole numbers

Postby 22/7 » Thu Nov 15, 2007 10:49 pm UTC

btilly wrote:OP means Original Poster. As in the person who started the thread.

In this discussion you're the OP.

Yes, but OP tends to get used when you're not thinking of someone specifically, but rather the initial idea presented in the thread. If I'm actually fairly familiar with the person, I'll refer to them using their handle (or some kind of shortened version of it) rather than calling them OP.
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Re: Multiplying pi to get whole numbers

Postby LoopQuantumGravity » Thu Nov 15, 2007 11:01 pm UTC

394372834342725903069943709807632345074473102456264
-----------------------------------------------------------------------
125532772013612015195543173729505082616186012726141

Gives Pi correct to 102 decimals :D.
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Re: Multiplying pi to get whole numbers

Postby btilly » Thu Nov 15, 2007 11:15 pm UTC

22/7 wrote:
btilly wrote:OP means Original Poster. As in the person who started the thread.

In this discussion you're the OP.

Yes, but OP tends to get used when you're not thinking of someone specifically, but rather the initial idea presented in the thread. If I'm actually fairly familiar with the person, I'll refer to them using their handle (or some kind of shortened version of it) rather than calling them OP.


That differs widely by person. Speaking personally, I'll often call someone I know well OP if my intent is to refer back in some way to what the discussion started off being about.

Why would I do that? Because even though I know the original poster, I don't assume that all of the people I am addressing know the original poster, or are paying close enough attention to realize that this was the person who started the conversation.
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Re: Multiplying pi to get whole numbers

Postby shill » Fri Nov 16, 2007 1:54 am UTC

LoopQuantumGravity wrote:394372834342725903069943709807632345074473102456264
-----------------------------------------------------------------------
125532772013612015195543173729505082616186012726141

Gives Pi correct to 102 decimals :D.


This one should be correct to 103 decimals:

314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798
-----------------------------------------------------------------------------------------------------------------------------------------------
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

(Please don't shoot me.)
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Re: Multiplying pi to get whole numbers

Postby 22/7 » Fri Nov 16, 2007 2:33 am UTC

shill wrote:
LoopQuantumGravity wrote:394372834342725903069943709807632345074473102456264
-----------------------------------------------------------------------
125532772013612015195543173729505082616186012726141

Gives Pi correct to 102 decimals :D.


This one should be correct to 103 decimals:

314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798
-----------------------------------------------------------------------------------------------------------------------------------------------
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

(Please don't shoot me.)

That's actually *EXACTLY* what I thought when I saw the "accurate to 102 decimals" thing.
Totally not a hypothetical...

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Re: Multiplying pi to get whole numbers

Postby btilly » Fri Nov 16, 2007 3:23 am UTC

LoopQuantumGravity wrote:394372834342725903069943709807632345074473102456264
-----------------------------------------------------------------------
125532772013612015195543173729505082616186012726141

Gives Pi correct to 102 decimals :D.


You just used a portion of the continued fraction representation, right?

For those who don't understand that comment, every number can be written in the form x0 + 1/(x2 + 1/(x3 + 1/(...))) with the sequence truncating if and only if the number is rational. And if you take a number that is irrational, take a few terms then truncate the sequence, you'll get a rational number that is a surprisingly good approximation.

If you do that for pi, the series starts off 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2,... and the resulting rational approximations you get start off 3, 22/7, 333/106, 333/106, 97258/30959 and so on.

Now the useless party trick is that given enough digits of the decimal representation of a rational number, you can find the continued fraction, truncate at whatever looks like roundoff error, and then you're very likely to be able to find the original decimal.

For instance let's put something like 0.662790697674... in your calculator. The leading term is clearly a 0. Now hit 1/x and get 1.50877192982551. The next term is a 1, so subtract 1. Now hit 1/x and get 1.96551724137565. Subtract 1 and hit 1/x again to get 1.03571428571821. Subtract 1 and hit 1/x to get 27.9999999969234.

So our continued fraction terms were 0, 1, 1, 1, 28. So
Code: Select all
0.662790697674...
  = 0 + 1/(1 + 1/(1 + 1/(1 + 1/28)))
  = 0 + 1/(1 + 1/(1 + 1/(29/28)))
  = 0 + 1/(1 + 1/(1 + 28/29))
  = 0 + 1/(1 + 1/(57/29))
  = 0 + 1/(1 + 29/57)
  = 0 + 1/(86/57)
  = 0 + 57/86
  = 57/86

(Mathematica has a function to convert decimals into fractions. This is how it does it.)
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Re: Multiplying pi to get whole numbers

Postby ShadowLurker » Fri Nov 16, 2007 5:38 am UTC

22/7 wrote:
shill wrote:
LoopQuantumGravity wrote:394372834342725903069943709807632345074473102456264
-----------------------------------------------------------------------
125532772013612015195543173729505082616186012726141

Gives Pi correct to 102 decimals :D.


This one should be correct to 103 decimals:

314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798
-----------------------------------------------------------------------------------------------------------------------------------------------
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

(Please don't shoot me.)

That's actually *EXACTLY* what I thought when I saw the "accurate to 102 decimals" thing.

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Re: Multiplying pi to get whole numbers

Postby SimonM » Fri Nov 16, 2007 7:47 am UTC

22/7 wrote:
shill wrote:
LoopQuantumGravity wrote:394372834342725903069943709807632345074473102456264
-----------------------------------------------------------------------
125532772013612015195543173729505082616186012726141

Gives Pi correct to 102 decimals :D.


This one should be correct to 103 decimals:

314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798
-----------------------------------------------------------------------------------------------------------------------------------------------
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

(Please don't shoot me.)

That's actually *EXACTLY* what I thought when I saw the "accurate to 102 decimals" thing.


Look how many more numbers you'd have to remember though!
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Re: Multiplying pi to get whole numbers

Postby gmalivuk » Fri Nov 16, 2007 7:50 am UTC

SimonM wrote:Look how many more numbers you'd have to remember though!

Not really. The continued fraction one requires remembering
394372834342725903069943709807632345074473102456264/125532772013612015195543173729505082616186012726141
and then computing. Whereas just remembering the digits outright requires remembering
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798
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Re: Multiplying pi to get whole numbers

Postby Twasbrillig » Fri Nov 16, 2007 8:01 am UTC

btilly wrote:
22/7 wrote:
btilly wrote:OP means Original Poster. As in the person who started the thread.

In this discussion you're the OP.

Yes, but OP tends to get used when you're not thinking of someone specifically, but rather the initial idea presented in the thread. If I'm actually fairly familiar with the person, I'll refer to them using their handle (or some kind of shortened version of it) rather than calling them OP.


That differs widely by person. Speaking personally, I'll often call someone I know well OP if my intent is to refer back in some way to what the discussion started off being about.

Why would I do that? Because even though I know the original poster, I don't assume that all of the people I am addressing know the original poster, or are paying close enough attention to realize that this was the person who started the conversation.


We try to talk to each other on a more personal level on the fora, Registered User #17229... I mean btilly. It's just a courtesy thing.

Scrolling up to the top of the page to check whom it is takes so much effort, I know, but that's the price to be paid.
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Re: Multiplying pi to get whole numbers

Postby btilly » Fri Nov 16, 2007 9:01 am UTC

gmalivuk wrote:
SimonM wrote:Look how many more numbers you'd have to remember though!

Not really. The continued fraction one requires remembering
394372834342725903069943709807632345074473102456264/125532772013612015195543173729505082616186012726141
and then computing. Whereas just remembering the digits outright requires remembering
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798


It is no coincidence that those two are almost exactly the same length. Information theory holds that perfectly compressed data looks like perfectly encrypted data looks like a random sequence. Per casual inspection, the digits of pi are random. So unless you do a whole lot of computing (enough to take advantage of the fact that they aren't really random), there is no more efficient way to state the first x digits of pi than to just write them down.

The representation that comes from continued fractions stores the exact same information, and so takes at least the same space. The amazing part is that it does not need more space (if you pick a random denominator then you'll need double). But it does not do better.

Incidentally I found http://planetmath.org/encyclopedia/Cont ... nOfPi.html and verified that the fraction above was indeed done with continued fractions. Here is the list of terms that was used:

3, 7, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1

And here is the program that I used to convert that into the fraction:
Code: Select all
#! /usr/bin/ruby

i = ARGV.map {|x| x.to_i}.reverse

n = 1
m = 0

i.each do |x|
  # Flip numerator and denominator
  n, m = m, n
  # Add x
  n += x*m
end

puts "#{n}/#{m}"
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Re: Multiplying pi to get whole numbers

Postby Cosmologicon » Fri Nov 16, 2007 2:16 pm UTC

Twasbrillig wrote:
btilly wrote:I'll often call someone I know well OP if my intent is to refer back in some way to what the discussion started off being about.

Why would I do that? Because even though I know the original poster, I don't assume that all of the people I am addressing know the original poster, or are paying close enough attention to realize that this was the person who started the conversation.

We try to talk to each other on a more personal level on the fora, Registered User #17229... I mean btilly. It's just a courtesy thing.

It must be an unspoken rule. This is the first I've heard of it. I didn't mean to hurt your feelings.
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Re: Multiplying pi to get whole numbers

Postby gmalivuk » Fri Nov 16, 2007 6:24 pm UTC

Cosmologicon wrote:
Twasbrillig wrote:
btilly wrote:I'll often call someone I know well OP if my intent is to refer back in some way to what the discussion started off being about.

Why would I do that? Because even though I know the original poster, I don't assume that all of the people I am addressing know the original poster, or are paying close enough attention to realize that this was the person who started the conversation.

We try to talk to each other on a more personal level on the fora, Registered User #17229... I mean btilly. It's just a courtesy thing.

It must be an unspoken rule. This is the first I've heard of it. I didn't mean to hurt your feelings.

It's not any kind of rule, spoken or not. No one needs to make sure to check who posted the original topic in order to address them by name.

In fact, that would sometimes be counterproductive. You may know that person well, but if I don't, and I just see "according to what Twasbrillig said", I might have to scroll up through the whole conversation to see what you're talking about. If you just say, "according to the OP", on the other hand, I'll either remember what the original post said, or just go straight back to the top and read it again myself.
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Re: Multiplying pi to get whole numbers

Postby Drone731 » Fri Nov 16, 2007 8:22 pm UTC

Am I missing something here? He said in the original post, "using pi to 9 decimal places", meaning that in his calculator pi is rational. so obviously if you pick the right numbers to multiply by you'll get a whole number.

That said, I don't really understand the question. Are you asking the chances of picking the winning number? In that case, I have no idea.

But maybe I'm just missing something.
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Re: Multiplying pi to get whole numbers

Postby Notch » Mon Nov 19, 2007 1:54 pm UTC

Cosmologicon wrote:It might be a very very *very* good approximation in absolute terms, but compared with some others it's not so hot. 245850922/78256779 is about 2.5 million times better, and it's shorter.


I kinda like 355/113 myself.
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Re: Multiplying pi to get whole numbers

Postby J Spade » Fri Nov 23, 2007 9:56 pm UTC

Well, π • π-1 = 1

…Sorry.
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Re: Multiplying pi to get whole numbers

Postby btilly » Fri Nov 23, 2007 11:11 pm UTC

J Spade wrote:Well, π • π-1 = 1

…Sorry.


That wasn't worth apologizing for. Compare with

I + π = no more π

Now that is apology-worthy!
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Re: Multiplying pi to get whole numbers

Postby Yakk » Fri Nov 23, 2007 11:26 pm UTC

mathmagic wrote:
Given floating point roundoff, it doesn't have to be that good an approximation.

I just have no idea how many digits Google or my calculator actually hold. I figured Google would hold enough to not have to round...


... this is a joke, right?
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Re: Multiplying pi to get whole numbers

Postby J Spade » Sat Nov 24, 2007 12:29 am UTC

btilly wrote:
J Spade wrote:Well, π • π-1 = 1

…Sorry.


That wasn't worth apologizing for. Compare with

I + π = no more π

Now that is apology-worthy!



*groans*
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Re: Multiplying pi to get whole numbers

Postby Mathmagic » Sat Nov 24, 2007 7:29 am UTC

Yakk wrote:
mathmagic wrote:
Given floating point roundoff, it doesn't have to be that good an approximation.

I just have no idea how many digits Google or my calculator actually hold. I figured Google would hold enough to not have to round...


... this is a joke, right?

In what way?

I figured that if an approximation such as the OP's approximation was accurate to x decimal places, I figured that it would have to be a pretty good approximation for Google to just read the answer out as pi. I didn't realize that Google's memory allocation only allowed for n decimal places, where n < x. :)

If it's a crime to not know off-hand the amount of decimal places that Google stores, then I'm guilty as charged.
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Re: Multiplying pi to get whole numbers

Postby SimonM » Sat Nov 24, 2007 7:48 am UTC

mathmagic wrote:
Yakk wrote:
mathmagic wrote:
Given floating point roundoff, it doesn't have to be that good an approximation.

I just have no idea how many digits Google or my calculator actually hold. I figured Google would hold enough to not have to round...


... this is a joke, right?

In what way?

I figured that if an approximation such as the OP's approximation was accurate to x decimal places, I figured that it would have to be a pretty good approximation for Google to just read the answer out as pi. I didn't realize that Google's memory allocation only allowed for n decimal places, where n < x. :)

If it's a crime to not know off-hand the amount of decimal places that Google stores, then I'm guilty as charged.


I think its because it sounded like "Google would hold enough to never have to round..." rather than to round approximations of pi when given as decimals
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Re: Multiplying pi to get whole numbers

Postby Mathmagic » Sat Nov 24, 2007 7:54 am UTC

SimonM wrote:I think its because it sounded like "Google would hold enough to never have to round..."

Haha, yes, I can see how it would be misinterpreted that way.

I guess you'd have to take it within context of my other posts, particularly my first:

mathmagic wrote:I'm guessing it's accurate enough to pi to have the same value as the amount of digits that the calculator/Google can store.
That... is really odd. Theoretically, that shouldn't happen since pi is irrational (ie. can't be expressed as a fraction), so saying that:

(94613256)*pi = 297236310

Is implying that:

pi = 297236310/94613256

Which is obviously not true. It just must be a very, very, *very* good approximation of pi; such a good approximation that both Google AND my TI-83 both just round up to 1 when you divide that fraction by pi. I'm guessing it's accurate enough to pi to have the same value as the amount of digits that the calculator/Google can store.
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