Thread View: uk.education.maths
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Started by Martin Ward
Sun, 16 Mar 1997 00:00
Pi day
Author: Martin Ward
Date: Sun, 16 Mar 1997 00:00
Date: Sun, 16 Mar 1997 00:00
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Pi day I saw a posting in one of the humour groups to the effect that 14 March is Pi Day because 3/14 was a good approximation to pi. Someone had replied that 3/14 was closer to 0.2142857... Any Briton and most other Europeans would consider 14 March as 14/3 any way, and 4.666666... is still not very close to pi. At this point any mathematically literate reader will think as I did: *22 July* should be Pi Day. 22/7 = 3.142857... is the good approximation to pi known since the days of the ancient Greeks. That made me think, what about e Day? This should be 19 July, since 19/7 = 2.714285 is close to e = 2.718281... The problem is to find the best rational approximation to e with denominator no more than 12 (and some constraints on the numerator that create no difficulty). Then there is Phi Day (phi is the so called golden ratio defined by 1/phi = phi - 1), this turns out to be 13 August. A calendar of interesting number days can be constructed, and may be of interest to maths students and of use to maths teachers. It is presented in the table below. The numbers marked * have an entry in "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells - look there if you wonder why pi/sqrt(18) is worth celebrating on 3 April and you do not happen to know about sphere packing. You may want to add more. What other interesting numbers are there in a suitable range? As an exercise sqrt(14), sqrt(15) etc can be found a day each... Some are not unique. I have presented all the possibilities but I think we should accept the canceled form as the day, so i^i Day is 1 May rather than 2 October. The latter could be (i^i Day)'. Martin Ward (5 Day)'' 1997 1/pi 0.3183098862 01-Mar * 2^sqrt(2) 2.6651441427 08-Mar * root 11 3.3166247904 10-Mar lg2(10) 3.3219280949 10-Mar * log of 10 to the base 2 pi/sqrt(18) 0.7404804897 03-Apr * 2^(1/3) 1.2599210499 05-Apr * cube root of two 2*pi 6.2831853072 25-Apr * i to the i 0.2078795764 01-May * i^i = exp(-pi/2) 6/pi^2 0.6079271019 03-May * zeta(3) 1.2020568753 06-May * =1/1^3+1/2^3+1/3^3+... root 13 3.6055512755 18-May 1/pi 0.3183098862 02-Jun * 1/zeta(3) 0.8319073919 05-Jun * =1/(1/1^3+1/2^3+1/3^3+...) 2^sqrt(2) 2.6651441427 16-Jun * root 8 2.8284271247 17-Jun root 10 3.1622776602 19-Jun root 11 3.3166247904 20-Jun lg2(10) 3.3219280949 20-Jun * log of 10 to the base 2 log(e) 0.4342944819 03-Jul * gamma 0.5772156649 04-Jul * e^(1/e) 1.4446678610 10-Jul * Steiner's number e 2.7182818285 19-Jul * pi 3.1415926536 22-Jul * phi-1 0.6180339887 05-Aug * =1/phi pi/sqrt(18) 0.7404804897 06-Aug * 2^(1/3) 1.2599210499 10-Aug * cube root of two phi 1.6180339887 13-Aug * phi^2 2.6180339887 21-Aug * Liouville's 0.1100010000 01-Sep * 1/pi 0.3183098862 03-Sep * root pi 1.7724538509 16-Sep * =gamma(1/2) root 5 2.2360679775 20-Sep * root 6 2.4494897428 22-Sep 2^sqrt(2) 2.6651441427 24-Sep * root 11 3.3166247904 30-Sep lg2(10) 3.3219280949 30-Sep * log of 10 to the base 2 i to the i 0.2078795764 02-Oct * i^i = exp(-pi/2) log(2) 0.3010299957 03-Oct * 6/pi^2 0.6079271019 06-Oct * ln(2) 0.6931471806 07-Oct 1/sqrt(2) 0.7071067812 07-Oct * =sqrt(2)/2 zeta(3) 1.2020568753 12-Oct * =1/1^3+1/2^3+1/3^3+... Brun's 1.9019500000 19-Oct * Brun's constant ln(10) 2.3025850930 23-Oct * 1/e 0.3678794412 04-Nov * pi/2sqrt(3) 0.9068996821 10-Nov * pi^2/6 1.6449340668 18-Nov * =zeta(2) root 3 1.7320508076 19-Nov * root 7 2.6457513111 29-Nov 1/pi 0.3183098862 04-Dec * pi/sqrt(18) 0.7404804897 09-Dec * 1/zeta(3) 2.2020568753 10-Dec * =1/(1/1^3+1/2^3+1/3^3+...) pi^4/90 1.0823232337 13-Dec * =1/1^4+1/2^4+1/3^4+... 2^(1/3) 1.2599210499 15-Dec * cube root of two root 2 1.4142135624 17-Dec * -- Martin Ward _________________________________________________________________________ "When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean - neither more nor less". "The question is", said Alice, "whether you *can* make words mean so many different things". "The question is", said Humpty-Dumpty, "which is to be master - that's all". "Through the Looking-Glass" - Lewis Carroll. _________________________________________________________________________ Turnpike evaluation. For information, see http://www.turnpike.com/
Re: Pi day
Author: Scott Phung
Date: Mon, 17 Mar 1997 00:00
Date: Mon, 17 Mar 1997 00:00
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Martin Ward <martin@ward-hull.demon.co.uk> wrote: MW> Pi day MW> MW> I saw a posting in one of the humour groups to the effect that 14 March MW> is Pi Day because 3/14 was a good approximation to pi. Someone had MW> replied that 3/14 was closer to 0.2142857... They probably meant 3.14, not 3/14. MW> Any Briton and most other Europeans would consider 14 March as 14/3 any MW> way, and 4.666666... is still not very close to pi. MW> At this point any mathematically literate reader will think as I did: MW> *22 July* should be Pi Day. 22/7 = 3.142857... is the good MW> approximation to pi known since the days of the ancient Greeks. It's the best approximation with a denomiator <= 12. MW> That made me think, what about e Day? This should be 19 July, since MW> 19/7 = 2.714285 is close to e = 2.718281... The problem is to find the MW> best rational approximation to e with denominator no more than 12 (and MW> some constraints on the numerator that create no difficulty). Uhm, the best thing you can do is to do continued fractions. I forgot how to do continued fractions off my head, but it'll give you sucessive *BEST* approximations with increasing denimators. Look in a number theory book. MW> Then there is Phi Day (phi is the so called golden ratio defined by MW> 1/phi = phi - 1), this turns out to be 13 August. MW> MW> Some are not unique. I have presented all the possibilities but I think MW> we should accept the canceled form as the day, so i^i Day is 1 May MW> rather than 2 October. The latter could be (i^i Day)'. Isn't it ironic how the imaginary number raised to the imaginary power equals a real number? (Involving the important constants pi, and e) :-) MW> Martin Ward MW> (5 Day)'' 1997 MW> <snip> (Impressive list, by the way) MW> Martin Ward Scott Phung
Re: Pi day
Author: Hagbard Celine
Date: Mon, 17 Mar 1997 00:00
Date: Mon, 17 Mar 1997 00:00
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In article <8WRuzAAXBILzEwKP@ward-hull.demon.co.uk>, Martin Ward <martin@ward-hull.demon.co.uk> wrote: >Pi day > >I saw a posting in one of the humour groups to the effect that 14 March >is Pi Day because 3/14 was a good approximation to pi. Someone had >replied that 3/14 was closer to 0.2142857... Well, maybe they were saying that 3.14, not 3/14 was a good approximation to pi. Just think it over.> -- ab jc ik bc ai kj R R R =R R R ij kf de ij dk ef Hagbard Celine
Re: Pi day
Author: Jill Russell
Date: Mon, 17 Mar 1997 00:00
Date: Mon, 17 Mar 1997 00:00
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In article <8WRuzAAXBILzEwKP@ward-hull.demon.co.uk>, Martin Ward <martin@ward-hull.demon.co.uk> writes suggesting that the following dates might be commemorated by maths teachers >1/pi 0.3183098862 01-Mar * >2^sqrt(2) 2.6651441427 08-Mar * >root 11 3.3166247904 10-Mar and lots more. What a good idea! We could make them into Bank Holidays, or at least school holidays. Children could study at home that day - the calendar. -- Jill
Re: Pi day
Author: David Madore
Date: Wed, 19 Mar 1997 00:00
Date: Wed, 19 Mar 1997 00:00
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Jaime Pina wrote:
Jaime Pina wrote:
> WRONG! If there are uninteresting numbers, there does not have to
> be a SMALLEST uninteresting number, just like there is no smallest number
> in the real numbers.
Here is a proof that all real numbers are interesting:
Let I be the set of interesting real numbers.
* I is not empty because 0, 1, e and pi are certainly in I.
* Any number sufficiently close to a given interesting number is
interesting.
(For example, a number which has the same first billion decimals as pi
is
clearly interesting.) Therefore I is open.
* A number which is the limit of a convergent sequence of interesting
numbers
is certainly interesting. Therefore I is closed.
But R is connected. Therefore I=R, and all real numbers are interesting.
QED.
David A. Madore
(david.madore@ens.fr,
http://www.eleves.ens.fr:8080/home/madore/index.html.en)
Re: Pi day
Author: Belinda Berry
Date: Wed, 19 Mar 1997 00:00
Date: Wed, 19 Mar 1997 00:00
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How do you actually calculate pi anyway. (without loooking it up in a book.) I suppose if you drew up a really accurate circle and did some measuring it might work, but nothing I draw could possibly get more than about 2 decimal places. -- Belinda Berry <BJBerry@onaustralia.com.au> >
Re: Pi day
Author: Ulrich Lange
Date: Wed, 19 Mar 1997 00:00
Date: Wed, 19 Mar 1997 00:00
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David Madore (madore@clipper.ens.fr) wrote:
: Here is a proof that all real numbers are interesting:
: Let I be the set of interesting real numbers.
: * I is not empty because 0, 1, e and pi are certainly in I.
: * Any number sufficiently close to a given interesting number is
: interesting.
: (For example, a number which has the same first billion decimals as pi
: is
: clearly interesting.) Therefore I is open.
: * A number which is the limit of a convergent sequence of interesting
: numbers
: is certainly interesting. Therefore I is closed.
: But R is connected. Therefore I=R, and all real numbers are interesting.
ROTFL :-)
Since your argument for I being open might be questioned by some people,
an alternative proof would be:
1) All natural numbers are interesting because a smallest uninteresting
natural number cannot exist.
2) Sums and products, as well as additive and multiplicative inverses
of interesting numbers are clearly interesting, hence all rational
numbers are interesting.
3) The limit of a sequence of convergent sequence of interesting numbers
is interesting, hence all real numbers are interesting.
However, I guess both of us used the axiom that algebra and topology are
interesting, which might also be questioned by some people :-)
--
Ulrich Lange Dept. of Chemical Engineering
University of Alberta
lange@gpu.srv.ualberta.ca Edmonton, Alberta, T6G 2G6, Canada
Re: Pi day
Author: chapm19
Date: Wed, 19 Mar 1997 00:00
Date: Wed, 19 Mar 1997 00:00
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In <5gmt4q$a40$1@nntp.ucs.ubc.ca>, buckland@unixg.ubc.ca (Tony Buckland) writes:
>In article <5gmrkk$d00$1@news-s01.ny.us.ibm.net>, <chapm19@ibm.net> wrote:
>>In <AF53F4E8966826FB9@brierley.demon.co.uk>, stephen@brierley.demon.co.uk (Stephen Brierley) writes:
>>Another idea for an imaginary date--
>>How about any date in the year 0 (zero)?
>
> If only there really had been a year zero.
> Then we wouldn't have to be concerned about whether the new
> century and millenium begin Jan 1 2000 or Jan 1 2001.
>
Personally, I am not losing much sleep over when the new millenium begins.
On the other hand, the people I contract with are real worried what will
happen to their computers on 1 Jan 2000. And we thought we only
needed to worry about Michelangelo's birthday.
Robert R. Chapman, Jr.
chapm19@ibm.net
Lynnwood, WA 98036 USA
**********************************************************
It's interesting how superstition thrives in unsettled times,
and how many are prepared to listen to it, at least with
half an ear.
--Dietrich Bonhoeffer
Letters and Papers from Prison, The Enlarged Edition
(New York: Macmillan Paperbacks Edition 1972), page 153
**********************************************************
Re: Pi day
Author: Brian Raiter
Date: Wed, 19 Mar 1997 00:00
Date: Wed, 19 Mar 1997 00:00
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> How do you actually calculate pi anyway. (without loooking it up in a > book.) I suppose if you drew up a really accurate circle and did some > measuring it might work, but nothing I draw could possibly get more than > about 2 decimal places. There are numerous web pages devoted entirely to pi. Most of them list numerous methods to calculate its value. Check yahoo. The one I remember off the top of my head: pi/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... This one, however, takes a very very long time to converge. There are much better ones. b
Re: Pi day
Author: G.W. Owen
Date: Wed, 19 Mar 1997 00:00
Date: Wed, 19 Mar 1997 00:00
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Belinda Berry (BJBerry@onaustralia.com.au) wrote: : How do you actually calculate pi anyway. (without loooking it up in a : book.) I suppose if you drew up a really accurate circle and did some : measuring it might work, but nothing I draw could possibly get more than : about 2 decimal places. : There are a number of ways. There are some infinite series that can be shown to add up to pi, or pi/2 or somesuch and taking as many terms as you like can give you as many decimal places as you want/need. Failing that, you can find a formula for the area of a circle, using integration, and then approximate it using Simpson's rule, taking sections as thin as necessary for the accuracy you want. -- Gareth Owen <G.W.Owen@keele.ac.uk> Pleidiol Wyf I'm Gwlad
Re: Pi day
Author: Tom Waters
Date: Thu, 20 Mar 1997 00:00
Date: Thu, 20 Mar 1997 00:00
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Tony Buckland (buckland@unixg.ubc.ca) wrote: : If only there really had been a year zero. : Then we wouldn't have to be concerned about whether the new : century and millenium begin Jan 1 2000 or Jan 1 2001. Does anyone remember whether the French revolutionary calendar had a year zero? Tom -- Thomas Waters twaters@use.usit.net 1021 East Oak Hill Avenue, Knoxville TN 37917 Dig And Be Dug In Return
Re: Pi day
Author: David Madore
Date: Thu, 20 Mar 1997 00:00
Date: Thu, 20 Mar 1997 00:00
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Tom Waters wrote:
> Does anyone remember whether the French revolutionary calendar had a year
> zero?
They did not. The dates were written in roman numerals, so they did not
make provision for a zero. The calendar wasn't supposed to be used to
refer to pre-revolutionary days, either, so there would be no negative
years (they would be written in the old gregorian calendar). Neither did
they have a month zero or a day zero in each month.
David A. Madore
(david.madore@ens.fr,
http://www.eleves.ens.fr:8080/home/madore/index.html.en)
Re: Pi day
Author: Martin Ward
Date: Fri, 21 Mar 1997 00:00
Date: Fri, 21 Mar 1997 00:00
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In article <01bc3440$d8a87040$LocalHost@default>, Belinda Berry <BJBerry@onaustralia.com.au> writes >How do you actually calculate pi anyway. (without loooking it up in a >book.) I suppose if you drew up a really accurate circle and did some >measuring it might work, but nothing I draw could possibly get more than >about 2 decimal places. Quite right, Belinda, not even the most accurate drawing and measuring could produce results good for many more places than that. Series are the traditional way of calculating numbers like pi to lots of places. For example: (pi^2)/6 = 1+(1/2^2)+(1/3^2)+(1/4^2)+... The series goes on for ever, but the terms get smaller and smaller and the sum is said to "converge", in this case "converge to pi squared over six". For a given degree of accuracy we can stop after a while. This particular series is easy to state but converges very slowly, so "after a while" means after a *long* while. For example, to get four decimal places of accuracy we have to add up 6918 terms. Fortunately there are other series which converge more rapidly. -- Best wishes Martin Ward __________________________________________________________________ "No-one is ever hurt by the truth - he is injured who continues in self-deception and ignorance." Marcus Aurelius __________________________________________________________________ Turnpike evaluation. For information, see http://www.turnpike.com/
Re: Pi day
Author: Martin Ward
Date: Fri, 21 Mar 1997 00:00
Date: Fri, 21 Mar 1997 00:00
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In article <5gmqsn$bs8@milo.vcn.bc.ca>, Mark Gibson <mgibson@vcn.bc.ca>
writes
>Martin Ward (martin@ward-hull.demon.co.uk) wrote:
>: Pi day
>:
>:
>: *22 July* should be Pi Day. 22/7 = 3.142857... is the good
>: approximation to pi known since the days of the ancient Greeks.
>:
>A very clever and entertaining post. I agree that July 22 should be Pi
>Day. Incidentally it is also the Deast of St. Mary Magdalene, patron saint
>of prostitutes and "fallen" women in the Calendar of the Catholic Chu7rch.
>
>But as for 22/7 being "close" to pi, "close" is a pretty bague term.
Yes, but the point is that it is as close a rational approximation as we
can get with a small denominator. In this case with denominator <= 12.
>Compared with the spiral nebula in Andromeda, Betelgeuse is pretty close
>to Earth but don't hold your breath waiting for some airline to announce
>weekend excursions there.
>
>3.14159265379 is a closer approximation to pi that I usually use. But that
>isn't very close compared with the approximation developed by the guy who
>used a computer to work it out to millions of sig figs.
>
>The most fascinating thing about all irrational numbers is that no matter
>how hard we try we can never really get "close" to them. The number itself
>always recedes over the horizon away from our attempts at precision.
>That's why IMHO there really can't be any Pi day, e day, sqrt(2) day, etc.
We can get as close as we like to an irrational in the sense that if you
say get within a millionth of pi, I can produce a rational number that
close; and if you say a billionth I can get that close too; and so on to
any degree of accuracy you want.
What we cannot do is state an irrational *exactly* as a fraction or
decimal.
>
>Incidentelly I'm sure it must have ocurred to someone that there are
>DEGREES of irrationality among the irrational numbers. For axample some
>irrational numbers when subjected to a simple operation can be mapped to a
>rational number. e.g. sqrt(2)*sqrt(2) = 2. Whilst as far as I know e*e and
>pi*pi, e*pi, etc are still irrational. Since this ocurred to a
>mathematical dunce like me it can't be that important but I though it was
>interesting, so there.
>
>MSG
>---
>THEY'RE COMING TO TAKE ME AWAY, HA-HA
>THEY'RE COMING TO TAKE ME AWAY...
You are quite right about this, and it is an important insight.
Irrational numbers like sqrt(2) are said to be algebraic, because they
are the roots of an algebraic equation (with rational coefficients). In
this case:
x^2 - 2 = 0
Irrational numbers like e and pi which are not the root of any such
equation are said to be transcendental. (It is rather hard to *prove*
that pi and e are transcendental).
--
Martin Ward
_________________________________________________________________________
"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".
"Through the Looking-Glass" - Lewis Carroll.
_________________________________________________________________________
Turnpike evaluation. For Turnpike information, mailto:info@turnpike.com
Re: Pi day
Author: Martin Ward
Date: Fri, 21 Mar 1997 00:00
Date: Fri, 21 Mar 1997 00:00
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In article <5gp93t$okk@nyx.cs.du.edu>, Charles Carroll <ccarroll@nyx.com> writes >"Belinda Berry" <BJBerry@onaustralia.com.au> writes: > >>How do you actually calculate pi anyway. (without loooking it up in a >>book.) I suppose if you drew up a really accurate circle and did some >>measuring it might work, but nothing I draw could possibly get more than >>about 2 decimal places. >>-- >>Belinda Berry >><BJBerry@onaustralia.com.au> > >One way to do it is with this series. This series doesn't converge >as quickly as some other series, And how! This takes over 5000 terms to reach even one decimal place of accuracy, even worse (in this respect) than the one I suggested! > but it's probably the simplest >one there is. Try it and see. > >pi = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 ... > > -- Martin Ward _________________________________________________________________________ "When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean - neither more nor less". "The question is", said Alice, "whether you *can* make words mean so many different things". "The question is", said Humpty-Dumpty, "which is to be master - that's all". "Through the Looking-Glass" - Lewis Carroll. _________________________________________________________________________ Turnpike evaluation. For information, see http://www.turnpike.com/
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