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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.
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