John Machin
(1680  1751)
Remember !
Slice of life
John Machin is a notsowellknown
mathematician. He was born in 1680 was a profesor in astonomy in
London. He discover in 1706 the formula =16arctan(1/5)4arctan(1/239), which, thanks to the
development of the power serie of arctan known since Grégory,
allowed him to get the above formula... But for the experience
historian of Pi, Machin played an important role, firstly because he
was the first to calculate 100 decimal using his formula, but mostly
because he opened the door of arctan formula research....
Around
The arctan formula are a simple and
fast method to work out the decimals of Pi. Knowing the expansion
arctan(x)= for x between 1 and 1, then all we
need is to find the combinations of arctan giving /4=arctan(1). The smaller the term inside the arctan
brackets is the faster the series associated with it converges. Today
still, we sometime check the calculation of the decimals of Pi with
this type of formula... It is true that the extration of square roots
is still tedious and a rational sequence very useful...
One of the last record to date (51 billion decimals) was checked with Gauss's formula.
The other famous arctan formula whose combination is equal to à (see historic) :
(convention x.arctan(1/y)>x*y )
16*54*239 
so, as we memorised, is
from Machin 1706 
20*7+8arctan(3/79) 
Euler 1755 
4*2+4*3 
Euler or Hutton 1776 no
one agrees... 
16*54*70+4*99 
Euler, him again ! 1764 
4*2+4*5+4*8 
L. von Strassnitzky 
8*3+4*7 
Charles Hutton 1776 then
Euler 1779 
8*24*7 
Hermann 
12*4+4*20+4*1985 
S. Loney 1893 then
Störmer 1896 
32*104*23916*515 
S Klingenstierna 1730 
48*18+32*5720*239 
by the great Gauss
himself ! 
48*38+80*57+28*239+96*268 
Gauss yet again... 
24*8+8*57+4*239 
Störmer 1896 
An by order of
usefulness... 
in easiness of Calculation

44*57+7*23912*682 
85,67% 
22*28+2*4435*139310*11018 
88,28% 
17*23+8*182+10*5118+5*6072 
92,41% 
88*172+51*239+32*682+44*5357+68*12943 
93,56% 
100*73+54*23912*207252*294324*16432 
96,38% 
12*18+8*575*239 
96,51% 
8*101*2394*515 
96,65% 
44*5320*4435*1393+22*444310*11018 
97,09% 
17*22+3*1722*6827*5357 
97,95% 
16*201*2394*5158*4030 
99,13% 
61*3814*5573*106817*345834*27493 
99,14% 
227*255100*682+44*2072+51*294327*12943+88*16432 
99,32% 
24*53+20*575*239+12*4443 
99,61% 
127*241+100*437+44*2072+24*294312*16432+27*28800 
99,92% 
4*51*239 
100% 
We measure the calculation cost of a formula such as Machin's by
1/log(5)+1/log(239).
That's the meaning of the percentage above...
I myself had fun in looking for a few formula and found with others...
128*107+128*122+28*239+96*268+48arctan(19/2167) and
732*530+732*563+128arctan(3/2611)+332arctan(27/64589)+48arctan(53/55479)+
+64arctan(6/15617)+28arctan(6/15617)+28*9703+100*14633
who have heavy calculation cost but with very fast convergence.
The most complete site on arctan (and who studies the usefulness of
those formula...) is www.ccsf.caltech.edu/~roy/pi.formulas.html
Precision
Let's try to estimate roughly the number of
terms that we need to calculate in the serie so to obtain d correct
decimals of Pi. We can observe from the expansion of the arctan in
series that we need to estimate n such that ,which after simplification comes down to n>. Given that, in the combination of arctans,
this is the term where b is the smallest that predominates, for
Machin's formula, we have then n>0,72d, which is a little well
respected according to the trials...
Proof
It would be tedious  and to be honest
useless!  to completly proof Machin's formula when it is the principle
that is mostly important.... We just need to know a few results so to
glimpse the full proof or the method to find similar formula.... Here
they are:
1) arctan(a)+arctan(b)= (by composition of
tan and by avoiding ab=1...)
2) (it follows from the above formula
(1) )
3) +arctan(x) (still obvious from (1) )
4) with integers a_{i },b_{i
},k
if and only if (a_{1}+ib_{1})(a_{2}+ib_{2})...(a_{n}+ib_{n})
has an zero imaginary part.
(we notice that this is the case for Machin's formula with a_{i}=5,
b_{i}=1 for i=1,...,16 and a_{i}=239, b_{i}=1
for i=17,...20 since (5+i)^{16}(239i)^{4}=681386607803576157184)
The formula comes from the fact that natural logarithm is defined in
complex by
ln(a+ib)= where p is relatif integer, and
the property ln(ab)=ln(a)+ln(b) does the job...
5) from the same style : with integers k, a,
b
if and only if (1i)^{k}(a+i)^{m}(b+i)^{n}
is real.
Trials
n replace infinity in the above series...
n=2 : we get 3,14182 (3)
n=10 : 3,141592653589793294 (16)
n=50 : 72 correct decimals
we have therefore a convergence of roughly 1.4n (close to
1/0.72=1,388...
founded above)
Acceleration of the convergence
Weirdly, if the good old Delta2 of'Aitken functioned well on Leibniz's
series of the same type, it seems that the terms of power (2k+1)
deorientate a bit the Delta2. It's use hence is a little bit less
useful than the calculation of a higher rank in the serie.
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