Development of Concepts in Math
Doug Ruby - July 24, 2002
“Futile is the labor of those who fatigue themselves
with calculations to square the circle.”
- Michael Stifel (1544)
“The state of our ignorance is more profound. We do
not know whether such basic constants as + e, / e, or
log are irrational, let alone transcendental”
- Bailey, Borwein, and Borwein (1986)
Why ?
Where did we leave off?
Archimedes finis…
Post-Elizabethan
Rigorous
through mid-1970’s
Non-Euclidean
Modern Developments
Finis
Sources
has been state of the art for over 5000 years!
I am using to re-visit math after 30+ years
Notion of radian and unit circle in trigonometry
Re-introduction to finite series and geometry
Re-learn calculus
More advanced math – Elliptical Integrals?
A personal travel with
is like a case of very good wine with no labels
Each encounter (bottle) is a surprise to be savored!
The best vintages are round and flavorful!
Francois Viete – (1540-1603).
Developed first infinite series (pre-calculus)
Calculated using 393,216 sided polygon
Sets stage for post-Elizabethan innovation….
Ludolph Van Ceulen (1540-1610)
Human “computer” of late 16th century
Spent most of his life calculating
know as “der Ludolphische Zahl” or the
Ludolphine number
Archimedes (220BC) – 6* 24 sided inscribed polygon
Viete (1593) – 6* 216 sided inscribed polygon
(note: contains 17 embedded square roots)
196608 * 2
2
2
2
2
2
2
2
2 ... 3
3.141592653 | 589 ...
Ludolph (1596-1610) – 6* 260 sided inscribed
polygon
Correct to 35(32?) decimal places
Took 62 embedded square roots!
Convergence Note: requires about 2 irrational terms (square roots) per decimal place
Wallis – 1655 - as rational series
Newton’s fluxions – 1655-56 (pub. 1742)
Royal Society founded – 1655
Gregory-Leibniz – 1671-74 - via arctangent series
Machin – 1706 - series using “good” arctan values
Jones – 1706 – “ ” first used
Euler – 1748 – Euler’s Theorem – eix= cosx+ i sinx (ei = -1 for x= )
Lambert – 1767 - is irrational
Shanks – 1873 - to 707 places using Machin
Lindemann – 1882 - is transcendental
Ferguson – 1945 - Shanks was wrong!
Ferguson – 1948 - to 808 decimal places
Rectangular Approximation
Gregory-Leibniz – (1671, 1674)
Machin’s Formula (1706)
Newton’s development of (1655, pub. 1742)
Compute
1
1
1 x 2 dx
2
Using rectangular approximation:
Provides following result:
i n 1
lim
n
i 0
2 4in 4i 2
n
n2
i n 1
lim
n
i 0
4
i(n i)
2
n
2
For n= 8 is:
1
8
0
7
12
15
Requires n= 100,000 to get
16
15
12
7
2.9957
to 7 decimal places!
Convergence Note: requires approx. 10n irrational terms per n decimal places
Gregory (1671) and Leibniz
(1674) establish arctan powerseries relationship
Uses power series for 1/ (1-x):
i
1
(x)i
1
i 0
Prove d(arctanx)/ dx = 1/ (1+ x2)
1
dx
1 x2
Derive power series for arctan(x)
1
dx
1 x2
For |x|<1
x
d arctan x
dx arctan x c
dx
x
x3
3
x5
5
x7
7
x9
9
x 11
11
... arctan( x)
For |x|< 1
Use arctan(1)= / 4
4
1
(1) 3
3
(1) 5
5
(1) 7
7
(1) 9
9
(1) 11
11
... arctan 1
Convergence Note: Not very “power”ful. Needs 105 terms for 7 decimal places. Has systematic error.
Machin derives more powerful series.
Calculates 100 digits of in 1706:
Derived using Trigonometric identities
for sum and difference of angles.
Used tan = 1/5
tan 4
Then use Gregory power series to
calculate for x=1/5 and x=1/239
4 arctan
4
tan 4
4
1
dx
1 x2
tan
x
tan
4
tan 4
4
x3
3
x5
5
1
5
arctan
120
tan 4 1
119
120
1 tan 4
1
119
1
x7
7
x9
9
Gains “power” as x -> 0. Has rational
coefficients with integer num./denom.
With help from Euler, primary method
for over 250 years!
Convergence Note: Very powerful. Computes
1
239
to 9 decimal places in 25 rational terms.
x 11
11
1
119
239
119
... arctan( x)
1
239
Newton’s Development (1655/ 1742)
Developed in 1655, pub. 1742
Uses arcsin function
1
1 x
For arcsin ½ =
/6
6 arcsin
1
2
6
1
2
Can also be developed using
f ( x) f (a ) f (a )( x a )
Taylor series
1
MacLaurin series (a= 0) 6 arcsin 1
6
2
2
provides:
Convergence Note: Also powerful. Computes
2
dx
arcsin x
1
2 3 23
f (a )( x a ) 2
2!
to 8 decimal places in 13 terms.
1
2 3 23
1 3
2 4 5 25
f (a )( x a ) 3
3!
1 3
2 4 5 25
Jones – 1706 – “ ” first used
Euler - 1748
Euler’s Thorem: eix= cosx+ i sinx or ei + 1= 0 for x=
Many formula’s for , 2/ 6, / 4, / 6
General case of arctan 1/ p and arctan x/ y
Most of terminology and symbolism used today
Lambert – 1767 - is irrational
Lindemann – 1882 - is transcendental
Shanks – 1873
to 707 places using Machin
Ferguson
Shanks was wrong! Only 527 correct (1945)
to 808 decimal places using mechanical
calculator (1948)
ENIAC – 1949 – 2037 decimal places
Various computer calculations using arctan
formulas to 500,000 places
Gauss-Legendre Arithmetic Geometric Mean (AGM)
Studied by LaGrange, Gauss, and Legendre
Algorithm published by Brent-Salamin – 1976
Simple iterative algorithm - uses sqrt
Converges quickly – 18 digits in 4 iterations
Convergence = 2n places per n iterations
Ramanujan – 1917 - Modular transformations of Elliptical Integrals
Borwein brothers (1980’s-90’s)
Quartic formula based on Ramanujan
Converges 4n places per n iterations
More complex computation
Chudnovsky brothers (1980’s-90’s)
Also use Ramanujan trnasformations
Built own Super Computer in NY apartment
For rigorous discussion see: http://numbers.computation.free.fr/Constants/Pi/piAGM.html
Fast HW FPU’s - 1968
First 1,000,000 digits – 1973 (Guilloud)
First 1,000,000,000 digits – 1989 (Chudnovsky)
Algorithm by Bailey, Borwein and Plouffe - 1996
allows the nth hexadecimal digit of
without the preceeding n- 1 digits
to be computed
Plouffe discovers new algorithm - 1997
computes the nth digit of
in any base
Current record – 206,158,430,000 digits – 1999 (Y.
Kanada)
Used Borwein Quartic and Gauss-Legendre AGM
Most of my goals are met:
Notion of radian and unit circle in trigonometry
(yes)
Re-introduction to finite series and geometry (yes)
Re-learn calculus (yes)
More advanced math – Elliptical Integrals? (no)
Personal travel with
Questions?
- (YES!)
Blatner, David; The Joy of ; Walker Publishing; 1997
Beckman, Petr; {a history of} ; St. Martin’s Press; 1971
Dunham, William; Journey through Genius; Penguin Books; 1990
Eves, H.; An Introduction to the History of Mathematics; Harcourt
College Publishers; 1990
Bailey, D, Borwein, P., and Jon M. Borwein, "Ramanujan, Modular
Equations, and Approximations to Pi or How to compute One Billion
Digits of Pi***", paper to Organic Math Workshop at Simon Fraser
University; 1995; Borwein Web Site
Tarnow, Danielle; “Ways to Estimate Pi”; Danielle's Math Links; 2002
The MacTutor History of Mathematics; St. Andrews University; St.
Andrews Web Site
Determination of Mathforum Series; MathForum Web Site
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