I recently “rediscovered” the calculations of Al-Khashi, from the 1400’s by using the same method, an approximation of pi by the circumference of a polygon. I found that this devolved onto the half-angle formula for sine, or equivalently for cosine, applied recursively. This is a trigonometric formula, but easily obtained by direct geometric construction.
I did all this on my own and I was amazed to see how closely I had followed Al-Kashi’s method. Of course it was very easy for me to carry the recursions much further, using the UNIX based extended precision calculator, bc .
Al-Kashi obtained 17 decimal digits of pi, and it is quite amazing to me how modern his thinking was concerning this recursive computation.
Watched a BBC produced series on Netflix called “The Story of Maths” (4 episodes) presented by Marcus du Sautoy. It was both fascinating and entertaining, and I am not even a mathematician.
And compared to using, say, an abacus, calculations in Hindu-Arabic numbers also allowed an “audit trail”, as Devlin points out: “An individual sitting in Pisa controlling a network of traders needed to be able to review the financial books on a regular basis.”
To fill that need, in 1202 Fibonacci (the son of a notary working for Pisan traders) published Liber Abaci, a compendium of Hindu-Arabic arithmetic and its practical applications to trade. The 600-page book introduces the numerals and explains how to use them for basic calculations.
alef (1) + tav (400) + alpha (1) + omega (800) = 1202
;-)
Speaking of introductions and basic calculations,
The Torah begins with the letter bet, and ends with a lamed (in the word Israel). This spells לב lev, heart. Sum total: 32 (bet = 2; lamed = 30)
Now as the numeric value of bet is 2, "bet lamed" can be read as "two lameds", which when placed face-to-face, form a heart.
Explanation here.
The first three letters of the Torah (ברא) are repeated in the second word bara ("created"):
בראשית ברא אלהים את השמים ואת הארץ׃
In the beginning created God the heavens and the earth.
...בראשית ברא
This word, bara, has a numeric value of 203 (2 + 200 + 1), which is the 6th Bell number, meaning that in a set containing 6 elements, there are 203 ways to partition the set. B6 = 203
Bell numbers are named for Eric Temple Bell. (Nice touch!)
Bell (בל) = bet lamed = 32 (bet = 2; lamed = 30)
203 is the mirror of 302, which can be read as 30 and 2, or 32, the numeric value of Bell, or its mirror, lev, heart.
In base 32, 203 which is the 6th Bell number (B6) = 6B
And B6 = 358 = Messiah (משיח)
Cf.
0, 1, 1, 2, 3, 5, 8, 13, 21...
God loves His creation. He loves to create. In the beginning, God ♥ the heavens and the earth. "Base 32", based on the heart, is based on 2 - "two lameds". Base 2: yes or no, on or off, open or closed.
The difference between Israel and the rest of the nations is that Israel said "Yes" to the Torah. Israel ישראל (541) + yes כן (70) = Torah תורה (611) ♥
Yes is 1; No is 0. Beis (בית) 2 = the letter bet, which is an open house.
בראשית ב
In the beginning, bet, in the beginning bara...
Thus the Torah begins by stating the obvious, as the first letter *is* the letter bet, and the first three letters are bet resh alef.
The sum of bere + bara = 406, which is the numeric value of the word tav (תו), the last letter of the alef bet. The three letters between the two "baras" - shin yud tav שית - permute to yesh tav, יש ת , "Here is a tav". Now that's pretty clever.
Some of the ancient mathematicians were amazing.