המספר e בתלמודים - "עישורייתא דבי רבי"

The number e (e = 2.71828…) is a very important and basic number in mathematics, much like pi (π = 3.14159…) and the square root of 2 (√2 =1.41421356...), but is much less known. e was allegedly discovered in the17th century and was defined by Leonard Euler in the 18th century. This number appears in many areas of mathematics and in varied calculations,including statistics and probability. One of the definitions of e is There is a notion that the number 1/e was unknown until the 17th century,even though it appears in many phenomena. However, as early as the third century the Jewish Sages, in their rulings of Jewish Law called “Rebbi’s Tenth”, estimated the number 1/e as a third and some, and even took note of this number’s unique qualities.The Babylonian and Jerusalem Talmud cover a very wide range of topics.In calculations on the subject “Rebbi’s Tenth” the sages of the Talmud are required to calculate expressions in the form of (1-1/n)^n. In one of these issues there is an estimate for the number 1/e, a little more than a third, and this is apparently the earliest estimate for this number.In this work, I will review and analyze all the issues (‘Sugyot’) in which acalculation is made that is connected to “Rebbi’s Tenth”:A. Inheritance of daughters after the death of their father (Ketubot 68 and the corresponding issue in the Yerushalmi), and in the book of in heritances written by Rav Saadya Gaon which performs the calculation explicitly.B. Visiting the sick (Nedarim 39)C. Making a rough step (Ta’anit 10, Brachot 43 and Shabbat 113)D. Construction of the temple menorah (Menachot 29)E. Things that take one out of five hundred from a person’s eyesight (Eruvin55, Pesachim 42)F. Putting a SEAH and taking a SEAH (Mishnah Mikvaot, Chapter 7Mishnah 2, Yevamot 82, Mishnah Trumot, Chapter 5 Mishnah 7)From these issues I will show that the Talmudic sages had extensiv emathematical knowledge as well as a strong intuition on calculations thatthey could not do using the mathematical tools of their time. The Sages did not define the number 1/e explicitly, but they correctly assessed its value and some of its properties.Using their knowledge and intuition, they came to very advanced conclusionsand performed evaluations of very complicated calculations. Apparently,they were also the first to pay attention to the specialty of the number 1/e,evaluate its size and use its features. However, there may be some errors inthe calculations of sages, in cases where they did not have the mathematical tools required to perform the calculations. These errors affect all of the abov eissues, and may even change different Halakhic laws. Although not all theirassumptions and calculations were correct, their advanced knowledge is quite surprising when considering the mathematical knowledge of the world at the time. In this work I will detail the assumptions made by Sages, thevarious calculations they did, and the effect of modern calculations on theseissues.According to Maimonides in The Guide for the Perplexed (Part C, End of Chapter 14), it is the flawed mathematics of those times that misled thesages. It remains to be discussed whether the Halakhic laws should bechanged with the advancement of mathematics.