G-D used 620 letters to write the entire Decalogue on the two tablets He gave to Moses on Mount Sinai. That’s because this magnificent summary of ethical behavior contains the key to all 613 mitzvot meant for the Jewish people as well as the 7 universal laws for mankind – 620 in all.
There has never been a better synopsis for righteous living than this essential text of Torah. Like all of Torah, no matter how much the times have changed the message of G-D is timeless.
At Mount Sinai God gave the Jewish people the Torah. The word "Torah" literally means "teaching." Included in this Torah were many laws that the Jewish people became obligated in performing from that point on. Over the 40 years in the desert, the Torah was written down onto a scroll by Moses. This became known as the Written Torah, because it was written down and sealed before their entry into the Land of Israel. (It is also commonly called the Five Books of Moses.)
Although the Written Torah contains the 613 mitzvot, it does not spell out exactly what they are. Along with the various laws that were commanded to the Jewish people, such as the commandment to wear Tefillin and to rest on Shabbat, came an oral explanation that explained exactly what the mitzvah entails. This is called the Oral Torah, as it was meant to remain as an "oral tradition" and passed on from parent to child throughout the generations. Some 1700 years ago, when the harsh Roman persecutions threatened the transmission, the Oral Torah was written down to form the Talmud.
Although the Written Torah does not state the exact number of the mitzvot, there is a clear tradition in the Oral Torah that the total is 613. The Talmud (Makkot 23b) states that God taught Moses 613 mitzvot, 365 negative ones ("don't do X") corresponding to the days of the (solar) year, and 248 positive ones corresponding to the limbs of the human being. The overall message is that the mitzvot serve the purpose of perfecting mankind, in both the dimensions of space and time.
It should be mentioned that although our tradition states clearly the totals, never does it tell us the precise list. This has engendered much discussion among the medieval scholars what precisely constitutes the list and the principles for counting mitzvot. For example, should we count the administration of each of the four types of death penalties as separate commandments or as a single one? The discrepancies are for the most part minor, but the precise rules for classification require much analysis. Some of the scholars to have compiled their own lists are Maimonides, Nachmanides, and R. Moshe of Coucy (known as SeMaG after his main work).
Another well-known hint to the 613 mitzvot dates from the story of Jacob. When Jacob was returning to the Holy Land from Laban's house in Aram, he sends Esau a conciliatory message. It began, "I have dwelled (GARTI) with Laban..." (Genesis 32:5). The Midrash ingeniously points out that the Hebrew word "garti" contains the same letters as "taryag" – the representation of the number 613 in Hebrew letters. Thus, Jacob's implied message was, "With Laban did I dwell, and the 613 mitzvot I observed." One implication of this is that the Patriarchs observed the Torah long before it was given at Sinai.
For a list of the 613 mitzvot, see: https://www.aish.com/jl/m/m/48945081.html or https://www.chabad.org/library/article_cdo/aid/756399/jewish/The-613-Commandments-Mitzvot.htm
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