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The singular of "Mitzvot" is Mitzvah. See that article for a general discussion of commandments in Judaism.
The Talmud (tractate Makkoth 23b) and Midrash calculate that the numerical value ( gematria) of the word "Torah" is 611. The Torah itself states that Moses transmitted the Torah from God to the Jewish people: "Moses commanded us the Torah as an inhertitance for the community of Jacob" ( Deuteronomy 33:4). However, there were two commandments which God directed straight at the Jewish people: the first two of the Ten Commandments; these are phrased in the first person. The grand total of 611 + 2 = 613.
Many Jewish philosophical and mystical works ( Baal ha-Turim, the Maharal of Prague and leaders of Hasidic Judaism) find allusions and inspirational calculations relating to the number of commandments.
In practice there is no one definitive list that explicates the 613 laws. The differences come about because in some places the Torah lists related laws together, so it is difficult to know whether one is dealing with a single law, which lists several cases, or several separate laws; Other "commands" in the Torah are restricted as one-time acts, and would not be considered as "mitzvot". In rabbinic literature there are a number of works, mainly by the Rishonim, that were composed to determine which commandments belong in this enumeration:
The most important of the above works is Sefer ha-Mitzvoth by Maimonides (Rambam). Maimonides went to great lengths to enumerate exactly which of the written Torah's ( Pentateuch) commandments can be considered fixed forevermore, in contradistinction to many "commands" that God makes in the Torah at various points but are restricted as one-time acts. He employs a set of fourteen rules (shorashim) which determine inclusion into the list. Nachmanides makes a number of critical points and replaces some items of the list with others.
The 613 commandments and their source in scripture, as enumerated by Maimonides:
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