OK, maybe we’d better step back a bit.
According to Orthodox Judaism, the written Torah (the commandments as written in the Pentatuch) was not the only instruction given to Moses at Mount Sinai. An oral explanation of these commandments was given as well. A simple perusal of the commandments should make it clear that the commandments, as written in the Torah, do not offer nearly enough detail on how they are to be observed. Some examples are:
Dueteronomy 6:9 tells us to write these words on the doorposts of our houses and gates. OK, how should they be written? Should they be written directly on the doorposts, or written on a scroll that is attached to the doorposts? Or some other method? Which words? The words of that paragraph? The whole of Dueteronomy? The whole of the Torah? Just that verse?
Leviticus 23:27 tells us that on Yom Kippur we are to “afflict ourselves.” How? Should we stand out in the cold? Fast? Whip ourselves? Avoid sexual relations? In what manner? Should we do so the whole day?
Leviticus 23:24 tells us that Rosh HaShanna is to be “a day of blowing.” OK, blow what? A trumpet? A shofar? And for how long are we to blow? The entire day? Should we set up shifts with blowers to blow the whole day?
Leviticus 23:40 tells us on Succos to take the “fruit of a beautiful tree” and other plants. What are we to do when we take them? How should we take them? Should we steal them from our neighbors? Should we hold them in our hands the entire seven day festival?
As you can see, the written Torah is lacking in detail regarding the commandments. This detail was included in an oral tradition that was passed down from generation to generation.
In later years, the oral tradition was in danger of being forgotten. Because of this, this oral tradition was committed to writing in the Talmud. However, the bulk of the Talmud consists of arguments regarding these details and how the commandments are to be performed. As such, there is no clear “statement from God” as you would expect to find in the Torah. The seven commandments are listed in the Talmud in Sanhedrin and they are as GilaB listed them.
The verse that they are derived from is Genesis 2:16. Now, I know what you are going to say. You’re going to say (after looking up the verse) “Hey, Zev, what are you talking about? That verse deals with the prohibition of eating from the Tree of Knowledge!” And you’d be absolutely correct. Without my copy of Sanhedrin in front of me, I can’t give you the exact breakdown of how the commandments are derived from this verse. Suffice it to say that there is a tradition in Orthodox Judaism that there are no extra words or letters in the written Torah. If there is a word that appears to be superflouous, it is there because something extra can be learned from it. It is from superflouos words/letters in the verse that the seven Noahide commandments are derived.
Zev Steinhardt