Gemara

(R. Zeira): If a Minchah is Ra'uy l'Bilah (could be mixed with the oil), Bilah is not Me'akev (even if it was not mixed, it is valid). If it is not Ra'uy l'Bilah (it is more than 60 Esronim, and there is not enough oil or room in the Keli to mix it), Bilah is Me'akev (it is invalid because it was not mixed).

Berachos 47b (R. Asi): We may include a baby in a crib for a Zimun.

Question (Mishnah): We do not include women, slaves or children in a Zimun.

Answer: R. Asi holds like R. Yehoshua ben Levi:

(R. Yehoshua ben Levi): We may not include a baby for a Zimun, but we may include him for 10 (for a Zimun with Hash-m's name).

(R. Yehoshua ben Levi): Nine Yisraelim and a (Kena'ani) slave join.

Question (Beraisa): A case occurred in which R. Eliezer found only nine in the Beis ha'Keneses, so he freed his slave to be the tenth.

Inference: He would not join if he were not freed!

Answer: There were only eight Yisraelim and two slaves. He freed one slave, and then the other joined for the Minyan.

48a: The Halachah is unlike all of these teachings, rather, like Rav Nachman, who said that if a minor knows to Whom we bless, he joins for a Zimun.

Rishonim

Rosh (Berachos 7:20): Rav Nachman said that if a minor knows to Whom we bless, he joins for a Zimun. He did not discuss joining for 10. This implies that he joins for 10 even if he does not know to Whom we bless. The entire Sugya shows that it is harder to join for a Zimun than to 10. Rav Hai Gaon says that a minor counts for a Minyan. He means only one. More than one does not join, like we say about slaves. It does not honor Hash-m to join more than one. R. Tam permitted to join a minor for (a Zimun with) 10 and for Tefilah even if he did not hold a Chumash, but he himself would not do so. I say that the Halachah does not follow R. Yehoshua ben Levi.

Rosh (Teshuvah 4:19): The Tzibur must be quiet and intend for the Berachos of Chazaras ha'Shatz and answer Amen. If there are not nine intending for the Berachos, in my eyes they are close to being Levatalah, for it was enacted to say them amidst 10. One should think that there are not nine without himself, and intend for the Berachos.

Hagahos Maimoniyos (Hilchos Tefilah 8:9): The Rambam (Teshuvah 529;in some editions, 202) says that if one started to pray by himself and he cannot answer, even so he joins, for he is in the Beis ha'Keneses. "V'Nikdashti b'Soch Bnei Yisrael" applies, for the Shechinah dwells among any 10 Yisre'elim. Someone praying is no worse than a baby, who joins according to R. Yehoshua ben Levi. The other opinion disagrees because he does not understand to Whom we bless, and he is not a Bar Mitzvah. However, it is proper to wait until he finishes his Tefilah, so he will merit to answer Amen. This is even if there are 10 without him. This is only if enough time will remain to pray.

Poskim

Shulchan Aruch (OC 55:6): If one of the 10 started to pray by himself and he cannot answer, or he is sleeping, even so he joins (to enable saying Kaddish).

Beis Yosef (OC 55 DH Kasuv b'Hagahos): My Rebbi Yakov Bei Rav learned from Hagahos Maimoniyos that if one was sleeping, he joins with nine for Kaddish and Kedushah. Since there are 10, the Shechinah dwells among them. Later I found that the Agur says that the Maharil (Teshuvah 150) was asked about if one of the 10 is sleeping, and he answered that R. Tam joined a baby. Even though the Rosh requires 10 hearkening, we find that the custom is to join people even though they are talking secular matters. Even though I am hesitant about this, perhaps we apply 'when mixing is possible, mixing is not Me'akev.' One may rely on my Rebbi.

Taz (4): How can R. Yakov learn about someone sleeping from one who is praying? There, even though he may not interrupt and answer, he can be quiet and listen, and Shome'a k'Oneh (one who listens is like one who answers), like we say later (104:7). This does not apply to someone sleeping! The Agur is even more astounding. First he wanted to learn from R. Yehoshua ben Levi, who says that a baby joins. The Halachah does not follow him! Even R. Tam, who rules like him, did not want to act upon this. How can we rely on him?! Also, perhaps someone sleeping is worse than a minor, for he has no Kedushah. The Beis Yosef (Siman 4) brought from the Zohar that (when one sleeps,) his Neshamah departs. Kedushah does not apply to him, for he tastes death and a spirit of Tum'ah rests on him. I think that even R. Yehoshua ben Levi would agree that he does not join. Since we can wake him, why should we join him in his sleep?! He is unlike a minor, for we cannot make him an adult. This is unlike Kol ha'Ra'uy l'Bilah Ein Bilah Me'akeves Bo. There, the question is after the event, whether something is Me'akev. It does not teach that like we may do something improper when we could fix it! After bringing the Rosh, how does he bring a proof from the custom to count people who talk? We do not bring a proof from lunatics whose sin is too great to bear! Heaven forbid to count such people when there is not a Minyan without them. All the more so we may not learn from them to someone sleeping! The Shulchan Aruch (124:4) rules that if there are not nine intending for the words of Chazaras ha'Shatz, the Berachos are close to being Levatalah. All the more so, one may not be lenient l'Chatchilah about someone sleeping.

Magen Avraham (8): Ral ben Chaviv says that the same applies if three or four are sleeping, for the majority remain (awake). I would not be so lenient, for the Maharil's primary proof was from a baby who joins, even though he cannot answer. Even though we do not hold like this, this is because the Shechinah dwells only on adults. The Kal va'Chomer cannot teach more than the source. If one baby can join, one sleeping person can join.

Mishnah Berurah (32): Even more than one can be praying, as long as the majority are not in Shemoneh Esre and they can answer. Some are stringent about more than one. Surely one may not join more than one sleeping person even in pressed circumstances, for many are stringent about even one sleeping person.

Mishnah Berurah (33): The Pri Megadim holds that the Shulchan Aruch allows joining a sleeping person also for Birkas ha'Mazon. The law of Kri'as Shema and Birkas Kohanim requires investigation. Surely someone sleeping does not join for Kri'as ha'Megilah, for 10 are needed to publicize the miracle, and a sleeping person does not contribute to this.

Mishnah Berurah (34): The Pri Chodosh holds that a sleeping person is like a lunatic, so he does not join, like the Taz says. Surely, l'Chatchilah one must wake him, at least to the point that he is only dozing.

Bi'ur Halachah (DH Oh): If one cannot wake him, it is not clear what to do. The Pri Megadim and Chayei Adam say that one who relies on the Shulchan Aruch will not lose, but but the Birkei Yosef and Derech ha'Chayim hold like the Pri Chodosh and Taz. Perhaps where they say Kaddish and Kedushah aloud, and everything else silently, one may join a sleeping person in pressed circumstances, for there is no concern for a Berachah Levatalah. The Pri Megadim connotes that this case is more lenient.

Kaf ha'Chayim (48): The Eliyahu Rabah answered the Magen Avraham's question, but many Acharonim hold like the Magen Avraham, so one should be stringent when possible. When it is not possible, one may rely on those who permit for Kaddish or Borchu. Chazaras ha'Shatz requires nine intending for the Berachos.

Birkei Yosef (6): The Rosh says that if there are not nine hearkening to the Chazarah, the Berachos are close to Levatalah. Maharil allows joining someone sleeping, for he rules unlike the Rosh, but the Shulchan Aruch rules like the Rosh! Perhaps the Beis Yosef holds primarily like his Rebbi; in Siman 124, he was stringent to bring the Rosh's opinion. It is best to be stringent and arouse him so he is only dozing.