Mitzvah to Daven
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- Some early authorities consider the mitzvah to pray as a biblical mitzvah based on the verse “ולעבדו בכל לבבכם” which Chazal explain as a reference to the service of the heart, ולעבדו connoting worship through prayer. Tefillah is listed as the fifth mitzvah in the Rambam's ספר המצות. However, many early authorities argue that the entire obligation is rabbinic and only biblical in times of great distress (בעת צרה). 
- One who lengthens his prayers will merit a longer life. 
- If a doctor is involved with healing a patient and there's an urgent need he shouldn't interrupt to say Shema or Shemona Esrei. However, if he can take a brief break and his involvement isn't critical for that time then he should do so in order to say Shema and Shemona Esrei.
- For example, someone who is a doctor healing a Jewish patient and missed the entire period of a Tefillah from beginning to end, since he was dealing with a mitzvah he doesn't need to recite Tashlumin afterwards.
- Women are obligated in saying Shmoneh Esrei of Shacharit and Mincha.  It is proper for women to accept the yoke of heaven by saying at least the first pasuk of Shema. 
- Women are not obligated in Arvit. If one wants to say it one should specify that it is done Bli Neder (so that it is not binding after 3 times of performing it). 
- Many poskim hold that women are exempt from saying Mussaf.
- A woman who is unable to say Shmoneh Esrei of Shacharit should at least say a short prayer which includes praise, a request, and a thanks of Hashem. 
Sequence of Importance
- A woman who has limited time for Shacharit should say parts of Tefillah according to the order of importance: 1) Shmoneh Esrei 2) first pasuk of Shema and Emet VeYatsiv before Shmoneh Esrei 3) Baruch SheAmar, Ashrei, and Yishtabach, 4) Birchot HaShachar, 5) Birkat HaTorah, 6) Birchot Kriyat Shema and the entire Shema and 7) Pesukei Dezimra. 
- According to the Sephardic minhag, women shouldn't recite the bracha of Baruch SheAmer, Yishtabach, or Birchot Kriyat Shema. 
- The minhag is that women don't say Tachanun. 
- According to Sephardim, a woman who doesn't have time should at least say Birchot Hashachar, Birchot HaTorah, and Shemona Esrei. It is also proper for them to also say Shema. 
- Women are exempt from davening with a minyan. Therefore, they shouldn't skip tefillot in order to catch up to the minyan.
- Regarding Kriyat Hatorah some say that women are obligated to listen to kriyat hatorah but the minhag is that women do not listen to kriyat hatorah.
- Once a child knows how to speak he should be taught to say the first pasuk of Shema. 
- A child of 6 or 7 should be taught to say the brachot of Kriyat Shema, Shema, and Shmoneh Esrei. 
- One should teach small children to answer Amen because once a child answers Amen, he has a place in Olam Haba. 
- There is a chinuch for a child to daven in a minyan but davening in a group of ten children isn't considered a minyan at all even for chinuch.
- A child should not serve as the Chazan for a minyan, especially for Shacharit and Mincha. Those who allow a child to lead the congregation for Arvit may have what to rely on, but ideally one should not do so.
- Regarding children performing Birkat Cohanim see Birkat Cohanim.
- Gentiles are not obligated in Tefillah but if they do pray they fulfill a mitzvah, and are considered to be like a person who is not commanded but volunteers to do so. 
- Deuteronomy 11:13/דברים פרק יא פסוק יג
- Tanit 2a
- Rambam Tefillah 1:1,ספר המצוות לרמב"ם מצות עשה ה
- Ramban on ספר המצות, Mishna Brurah 106:4
- Sh"t Rivivot Ephraim 1:50:7 says that although the gemara Brachot 54b says this in reference to the amida, it is true for all prayers
- Nishmat Avraham 38:6. Dirshu 93:8 also cites Chut Shani Chol Hamoed v. 1 p. 328 as agreeing.
- Rav Shlomo Zalman Auerbach cited in Nishmat Avraham 38:6 writes that someone who is healing a Jewish patient is doing a mitzvah and as such he is exempt from Tefillah and if he's involved with that mitzvah from the beginning of the time of the Tefillah to the end then he doesn't even need to recite Tashlumin in accordance with the Derisha YD 341:3. He also cites Rav Zilberstein who explains that even if he doesn't have intention for a mitzvah nonetheless he is practically doing the mitzvah of Hashavat Aveidah and as such the exemption applies.
- S”A 106:1, Mishna Brurah 106:4. See also Sh”t Yechave Daat 3:7 who writes that women are obligated in one Shmoneh Esrei a day but if they say three a day it is praiseworthy. Tefillah KeHilchata (chapter 1 note 17) quotes Rav Ben Tzion Abba Shaul who says that women are obligated in all Tefillot except Arvit.
- S”A and Rama 70:1, Mishna Brurah 70:5
- Mishna Brurah 106:4, Tefillah KeHilchata 1:11. Rabbi Hershel Schachter in a shiur on yutorah.org ("Women at Prayer", min 10-16) agrees and explains that women are not obligated to say Arvit since according to the gemara, Arvit is optional and although men have accepted it upon themselves to say Arvit, making it an obligation for men, the minhag however does not include women and therefore Arvit is not an obligation for them.
- Mishna Brurah 106:4 quotes the Tzlach (Brachot 26a s.v. VeShel Musafin) who says that women are exempt from Mussaf, while the Magen Giborim disagrees. Rabbi Hershel Schachter in a shiur on yutorah.org ("Women at Prayer", min 10-16) explains that the reason women are exempt, is that Mussaf is not about asking for mercy, which is the basis for women's obligation to pray (Gemara Brachot 20b). Additionally, Mussaf was instituted because of the korbanot Mussaf and women were not obligated in donating the Machasit HaShekel which was used for communal korbanot such as the Mussaf korbanot.
- Tefillah KeHilchata 1:10 based on Magen Avraham 106:2
- Tefillah KeHilchata (chapter 1 note 21).
- Yalkut Yosef (Dinim L'isha V'lbat 8:1-2). Tefillah KeHilchata (chapter 1 note 21) quotes Rav Ben Tzion Abba Shaul as saying that women may say Pesukei Dezimra and brachot of Kriyat Shema with Shem UMalchut and Sh”t Yabia Omer 2:6 holds that these brachot should be made without Shem UMalchut.
- Piskei Teshuvot 131:1, Tefillah KeHilchata 15:5
- Yalkut Yosef (Otzar Dinim Lisha Velebat p. 68, 77, 104)
- Shvut Yakov 3:54, Leket Halachot 3:577 based on Teshuva M'ahava 2:229:10, and Rav Shlomo Zalman Auerbach in Halichot Shlomo (Tefillah ch. 5 fnt. 4 p. 61). Rav Eliezer Melamed (Peninei Halacha ch. 20) explains that since women aren't obligated in mitzvot that are time bound they are exempt from joining the minyan. (Another link here.) Even though Rav Yakov Ariel (Bohela Shel Torah v. 2 ch. 26 p. 100) wrote that women are obligated in davening in a minyan he explained himself that he meant that torahland.org.il women are exempt but it is ideal for women to daven in a minyan. The fact that they are exempt is also evident in the fact that they can't join for the requisite quorum for a minyan (Tosfot Brachot 45b, Shulchan Aruch O.C. 55:1, Mishna Brurah 55:3, Ishei Yisrael 15:6, Tefillah Khilchata 8:27).
- Rav Shlomo Zalman Auerbach in Halichot Shlomo (Tefillah ch. 5 fnt. 4 p. 61) holds that there's no obligation at all for a women to daven with a minyan and therefore she shouldn't skip to catch up with the minyan. Avnei Yishfeh Tehillah 16:6 quotes Rav Shlomo Zalman but also quotes Rav Elyashiv and Rav Shmuel Wosner who disagrees that women can skip pesukei dzimra in order to daven with a minyan since it enhances her davening.
- Magen Avraham 282:6, Mishna Brurah 282:13. Avnei Yishfeh Tehillah 16:6 explains that even the Magen Avraham who says that woman are obligated to listen to kriyat hatorah which needs a minyan that is only because it is part of a public reading that she understands and gains from as we find by hakel which includes women. However, davening Shemona Esrei with a congregation is something that since they can't form or contribute to the minyan, chazal never obligated to daven with a minyan.
- Mishna Brurah 70:7 notes that this does not have to be said in the proper time of shema
- Mishna Brachot 20a says that children are exempt from reciting shema. Rashi explains that this refers to children that have reached the age of chinuch. Rabbenu Tam argues that this refers to children that have not reached the age of chinuch, but once they have reached the age of chinuch, they should be taught to say shema. Shulchan Aruch 70:2 quotes both opinions and says it is appropriate to be strict like Rabbenu Tam.
M.B. 70:6 notes that this means one should train the child to recite shema at the proper time with all the brachot
- Tefillah KeHilchata 1:12-13. M.B. 106:5 says this means shachrit and mincha. Rav Simcha Bunim Cohen (Children in Halacha page 19) says that the custom is not to train children to daven mariv.
- Rav Simcha Bunim Cohen (Children in Halacha page 19) quotes Chinuch Yisroel page 77
- M.B. 106:5
- Rama 124:7
- Rav Soloveitchik (Nefesh Harav p. 113) explained that for chinuch it needs to be a halachically valid structure for the chinuch to be effective (based on Ritva Sukkah 2b).
Rav Moshe Feinstein (Igrot Moshe O.C. 2:98) says that one must stop children if they are doing dvarim shebikdusha with a minyan of children as that is prohibited.
- Shu"t Yabea Omer 9:100:4, Yalkut Yosef Dinei Chinich Katan pg 61
- Shulchan Aruch 53:10, Yalkut Yosef Dinei Chinich Katan pg 61
- Mishna Brura 53:30
- Sh”t O.C. Igrot Moshe 2:25 writes that non-Jews are not obligated to pray to Hashem but if they do they fulfill a mitzvah and receive reward as someone who is not commanded but nonetheless, volunteers. See Sh"t HaRambam 148 who writes even regarding Brit Milah that non-Jews can volunteer to perform a mitzvah.