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Must one wear a pair of Tzitzis throughout the day? 
There is no legal obligation for a person to purchase a four cornered garment in order to become obligated in Tzitzis. The obligation of Tzitzis only applies in a case that one desires to wear a four cornered garment, in which case he is then obligated to tie Tzitzis to its corners. [If however he does not desire to wear a four cornered garment, he does not transgress any command.] Nevertheless, it is good and proper for every person to be careful and vigilant to wear a pair of Tzitzis throughout the entire day [and night, and at all times, with exception to when he is in the bath] in order to remember the Mitzvos constantly. [In truth, this is more than a mere act of piety, but is an encouraged Biblical responsibility, and one who is not careful to do so is punished by Hashem during times of anger. Furthermore in today’s times, some Poskim rule that the constant wearing of Tzitzis is an absolute obligation, due to the age old custom to do so. Others rule there is a Rabbinical obligation to wear Tzitzis at all times. Certainly, one who does not wear Tzitzis loses the reward given for doing so, and forfeits the ability to fulfill five positive commands.]
Immediately upon awakening: Directly and immediately after washing hands, when one’s hands are clean and he is able to say a blessing, one should wear a pair of Tzitzis. [See Q&A below, and Halacha 10 regarding wearing of Tzitzis at night and while sleeping]
During prayer: During the reading of morning Shema, it is a requirement to wear Tzitzis, as explained in the next Halacha.
Not to walk four Amos without Tzitzis: Many are accustomed not to walk a length of four Amos [190 cm] without Tzitzis.
Although there is no Biblical obligation to specifically wear a four cornered garment with Tzitzis, nevertheless one must do so in order to remember the Mitzvos constantly. One is to wear a Tallis Katan at all times [with exception to when he is in a bathhouse].
The reason the Torah does not command one to wear a pair of Tzitzis:
Wearing Tzitzis represents the 613 commands and hence expresses one’s acceptance of the yoke of heaven. It is for this reason that the Torah does not command us to wear it, as accepting the yoke of Heaven must be performed by every individual on his own accord, and not through force or command. Thus in truth, the reason for the lack of obligation in wearing a four cornered garment with Tzitzis is not because doing so is optional, but because it touches such a fundamental point that it can only be fulfilled by the person on his own accord. Hence in essence, the Mitzvah of wearing Tzitzis throughout the day is not just a mere proper act but is the start of all of one’s religious service, and from perspective of man should be viewed as a complete obligation.
The punishment for not wearing a pair of Tzitzis:
Is one to be careful to wear a pair of Tzitzis also at night?
Yes. See Halacha 10 in Q&A!
Is one to wear Tzitzis when sleeping at night?
Yes. See Halacha 10 in Q&A!
Is one to be careful to wear a Tallis Katan while working a laborious job on a hot day, or while playing sports?
Yes. If one fears that the Tallis Katan will become dirty due to the above activity then he should purchase a second pair to wear during those times.
Is one to wear a Tallis Katan during marital relations?
Some write that one is to do so.
If one woke up in the morning without Tzitzis, may one wear it prior to washing hands?
One is to first wash hands near his bed and then put on the Tzitzis.
One woke up before daybreak: Even in the event that one woke up before morning, he is nevertheless to wear the Tzitzis without a blessing after washing hands, and then say a blessing after morning. Some Poskim however are lenient in such a case.
Having an extra pair of Tallis Katan:
It is proper that one own an extra pair of Tzitzis, just in case one of his Tzitzis becomes invalid. When traveling, one should bring with him two pairs of Tzitzis.
 24/1; Michaber 24/1
 Arizal in Shaar Hakavanos Derush Arvis; Kaf Hachaim 21/15-16
 The reason: The following is a parable for understanding why one should constantly wear Tzitzis. A certain individual was warned by a friend regarding a certain matter. The individual went ahead and made a knot around his belt area in order so he constantly remembers the warning. The knots of the Tzitzis serve a similar purpose; to remind us of the Mitzvos of Hashem at all moments. It is for this reason that the Tzitzis contains five knots, as they correspond to the five books of the Torah. We tie the Tzitzis to every corner in order so that whatever corner we turn to, we will remember the Mitzvos. [Admur ibid; Michaber ibid]
 Piskeiy Dinim Tzemach Tzedek 2/6 “To sit one even one moment without Tzitzis, no person would do so today G-d Forbid. The entire world is careful in Tzitzis, and it is not just a mere act of piety but rather a Biblical command”; Gr”a 24/1 in name of Rabbeinu Yonah that “the main Mitzvah of Tzitzis is to always wear it”; See also Baal Haitur Tzitzis 2/1 and Igros Kodesh 8/360
 Menachos 41a, stated to Rav Ketina by an angel; Rokeiach 361 “Since one is punished for not wearing Tzitzis therefore everyone must do so.” Tosafus Pesachim 113b “One is punished for not purchasing Tzitzis and making himself obligated in it”; Admur in Hilchos Talmud Torah 3 KU”A 1 “The punishment for nullifying a Mitzvah even though one is exempt, such as Tzitzis.” However see Rashi ibid and other Rishonim ibid that explain that only when one goes out of his way [Tatzdiki] to avoid the Mitzvah of Tzitzis is he punished and not when the chance is simply not readily available. The Ashel Avraham Butchach 21 explains that the punishment mentioned in the Menachos 41 only refers to one who nullifies the Mitzvah constantly. However one who usually fulfills the Mitzvah, and on occasion does not do so, is not included within the punishment. Certainly then one who does not wear a pair of Tzitzis at night has done nothing wrong and will not be punished. [ibid]
 Igros Moshe 4/4
 Daas Torah 8 in name of Or Zarua Birchas Hamotzi 140; See also Chinuch 386
 Shesilei Zeisim 24/1; Kaf Hachaim 24/1
 8/1; Tur 8/1; Levush 8/1; M”A 8/1
 The reason: As every person is required to be careful and hasten himself to precede the fulfillment of the Mitzvah as much as possible. [ibid] Alternatively, the reason is in order so one does not walk four Amos without Tzitzis. [P”M 8 A”A 1; Chayeh Adam 11/1] The reason it is worn after Netilas Yadayim is so he can say the blessing upon wearing it. [Kaf Hachaim 8/1] Alternatively, the reason is because it is forbidden to touch clothing prior to washing hands. [P”M 8 A”A 1; Kaf Hachaim ibid; See Admur Kama 1/7; Basra 1/7; Siddur; Seder Hayom; Peri Eitz Chaim; Beir Heiytiv 4/1; Ben Ish Chaiy Toldos 6; Olas Tamid p. 17a; Tzvaas Rebbe Eliezer Hagadol; Sheivet Mussar 16]
 See Even Ezra Shelach 15/39 that there is a greater obligation to wear Tzitzis throughout the day than during Shema and prayer as the entire purpose of the Mitzvah is to remind one of Mitzvos throughout the day when he is not involved in prayer.
 See Admur 8/20; Siddur Admur [letter 26 in Raskin]; Beir Heiytiv 12/6; P”M 8 A”A 1; Chayeh Adam 11/1; M”B 12/13; Poskim in Piskeiy Teshuvos 8/2 footnote 17
 Sefer Hasichos 1989 2/541
 Menachos 41a
 Tosafus Pesachim 113b; See Admur in Hilchos Talmud Torah 3 KU”A 1 “The punishment for nullifying a Mitzvah even though one is exempt, such as Tzitzis.”
 Tzitz Eliezer 14/49 in name of Mahari Algazi [regarding a hot day]; Rebbe in Igros Kodesh 17/27 [brought in Shulchan Menachem 1/61, regarding working a laborious job]; Az Nidbaru 2/55 [regarding exercise]
 Rebbe ibid
 Kaf Hachaim [Falagi] 10/7; Kaf Hachaim 21/16 based on Arizal that stated “One is never to remove the Tallis Katan except for when he is in the bathhouse.” The following Poskim rule it is permitted [but not necessarily obligated] to do so: Elya Raba 21/4; Chayeh Adam 11/37
 Admur 8/1; Tur 8/1; Levush 8/1; M”A 8/1; Kaf Hachaim 8/1
Opinion of Admur in 8/20: In 8/20 Admur mentions that one is allowed to place Tzitzis on him before washing in order not to walk 4 Amos without Tzitzis. Thus we see that at times getting dressed is allowed prior to washing if it is being done for the sake of a Mitzvah. However in truth no proof can be brought from this Halacha as it is going in accordance to the Talmud which permits touching clothing before washing. This is proven from the fact Admur there did not mention the prohibition of walking four Amos to reach the Tzitzis. [so also writes footnote 97 in new Shulchan Aruch Harav] Vetzaruch Iyun.
 The reason: The reason it is worn after Netilas Yadayim is so he can say the blessing upon wearing it. [Kaf Hachaim 8/1] Alternatively, the reason is because it is forbidden to touch clothing prior to washing hands. [P”M 8 A”A 1; Kaf Hachaim ibid; See Admur Kama 1/7; Basra 1/7; Siddur]
 Machatzis Hashekel 8/1; Aruch Hashulchan 8/3; Kaf Hachaim 8/1
 Chayeh Adam 11/1
 Kitzur SHU”A 68/6