1) LESS THAN A "K'ZAYIS" OF CHAMETZ
QUESTION: The Mishnah (45a) states that one is obligated to destroy the Chametz in the crevices of a kneading basin if the Chametz is the size of a k'Zayis. If the Chametz in the basin's crevices is less than a k'Zayis, one is not obligated to destroy it, because it is Batel.
The Gemara explains that half of a k'Zayis of dough is Batel if it is either on the bottom of the basin or on the "middle part of the inside of its walls." If, however, the Chametz is on the rim or on the "top of the inside of its walls" or on the "outside of the walls," then even half of a k'Zayis must be destroyed and is not Batel to the basin (see RASHI DH Ela).
The Gemara continues and says that if there are two half-k'Zeisim on the inside walls of the basin and they are not connected at all (by a thread of dough), they are both Batel. Ula adds that this leniency applies only with regard to Chametz in the crevices of a basin. When the Chametz is in a house, two half-k'Zeisim must be disposed of even if they are not connected. This is because it is possible that one will sweep the separate half-k'Zeisim together into one pile when he sweeps the house, and thereby they will combine to form a full k'Zayis in one place.
Why does the Gemara need this reason to explain why two half-k'Zeisim of Chametz in a house must be destroyed? The Gemara earlier teaches that if the two half-k'Zeisim would be on the outside of the walls of the basin, they would not be Batel to the basin and one would be required to get rid of them. The same certainly applies when they are on the floor of a house; they are not Batel to anything, and thus each half-k'Zayis by itself must be destroyed!
(a) The MAHARAM CHALAVAH writes that in the case of a house, one might have thought that small pieces of Chametz that are less than a k'Zayis do not have to be disposed of, because the Halachah of "Pirurin" (6b) dictates that small, insignificant crumbs of Chametz are Batel. In the kneading basin, though, such small pieces of dough are still in good, usable condition, and thus they are still significant and are not Batel. The Gemara concludes that even in a house, small pieces of dough are not Batel because they might come together to make a k'Zayis when one sweeps the house. Since they might come together and become significant, they are not Batel even when they are separate. When the Gemara earlier (6b) says that small crumbs are Batel, it refers to dried crumbs that will never become fit for use. Alternatively, as the PISKEI RI'AZ explains, the Gemara earlier refers to very small crumbs, and there is no concern that they will join together to become a k'Zayis. The MAGEN AVRAHAM (OC 442:2) gives the same explanation.
Similarly, RABEINU DAVID answers that small pieces of Chametz are normally Batel; only when they are on a kneading basin are they not Batel, because of the concern that they might join together to make a Shi'ur. Rabeinu David explains that pieces of Chametz on the outside of the basin must be disposed of because of the concern that they might come in contact with each other and form a k'Zayis of Chametz. The Gemara concludes that a house has the same status as the outside of a basin (and the small pieces of Chametz in a house are not Batel due to the fact that they are so far apart from each other).
(b) The MAGID MISHNEH (Hilchos Chametz u'Matzah 2:16) infers from the words of the RAMBAM that the requirement that half of a k'Zayis on the floor of a house be disposed of applies only when the Chametz is not caught between the floor boards. If the pieces of Chametz are between the floor boards and effectively fill up cracks in the floor, then they are Batel, just like the pieces of dough on the inside of a kneading basin which serve a purpose for the basin and are Batel.
(c) The RAMBAN (in Milchamos) and RABEINU DAVID note that according to the RIF, the Gemara here follows the view of Rav Huna who, earlier, did not reconcile the two conflicting Beraisos by saying that one Beraisa refers to places in the basin which are "Makom Lishah" while the other Beraisa refers to places which are not "Makom Lishah." Rather, Rav Huna explained that the two Beraisos argue; one maintains that a half-k'Zayis on the inside of the wall of the basin is Batel, while the other maintains that it is not Batel and must be destroyed. The Gemara here (Rav Nachman), which discusses a half-k'Zayis on the floor of a house, follows the view of the Beraisa that says that a half-k'Zayis is Batel even when it serves no other purpose.
(d) TOSFOS (45a, DH Kan) explains "Makom Lishah" differently. Tosfos says that when pieces of dough are not in the "Makom Lishah," the Halachah is less stringent. The Chametz there is Batel, because the dough inside the basin will not come into contact with it, and there is no concern that one might eat it. Accordingly, a half-k'Zayis on the floor of a house certainly should be Batel (if not for the concern that they might be swept into one pile to form a k'Zayis).
(e) RABEINU YECHIEL (cited by TOSFOS RABEINU PERETZ) and TESHUVOS PNEI YEHOSHUA (#15) suggest that small pieces of Chametz are Batel, but not because of the Halachah of "Pirurin" (6b; see (a) above). Rather, they are Batel because they are a half of a Shi'ur ("Chatzi Shi'ur"), and the prohibition of Bal Yera'eh does not apply to Chametz less than the minimum Shi'ur.
However, according to this answer, why should a half-k'Zayis not be Batel when it is on the rim or the outside of the kneading basin? The Teshuvos Pnei Yehoshua concludes that it must be that the Gemara is discussing a basin that has lots of pieces of Chametz around its outside, which all add up to a k'Zayis. Therefore, they must be disposed of, because the basin serves to join them together to make a k'Zayis. However, when there are multiple pieces of Chametz less than a k'Zayis on the floor of a house, it is not necessary to dispose of them, because there is nothing to join them together to make a k'Zayis. Therefore, the Gemara needs to explain that the reason why such pieces must be destroyed is because one might sweep them together into one pile. This is also the way the KORBAN NESANEL (3:2:100) understands the Gemara.
According to this explanation, however, why does the Isur of Bal Yera'eh not apply to half of a Shi'ur? Chatzi Shi'ur is prohibited mid'Oraisa (as the Gemara in Yoma (74a) teaches with regard to one who eats a Chatzi Shi'ur of food on Yom Kippur). Why does the Gemara here assume that Chatzi Shi'ur of Chametz is not included in the Isur of Bal Yera'eh?
The CHACHAM TZVI (#86) explains that Chatzi Shi'ur prohibits only eating partial Shi'urim of forbidden food items. When one eats the item he gives it importance, regardless of how small it is, and therefore it is forbidden. Chatzi Shi'ur does not apply to a prohibition that does not involve eating, such as Bal Yera'eh.
The SHA'AGAS ARYEH (#81) gives a different reason for why the principle of Chatzi Shi'ur does not apply to Bal Yera'eh. Why is a Chatzi Shi'ur of a prohibited food forbidden? The Gemara in Yoma explains that it is forbidden because a small amount of food is "Chazi l'Itzterufi" -- it is fit to join together with other small amounts of food and become a proper Shi'ur. This means, according to one way of understanding, that one may not eat a half-Shi'ur, because he might eat another half-Shi'ur. He will thereby consume an entire Shi'ur and will transgress the Isur d'Oraisa retroactively through the first half-Shi'ur that he ate. Anything which can lead to the transgression of an Isur retroactively is deemed forbidden now as well.
The prohibitions of Bal Yera'eh and Bal Yimatzei are different. It is not possible for Chatzi Shi'ur to be forbidden, because in order to transgress Bal Yera'eh and Bal Yimatzei, the full Shi'ur of a k'Zayis of Chametz must be in one's house all at one time. One transgresses Bal Yera'eh without doing any action with the prohibited item; he transgresses only when the full Shi'ur is in his possession. In contrast, one transgresses an Isur Achilah, a prohibition against eating a forbidden food, even when he performs two acts at two different times (within Kedei Achilas Peras) with two partial Shi'urim which join together to make one full Shi'ur. When no action is involved in the transgression, no temporal element can join two partial Shi'urim; one must have the entire Shi'ur at once in order to transgress.
Accordingly, a Jew is permitted to own a "Chatzi Shi'ur" on Pesach, because the "Chatzi Shi'ur" of Chametz cannot become forbidden retroactively. If another "Chatzi Shi'ur" of Chametz is brought into the house later, only from that point on does the first "Chatzi Shi'ur" become forbidden. (See also Insights to Pesachim 29:3 and Yoma 74:2.)
Many Acharonim reject this approach. They argue that there are several reasons why Bal Yera'eh should be prohibited even with a Chatzi Shi'ur. First, the MINCHAS CHINUCH (11:12) says that even though the logic of the Sha'agas Aryeh explains why there is no Isur d'Oraisa of Chatzi Shi'ur for Bal Yera'eh, there should still be an Isur d'Rabanan of Chatzi Shi'ur. Second, the MAHARAM CHALAVAH writes that we know that the purpose of the prohibition of Bal Yera'eh, of not having Chametz in one's possession, is to prevent one from eating Chametz on Pesach. Consequently, even if the prohibition of Chatzi Shi'ur applies only to Isurei Achilah, forbidden food items, the Shi'ur of Bal Yera'eh is based on the Shi'ur of Achilah, and therefore Chatzi Shi'ur should be Asur mid'Oraisa for Bal Yera'eh just as it is Asur for Achilah. (See also TAZ OC 442:5.)
2) HALACHAH: HOW EXTENSIVE MUST THE SEARCH FOR CHAMETZ BE
QUESTION: The Gemara teaches that one is obligated to dispose of crumbs of Chametz that are in one's home even when they are less than the size of a k'Zayis. How extensive does one's Bedikas Chametz have to be?
ANSWER: The ROSH here does not record all of the Halachos discussed in the Gemara with regard to the small amounts of Chametz that are in the kneading basin. Rather, he writes that it is not important to discuss all of the intricacies of the Halachah, because "Yisrael Kedoshim Hem" -- the Jews are a holy nation, and their practice is to be stringent and to rub off any possible traces of Chametz from the walls of their homes, from the chairs and the like. The Rosh cites support for this practice from the Yerushalmi that states that even Chametz that was made as part of the floor must be destroyed unless it is not fit for animal consumption.
In practice, many families follow the custom to clean and dispose of any possible traces of Chametz in their homes. However, the Sages have stated that during the weeks and days before Pesach, one should ensure that the members of his family, especially his wife, do not impede their enjoyment of the Yom Tov as a result of excessive toil.