The purpose of bedikat ḥametz is to find pieces of ḥametz that are a kezayit or larger, for one only violates bal yera’eh and bal yimatzei if there is a kezayit or more of ḥametz in one place in his home. The prohibition of bal yera’eh and bal yimatzei does not apply to less than a kezayit of ḥametz. Any part of the house where people sometimes bring ḥametz must be searched. Where children are present, one must search anywhere that the children reach; however, one need not search closets or high shelves that the children cannot reach.
Some poskim are stringent, maintaining that the purpose of the search is to ensure that not a single edible crumb of ḥametz remains in one’s possession, for if even one crumb remains, somebody might mistakenly eat it on Pesaḥ and thus transgress a Torah prohibition. Even though there is no punishment for eating less than a kezayit of ḥametz, the Torah nonetheless prohibits it. According to this position, one must search the entire house meticulously and keep an eye out for even small crumbs that might be ḥametz. Such a search in a normal house should take at least two hours. However, even according to stringent opinions, one need not search for crumbs so small that they are not recognizable as food. Likewise, there is no need to search for crumbs so filthy that they are inedible. For example, it is not necessary to inspect the cracks between floor tiles, because the crumbs there are repulsive and not fit to be eaten.
In practice, the halakha follows the lenient view because the obligation to search for ḥametz is rabbinically ordained. According to the Torah, one who mentally nullifies his ḥametz has already avoided the prohibition of ḥametz and need not search his home. It is the Sages who ordained that, in addition to the bitul, we must seek and destroy ḥametz. Whenever there is a disagreement about a rabbinic enactment, the lenient opinion is generally preferred.
All this applies to the house in general, but any place that comes into contact with food during Pesaḥ must be thoroughly cleaned, so that not even a crumb remains, for even the slightest amount of ḥametz renders food forbidden on Pesaḥ. Therefore, countertops, tables, and cabinets must be cleaned so well that not a single crumb of ḥametz remains.
At first glance, it seems possible to connect this dispute to a disagreement about the purpose of bedikat ḥametz. According to Rashi, the purpose of bedikat ḥametz is to prevent one from transgressing the Torah prohibitions of bal yera’eh and bal yimatzei, which, according to almost all authorities, one does not violate with less than a kezayit of ḥametz. Sha’agat Aryeh explains that small amounts of ḥametz do not combine (“lo ḥazi le-itztarufei”) to cause one to transgress bal yera’eh and bal yimatzei, unlike small amounts of forbidden foods, which combine to the size of a kezayit (according to Ḥakham Tzvi, since there is no action involved in the violation of bal yera’eh and bal yimatzei, the principle of “aḥshevei” does not apply). On the other hand, according to Tosafot, since the purpose of bedikat ḥametz is to prevent one from eating any ḥametz that remains in his house on Pesaḥ, it would seem that one would be required to check for even small crumbs of edible ḥametz, for one who eats them violates a Torah prohibition. And since we follow both Rashi’s and Tosafot’s reasons, and Ran also writes both, perhaps one must be stringent in this matter.
However, it seems that even according to Tosafot we can say that the Sages ordained bedika in addition to bitul out of concern that one will not wholeheartedly annul the ḥametz in his heart or that he will find a piece of cake and end up eating it. This applies only to a piece that is the size of a kezayit (at least its size nowadays) which has a certain amount of significance and can conceivably stimulate appetite (and which incurs karet). Even if one were to find, say, a small crouton somewhere in his house, it is not likely that he will eat it, since it was found where food is not normally stored. (And even if he does eat the crouton, he arguably did not intend to eat it; rather, he was cleaning, and instead of throwing the crouton in the garbage, he put it in his mouth. Such an action is not a Torah prohibition, and not something for which the Sages would mandate bedikat ḥametz. Nevertheless, perhaps one who normally eats these types of crumbs should be stringent in accordance with the ruling of Ḥazon Ish.)
See also Hilkhot Ḥag Be-ḥag ch. 6 n. 2, which discusses the two opinions in this matter and proves the lenient approach from the implication of Pesaḥim 4a that bedikat ḥametz lasts for less than an hour. And see Sidur Pesaḥ Ke-hilkhato §13 n. 39, which rules that one must check for crumbs. However, since this is a dispute about a rabbinic injunction, practice follows the lenient approach. (This is also the view of Or Le-Tziyon 1:32 and R. Elyashar’s Yevakesh Torah, ch. 9, based on the majority of poskim.) Nevertheless, one must differentiate between places that come into contact with ḥametz and the rest of the house, similar to the distinction made regarding kashering kelim and bedikat ḥametz.