Gemara

47b (Rav): All years mentioned in this Perek must be (full,) from day to day.

(Beraisa): The years of a boy or girl must be full years.

48a (Rav Gidal citing Rav): This refers to the years of Erchin.

Bechoros 49a (Mishnah): If Reuven's Bechor died after 30 days, even if Reuven had not yet given five Shekalim to a Kohen, he must give now.

If he died on day 30, this is like dying within 30 days. (If Reuven already gave, the money is returned).

R. Akiva says, if he already gave, it is not returned. If he did not yet give, he need not give.

Chachamim learn a Gezeirah Shavah "Chodesh-Chodesh" from the counting of the Bechoros in the Midbar. Just like only Bechoros above 30 days were counted, also Pidyon ha'Ben applies only after 30 days.

R. Akiva is unsure whether we learn from there;

It says "va'Ma'alah" regarding Erchin to teach that a 30-day old is like less than 30 days (he has no Erech). We do not learn this from Bechoros (in the Midbar). Therefore, Erchin and Bechoros are Shnei Kesuvim. We do not learn from them to other places. However, perhaps Pidyon ha'Ben is considered the same matter! (The Bechoros were counted for the sake of redemption.)

Yevamos 41 (Mishnah): No woman may become Mekudeshes or married for three months, whether she was divorced or widowed, from Nisu'in or Kidushin.

43a: A case occurred in which a man made a Kidushin feast on the 90th day, and Rava canceled it.

48b (Beraisa): "A month of days" (regarding Eshes Yefas To'ar) refers to 30 days;

R. Shimon ben Elazar says, it is 90 days. "Month" is 30 days, "days" are 30, and "after this" is another 30.

Megilah 5a (R. Aba): "A month of days" teaches that a month is always composed of full days.

Sanhedrin 11a (Beraisa): We add 30 days to make an Ibur Shanah;

R. Shimon ben Gamliel says, we add a month.

(Rav Papa): R. Shimon ben Gamliel means that we may add 29 or 30 days.

Kesuvos 70a (Mishnah): If one vowed that his wife may not benefit from him, up to 30 days, he gets someone to finance her. After this, he divorces her.

R. Yehudah says, if he is a Yisrael, he keeps her wife for one month. If he vowed for two months he must divorce her. A Kohen keeps her for two months.

71a Question: Also the first Tana said that a Yisrael keeps her for one month!

Answer #1 (Abaye): True! R. Yehudah comes merely to teach about a Kohen.

Answer #2 (Rava): The first Tana says allows a full month of 30 days. R. Yehudah allows only a short month of 29 days.

Rishonim

Rambam (Hilchos Bechoros 11:17): One is obligated to redeem from when 30 days are completed, for it says "u'Fduyav mi'Ben Chodesh Tifdeh." If the son died within 30 days, even on day 30, he is exempt.

Tosfos (Sanhedrin 11a DH Raban): Our Gemara connotes that a Stam Chodesh is 29 days. Kesuvos 71a connotes similarly. However, in Yevamos we say that it is 30 days! Similarly, in Nazir (5a) we ask that Chodesh is 30 days. I answer that the Torah uses Chodesh (to mean 30 days) unlike Chachamim (who mean 29). Alternatively, here and in Kesuvos, since the first Tana says 30 days, this shows that the other Tana refers to a Chaser month.

Tosfos (Kesuvos 71a DH Rebbi): In Yevamos, R. Yehudah says that a Chodesh of the Torah is 30 days. We say similarly in Nazir. For mid'Rabanan laws, a month is Chaser, like it says in Erchin. In Rosh Hashanah, R. Shimon ben Gamliel says that a Chodesh may be Chaser or full. Here we ask that he says the same as the first Tana! Also Abaye, who says that he comes to teach about a Kohenes, holds that it is 30 days! The Ri answers that since he teaches that one may keep a Kohen for two months, he mentioned Chodesh. Regarding the three months of Havchanah and nine months of pregnancy, they are 30 days. (Even so, it is called three months,) because it is burdensome to explicitly say 90 days and 270 days. Or, we can say that even Chachamim use Chodesh to discuss 30 days when there is no proof otherwise. When there is a proof is different.

Poskim

Shulchan Aruch (YD 305:11): A Bechor cannot be redeemed until 30 days passed.

Bach (14): It is clear from Bechoros that we do not redeem until day 31. Semak (245) says that one must redeem the Bechor on the 31st day. If he redeemed him on day 30 or earlier, he is not redeemed. However, we do not necessarily require the 31st day. As long as 30 days me'Es la'Es passed, we may redeem. Chachamim say that if he died on day 30, it is like before. They learn from a Gezeirah Shavah Chodesh-Chodesh from the Midbar. If so, we require only a full month, i.e. 29 days, 12 hours and 793 parts (out of 1080 in an hour). Semak said the 31st day, for then one may always redeem. On day 30, one must make an exact calculation, and perhaps someone will err. Therefore, we redeem on day 31, to avoid mistakes.

Shach (12): If day 31 is a fast day, if 29 days (and 12 hours...) passed on day 30, we redeem then.

Magen Avraham (OC 339:8): The Bach and Shach permit redeeming after 29 days (and 12 hours...). One may not rely on them. The Gemara said that if he died on day 30, this is like the day before. We asked their reason, and answered that they learn from the Midbar. What was the question? A month did not pass yet! Rather, we asked about after 29 days (and 12 hours...) passed, and answered that we learn from the Midbar that one must wait until day 31. When a baby died on day 31, we do not distinguish between before or after 29 days (and 12 hours...) Also, Tosfos (Sanhedrin 11a DH Raban, Yevamos 48 DH Yerach) says when the Torah says 'Chodesh', it is 30 days. The other answers say so even regarding when Chachamim said 'Chodesh.' Rosh Hashanah 19b and Megilah 5a connote like this.

Panim Me'iros (1:3): A case occurred in which day 30 of a Bechor was on Friday; 29 days (and 12 hours...) were completed in the morning. People wanted to redeem him, like the Magen Avraham and Bach. I disagree. In (Erchin and) Sanhedrin, Rashi explains that R. Shimon ben Gamliel says that a Stam Chodesh is 29 days. Tosfos asked from Yevamos, in which we say that it is 30 days. Tosfos' second answer is primary, that when there is no proof otherwise, also Chachamim use Chodesh to denote 30 days. A proof is from Yevamos. A man made a nuptial feast on the 90th day, and Rava canceled it. There, there is no proof (from the Mishnah how long the months are). According to Tosfos' first answer, the months should be 29 days. Why did Rava cancel the feast?! Tosfos in Kesuvos said that Chachamim said three months because it is burdensome to say 90 days. This is a poor answer.

Panim Me'iros: The Bach brought from the Yere'im that month of the Torah are lunar months. We have a tradition from R. Gamliel that it is at least 29 days (and 12 hours...). In Nidah we say that the years of a boy or girl must be full years regarding Erchin, i.e. mi'Ben Chodesh. Chachamim said that Ben Chodesh for Erchin must be day to day; the same applies to Pidyon ha'Ben. This is astounding. In Erchin 18a, we say that day 30 is like before. Rashi explains that if one pledged the Erchin of a baby on day 30, his words have no effect. This is even if 29 days (and 12 hours...) passed. If not for the Gezeirah Shavah, we would have said that the baby is no longer a Safek Nefel and it is considered a month old. We cannot say that the Shi'ur is 30 days me'Es la'Es. The Gezeirah Shavah obligates "v'Ma'alah" (above a month)! The Gemara requires me'Es la'Es regarding years. One who was redeemed on day 31, even after 29 days (and 12 hours...), was not redeemed at all, and he must be redeemed again.

Kesav Sofer (YD 151): I saw a manuscript of my grandfather R. Akiva Eiger that he was unsure about the Magen Avraham. If a baby was born at Tzeis ha'Kochavim, 29 days (and 12 hours...) are finished when it gets light in the morning of the 30th. The Magen Avraham is stringent to wait until the 31st. This is based on Megilah 5a, which says that we count days towards months, but we do not count hours. Here, the Torah says "Ben Chodesh"! The Safek is when he was born before night. Then, part of the day counts like the entire day, but 29 days (and 12 hours...) are not completed until it is light on the morning of the 31st. Do we say that from the beginning of the night of the 31st, one may redeem him, even before 29 days (and 12 hours...), since part of the day is like the entire day? Or, perhaps we need a full month of the moon! Or, since we do not count hours for months, one must wait until day 32! It is difficult to say so. This requires investigation. My father (the Chasam Sofer) asked R. Akiva Eiger about this, and did not resolve it. On the Magen Avraham, he wrote a marginal comment to see the Chacham Tzvi, Perach Mateh Aharon and Sidrei Taharah 189. Panim Me'iros rejected the Bach and Shach. Also the Ya'avetz did; he said that we are not concerned at all for 29 days (and 12 hours...). Rather, we require 30 days and part of the 31st day, which is like the entire day. He connotes that this is like his father (the Chacham Tzvi). Really, the Chacham Tzvi connotes that we also require 29 days (and 12 hours...). He connotes that he is always more stringent than the Bach and Shach. If there is any Safek, we should not be lenient, for we can redeem the next day.

Pischei Teshuvah (17): Tevas Gome says that if a baby was born a half-hour before Motza'ei Shabbos, we may not redeem immediately on Monday morning, until 29 days (and 12 hours...) have passed in the summer, for the night is short (less than 12 hours).