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May one use the secular dates verbally or in writing?
The years: Some Poskim rule it is Biblically forbidden to write secular dates. Other Poskim however rule it is permitted. Practically, the custom is to be lenient as did many Gedolei Yisrael. However, those who are meticulous only write the last two digits, or write “Liminyanam/for their count”, or also write the Hebrew year.
The months in numbers: It is questionable whether it is permitted to write or say the secular month in numbers, such as 01 for January. The reason for this is because some Poskim rule that the Torah commanded us to count Nissan as the first month. Accordingly, if one counts any other month as the first one transgresses the Biblical command. Practically, it is best to be stringent to avoid using numbers for the secular months. One is to rather use the English names if he needs to.
The months in English names: It is permitted to call the months by their secular names.
Weekdays: One may call the weekdays by their English name.
 Piskeiy Teshuvos 156:3
 Maharam Shick Yoreh Deah 171; See also Chasam Sofer in Toras Moshe Bo and Drashos for 7th of Av; See Beir Moshe 8:19; Kinyan Torah 1:55; Yabia Omer 3:9
 The reason: The reason is because the years are counted from when their false deity, yoshka, was born and it is forbidden to mention the name of a deity. [ibid]
 Mishmeres Shalom Semachos 84; Kinyan Torah 1:55; Yabia Omer 3:9
 The reason: As the prohibition of mentioning the name of a deity is only applicable when the words explicitly imply it. [Poskim ibid; Hagahos Maimanis Avoda Zara 5; Darkei Moshe Y.D. 147]
 Shut Rama 51; Maharam Pawa 36; Chavos Yair 184; Chasam Sofer E.H. 1:43
 Mishmeres Shalom ibid; See Darkei Chaim Veshalom 1010
 Ramban Bo ibid; However see Rashba on R”H 7a and Ran and Ritva there that imply that no such Mitzvah exists, as they question what practical ramification is there in knowing which month is the first month, and they do not answer as above; See also Baal Hikkarim that this Mitzvah no longer applies today; Likewise the Rambam does not list this Mitzvah anywhere in his Mishneh Torah or Sefer Hamitzvos.
 Minchas Chinuch Mitzvah 311
 Toras Moshe Parshas Bo; Iyun Yaakov Megillah Pirush Hakosev; Mishneh Sachir 1:120; Yabia Omer 3:9
 Iyun Yaakov Megillah 2 Pirush Hakosev; Yabi Omer ibid; Piskeiy Teshuvos 156:3; However, see Tzitz Eliezer 8:8
 Iyun Yaakov Megillah 2 Pirush Hakosev; Yabia Omer ibid; However, see Tzitz Eliezer 8:8 that one is to avoid doing so being that these names are rooted in idolatry.
 Igros Kodesh 2:361 [printed in Shulchan Menachem 1:93]; See Piskeiy Teshuvos 156 footnote 18