בחדש בראש חדש כמו מחר חדש. בחדש, “on the first of the (third month);” the word חדש is used here as in Samuel I 20,18: מחר חדש, “tomorrow is new moon.”
השלישי לפי שנשבו ונשתחררו ונתגיירו, וכן שנו רבותינו הגיורת והשבויה והשפחה שנפדו ושנתגיירו ושנשתחררו צריכות להמתין שלשה חדשים. השלישי, it is called thus because it was when the Israelites were enslaved, נשבו, when they were liberated, נשתחחרו, and when they converted to Judaism, נתגגיירו. The latter word is the reflexive mode of גר, alien, stranger. Our sages in the Talmud Ketuvot 37 taught us: different categories of female converts, including those who were prisoners and have been released and converted have to wait three months before they can marry a Jew, to eliminate the suspicion than a child born to them was sired by a gentile before they had converted.
ביום הזה שנסעו מרפידים בראש חדש סיון שהיה ביום שני בשבת היינו אליבא דרבנן, כיצד קים לן בפרק ר׳ עקיבא דבשבת היה מתן תורה לדברי הכל. ורבנן אמרי בששה בחדש נתנה, א״כ ר״ח סיון היה ביום שני בשבת ואייר דההיא שתא מלויי מליוהי. וא״ת א״כ מצינו שבעה שבועות ויום אחד מפסח לעצרת, אלא י״ל לא נצטוו על ספירה עד לאחר ביאה. ביום הזה, ”on this day;” the day referred to is the day they broke camp in Refidim. It was the first day of Sivan which occurred on a Monday, according to the opinion of the majority of the scholars. There was unanimity among the scholars that the day of the revelation was Sabbath. (Shabbat 86) According to the majority of the Rabbis who hold that that the revelation occurred on the sixth day of the month of Sivan, the first day of that month had to have been on a Monday. This is based on the month of Iyar during that year having had 30 days. If you were to argue that if this is so then 50 days as opposed to 49 days, i.e. seven weeks, had elapsed since the Exodus, the answer is that the commandment regarding when the omer is to be offered was not given to the people (or obviously could not apply) until they had taken possession of the land and planted a crop.when the omer
ביום הזה באו מדבר סיני אמר רבי לוי משל לבן מלך שעמד מחליו אמר פדגוגו ילך לו לאסקולא אישקול״א בלע״ז פי׳ בית שמלמדים בו תינוקות. ביום הזה באו מדבר סיני, “on this day they arrived at the desert of Sinai.” Rabbi Levi in Pessikta de Rav KahaneMandelbaum edition page 205), relates a parable [to explain that the Torah describing this detail conveyed a lesson worth taking to heart, Ed.] A king had a son who had recovered from a serious illness. He was told by a pedagogue that the son should go an elementary school forthwith. The father protested by saying to the pedagogue that his son was still too frail to go to school and be subjected to a strict regimen. He agreed to send him to school after a recovery period of three months. He intended to pamper him during these three so that he would recuperate faster.