In all our measurement systems the larger unit consists of several smaller units - one meter is 10 dm, 1dm = 10cm, 1cm = 10 mm and so on. Unfortunately we cannot do this with calendar while at the same time preserving the natural calendar units, because the length of lunar month is between 29 and 30 days and the length of a year is between 12 and 13 lunar months.
We can, of course, have something like 29.5 days months and 12.5 months years, but this kind of calendar will be very difficult to use. Imagine what life would be like if each month starts in the middle of the day, at a different time of a day; and the year would start in the middle of a month and the date of the New Year would be constantly shifting from year to year. To avert this catastrophic condition, which would render the calendar unusable, the calendars must adjust the length of a month to whole days and the length of a year to whole months.
The Jewish tradition also requires the use of the whole days and months. Rabbi Gedaliah Lipschutz (better known as Tiferes Yisroel / *** ***) in his work "The Paths of the Heaven" / "*** ****" mentions that we derive this rule from two passages in Torah. The passage in the Book of Numbers 11:20 "Until a month of days" / "**** ****" is understood to indicate that the month must be composed exclusively of whole days and may not contain any additional hours or parts of a day. Another passage in the Book of Exodus 12:2 "For the months of the year" / "*** ****" is understood to indicate that a year must be composed exclusively of whole months and may not contain any additional days.