# Mathedu

Learn math by practice

# Re-engineering Math

## The Measurement of Time I: the dates in the calendar

As we saw in a previous post, the metric system for the measurement of lengths is a decimal system, and the time measurement system is not.

First, the basic time units are NOT the hour or the second, but the day, the month and the year:

• The day is the time by which the earth spins on its axis completely.
It used to be measured by the time between 2 instants at which the sun is at its highest position in the sky.
• The year is the time by which the earth turns around the sun by a full orbit.
It used to be mearured by the time between 2 "shorter days", 2 Winter Solstices.
• The month used to be the time by which the Moon turns around the earth a complete revolution, or the time between 2 "full moons".

The rough conversions between these units are so:

• A year is very close to 365 days, the difference being corrected by the system of leap years.
• A "Lunar month" is about 29 days and a year is roughly 12 Lunar months
Our current calendar contents 12 months of 30 or 31 days each, except February that is 28 days (or 29 for the leap years), the total being 365 days (or 366 the leap years).

Thus, if we wish to convert years in months, we have to multiply the number by 12 (5 years are 5x12=60 months). We may also convert months in days, but only roughly, by a multiplication by 30 (3 months are roughly 90 days).

There is another useful "small" unit of time: the week, that has no astronomic sense, but that is exactly of 7 days. For our schedules, it may be useful to convert months in weeks: one month is roughly 4 weeks.

But here encountered we a common missuse of the conversion laws. We could say that 3 months are 3x4=12 weeks, thus 12x7=84 days, that is quite far from 90 days!

As a matter of fact, a quarter of 3 months is closer from 13 weeks, as 13x7=91.

So that a year of 4 quarters is about of 4x13=52 weeks, that is well known: a year of 365 days is 52 weeks and 1 day, and a leap year of 366  days is of 52 weeks and 2 days. This is because 7x52=364.

So, we can see that the conversion laws are rather whimsical, especially when used with approximated values…