Time & Calendar

Calendar rules, weekday and date algorithms, clock and astronomical timekeeping.

Time & Calendar formulas help you make sense of dates, clocks, and the rhythms of both everyday life and the sky. From finding leap years and day-of-week patterns to converting time zones, counting days, and working with Julian or Unix time, these formulas are useful for students, exam practice, and real-world tasks in scheduling, computing, astronomy, and data handling.

Leap Year Rule

\text{leap} \iff (y \bmod 4 = 0 \;\wedge\; y \bmod 100 \neq 0) \;\vee\; (y \bmod 400 = 0)

Tests whether a Gregorian year has 366 days.

Clock Hand Angle

\theta = |30H - 5.5M|

Angle between the hour and minute hands of an analog clock.

Zeller's Congruence

h = \left(q + \left\lfloor\tfrac{13(m+1)}{5}\right\rfloor + K + \left\lfloor\tfrac{K}{4}\right\rfloor + \left\lfloor\tfrac{J}{4}\right\rfloor + 5J\right) \bmod 7

Finds the day of the week for any Gregorian date.

Day of the Week from a Birthdate

W = \left(d + \left\lfloor 2.6m - 0.2 \right\rfloor + y + \left\lfloor\tfrac{y}{4}\right\rfloor + \left\lfloor\tfrac{c}{4}\right\rfloor - 2c\right) \bmod 7

Gauss's weekday formula — find which day of the week any birthdate fell on.

Julian Day Number

JDN = \left\lfloor 365.25(Y+4716) \right\rfloor + \left\lfloor 30.6001(M+1) \right\rfloor + D + B - 1524

Continuous day count used by astronomers to compare dates.

Days Between Two Dates

\Delta = JDN_2 - JDN_1

Number of days separating two calendar dates.

Age in Years

A = \left\lfloor \dfrac{JDN_{today} - JDN_{birth}}{365.2425} \right\rfloor

Whole years elapsed since a birth date.

Unix Timestamp

t = 86400\,d + 3600h + 60m + s

Seconds elapsed since 1970-01-01 00:00:00 UTC (the epoch).

Sidereal Day

T_{sid} = \dfrac{T_{sol}}{1 + T_{sol}/T_{yr}}

Earth's rotation period measured against the stars, not the Sun.

ISO Week Number

W = \left\lfloor \dfrac{d_o - w_d + 10}{7} \right\rfloor

ISO-8601 week of the year (weeks start on Monday).

Local Time from UTC

t_{local} = t_{UTC} + \Delta

Convert Coordinated Universal Time to a local zone.

Equation of Time

E \approx 9.87\sin 2B - 7.53\cos B - 1.5\sin B

Difference between sundial (apparent) and clock (mean) time.

Period & Frequency

T = \dfrac{1}{f}

A repeating event's period is the reciprocal of its frequency.

Ordinal Day of Year

N = \left\lfloor \dfrac{275M}{9} \right\rfloor - k\left\lfloor \dfrac{M+9}{12} \right\rfloor + D - 30

Day-of-year number (1–366) for a given date.

Easter Date (Computus)

d = (19a + b) \bmod 30,\quad a = y \bmod 19

Core step of the Gregorian algorithm locating the Paschal full moon.