📆 Month Lengths

30 days: Apr, Jun, Sep, Nov

31 days: Jan, Mar, May, Jul, Aug, Oct, Dec

February: 28 days (29 in leap years)

🔢 Leap Years

Divisible by 4: Leap year[web:250]

Divisible by 100: NOT leap (exception)

Divisible by 400: IS leap year[web:253]

⏰ Solar Year

365.2422 days per year[web:12]

365.25 days (Julian calendar)

365.2425 days (Gregorian average)

Formula 1: Leap Year Determination

Algorithm to determine if year is a leap year:[web:250][web:253]

\[\text{Leap Year} = \begin{cases} \text{True} & \text{if } (y \bmod 4 = 0) \land (y \bmod 100 \neq 0) \\ \text{True} & \text{if } y \bmod 400 = 0 \\ \text{False} & \text{otherwise} \end{cases}\]

Example: 2000 is leap (÷400), 1900 is not (÷100 but not ÷400), 2024 is leap (÷4).

Formula 2: Days Between Dates (Simple)

Basic calculation for days between two dates:[web:244][web:257]

\[\Delta D = |D_2 - D_1|\]

Where \(D_1\) and \(D_2\) are dates. Most programming languages calculate as milliseconds since epoch.

Formula 3: Count Leap Years Between Dates

Number of leap years before year \(y\):[web:247]

\[L(y) = \left\lfloor\frac{y}{4}\right\rfloor - \left\lfloor\frac{y}{100}\right\rfloor + \left\lfloor\frac{y}{400}\right\rfloor\]

Leap years between two years: \(L(y_2) - L(y_1 - 1)\)

Formula 4: Adding Days to Date

Calculate new date by adding days:[web:248][web:254]

\[D_{\text{new}} = D_{\text{start}} + n_{\text{days}}\]

Account for month boundaries and leap years when manually calculating.

Formula 5: Days in Month

Calculate days in any month:

\[D_m = \begin{cases} 31 & \text{if } m \in \{1,3,5,7,8,10,12\} \\ 30 & \text{if } m \in \{4,6,9,11\} \\ 29 & \text{if } m=2 \land \text{leap year} \\ 28 & \text{if } m=2 \land \text{not leap year} \end{cases}\]

Where \(m\) is month number (1-12).

Formula 6: Age Calculation

Calculate exact age from birth date:

\[\text{Age} = Y_{\text{current}} - Y_{\text{birth}} - \begin{cases} 1 & \text{if birthday hasn't occurred yet} \\ 0 & \text{if birthday has occurred} \end{cases}\]

Compare month and day to determine if birthday has occurred in current year.

📆 Project Management

Calculate project timelines, milestone dates, and deadline tracking. Essential for Gantt charts, critical path analysis, and resource planning.

💼 Business & Finance

Payment due dates, interest calculations, contract terms, billing cycles, and fiscal period analysis all rely on precise date arithmetic.

🎂 Age & Anniversary

Calculate exact age in years/months/days, count days until birthdays or anniversaries, track relationship milestones and life events.

🏥 Healthcare

Pregnancy due dates, medication schedules, appointment reminders, patient age calculations, and medical record chronology.

📚 Historical Research

Calculate time spans between historical events, verify dates in different calendar systems, analyze chronological sequences and timelines.

✈️ Travel Planning

Trip duration calculations, visa validity periods, passport expiration tracking, and itinerary planning across time zones.

📅 October 1582 "lost" 10 days! Countries switching to the Gregorian calendar skipped from Oct 4 to Oct 15 to realign with the seasons.

🎂 People born on Feb 29 are called "leaplings" or "leapers" - they technically only have birthdays every 4 years!

🌍 Not all countries use the Gregorian calendar! Ethiopia uses a calendar 7-8 years behind ours, and Islamic countries follow a lunar calendar.