Question

A ycar with 366 days is called a leap year. Leap years are necessary to kecp the calendar synchronized with the sun because the earth revolves around the sun once every \(365.25\) days. Actually, that figure is not entirely precise, and for all dates after 1582 the Gregorian correction applics. Usually years that are divisible by 4 are leap years, for example 1996 . However, years that are divisible by 100 (for example, 1900 ) are not leap years, but years that are divisible by 400 are leap years (for example, \(2000)\). Write a program that asks the user for a year and computes whether that year is a leap ycar. Use a single if statement and Boolean operators.

Short answer

Use the rule: divisible by 4 but not 100, or divisible by 400.

Step-by-step solution

Understanding the Leap Year Rules

We need a program that determines whether a given year is a leap year. A year is a leap year if it is divisible by 4. However, if a year is divisible by 100, it is not a leap year unless it is also divisible by 400.

Setting Up the Inputs

Ask the user for the input year. This is usually done with an input statement in programming languages. For Python, you would typically use `year = int(input('Enter a year: '))` to read an integer year from the user.

Writing the Conditional Statements

To determine if a year is a leap year, we need to check multiple conditions using a single `if` statement. This involves using Boolean operators (`and`, `or`). The condition can be structured as follows: - If the year is divisible by 4 but not divisible by 100, or if it is divisible by 400, then it is a leap year. In Python, the conditional check could be written as: `if (year % 4 == 0 and year % 100 != 0) or (year % 400 == 0):`.

Returning the Result

Based on the condition in Step 3, print the result indicating whether the year is a leap year or not. If the condition evaluates to `True`, print 'Leap Year', otherwise print 'Not a Leap Year'. For Python, you could use: ```python if (year % 4 == 0 and year % 100 != 0) or (year % 400 == 0): print('Leap Year') else: print('Not a Leap Year') ```

Key concepts

Gregorian Calendar

The Gregorian Calendar, introduced by Pope Gregory XIII in 1582, is a solar calendar used almost globally today. This calendar was an improvement over the Julian calendar, which did not accurately align with the solar year, causing dates to drift over time. The Gregorian reform corrected this by introducing the concept of leap years, thus maintaining synchronization with Earth's revolutions around the Sun.

In the Gregorian Calendar, most years are 365 days long. However, to account for the extra 0.25 days each year, an additional day is added every four years, called a leap year, making that year 366 days long. But, simply adding a day every four years would over-correct, because the precise astronomical year is about 365.2425 days, not 365.25 days. To adjust for this, the Gregorian calendar includes a rule for century years: only one out of every four century years is a leap year. Thus, a century year must be divisible by 400 to be a leap year. For example, the year 2000 was a leap year, but 1900 was not.

Boolean Operators

Boolean operators are fundamental in programming and logic as they allow condition testing with true or false values. In Python, these operators are

• `and`
• `or`
• `not`

They enable combining multiple conditions to form complex logical expressions.

A Boolean expression evaluates to `True` or `False`. For instance, in the context of leap year calculation, we use these operators to check if a year is divisible by certain numbers: 4, 100, and 400.

The `and` operator results in `True` only if both expressions on its sides are `True`. In our year check, `(year % 4 == 0 and year % 100 != 0)` ensures a year is divisible by 4 but not 100.

The `or` operator results in `True` if at least one of its operands is `True`. Thus, `(year % 4 == 0 and year % 100 != 0) or (year % 400 == 0)` evaluates to `True` if at least one of these conditions is satisfied.

Python Conditional Statements

Python conditional statements enable decision-making in code by performing actions based on conditions that evaluate to `True` or `False`. The `if` statement is a fundamental part of this. It executes code within it if the condition is `True`, and can be paired with `else` to handle cases where the condition is `False`.

In a leap year calculation program, this looks like: ```python if (year % 4 == 0 and year % 100 != 0) or (year % 400 == 0): print('Leap Year') else: print('Not a Leap Year') ``` Here, the `if` statement checks the leap year conditions using our Boolean expression. If the expression evaluates as `True`, indicating the year meets leap year criteria, the code prints `'Leap Year'`. Otherwise, it moves to the `else` block to print `'Not a Leap Year'`. This simple structure allows Python to determine the correct path based on logic and execute actions rapidly and accurately.