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The floor function, denoted by \( \lfloor x \rfloor \), calculates the largest integer that is less than or equal to a given real number \( x \). This function always rounds down the number to the nearest integer.
For example:

• \( \lfloor 3.7 \rfloor = 3 \)
• \( \lfloor -2.3 \rfloor = -3 \)

Understanding the floor function is crucial when working with inequalities involving real numbers and integer constraints. In the context of our exercise, using the floor function helps to find the maximum integer within a defined boundary.

The ceiling function, denoted by \( \lceil x \rceil \), finds the smallest integer that is greater than or equal to a given real number \( x \). This function always rounds up the number to the nearest integer.
For example:

• \( \lceil 3.2 \rceil = 4 \)
• \( \lceil -1.7 \rceil = -1 \)

The ceiling function is vital for identifying integer thresholds in mathematical problems involving real numbers. In our specific case, it helps determine the smallest integer greater than the lower boundary of an inequality.

Real numbers include all the numbers that can be found on the number line, encompassing both rational and irrational numbers.
- Rational numbers are numbers that can be expressed as fractions, such as \( \frac{1}{2} \) or \( 3 \).
- Irrational numbers cannot be expressed as simple fractions and include numbers like \( \sqrt{2} \) and \( \pi \).
Real numbers are denoted by the symbol \( \mathbb{R} \) and are fundamental in various mathematical theories and applications.
In the given inequality \( a < n < b \) where \( a \) and \( b \) are real numbers, understanding real numbers is crucial because it shows the continuum of values that \( a \) and \( b \) can take, impacting the resulting integer solutions.