In this Grade 4 AMC Math lesson from AoPS Introduction to Algebra, students learn to evaluate the floor function and ceiling function, using the notation ⌊x⌋ and ⌈x⌉ to round real numbers down or up to the nearest integer. The lesson covers special cases with negative numbers, the fractional part notation {x}, and the key identity ⌊x⌋ + {x} = x. Students also practice graphing these step functions and applying them to expressions involving fractions and square roots.
Section 1
Floor and Ceiling Functions
The floor and ceiling functions are used to convert decimal numbers to integers:
Unlike traditional rounding, floor and ceiling functions always round in a specific direction regardless of the decimal part.
Section 2
Fractional Part Function
The fractional part function represents the non-integer portion of a real number :
Key properties: for all real numbers , and if and only if is an integer.
Section 3
Floor Function and Fractional Part Relationship
The floor function and fractional part are related by the fundamental equation:
This can also be written as:
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Section 1
Floor and Ceiling Functions
The floor and ceiling functions are used to convert decimal numbers to integers:
Unlike traditional rounding, floor and ceiling functions always round in a specific direction regardless of the decimal part.
Section 2
Fractional Part Function
The fractional part function represents the non-integer portion of a real number :
Key properties: for all real numbers , and if and only if is an integer.
Section 3
Floor Function and Fractional Part Relationship
The floor function and fractional part are related by the fundamental equation:
This can also be written as:
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter