Which statement explains why the value of $\lfloor 2.4\rfloor$ is 2 but the value of $\lfloor-2.4\rfloor$ is -3?
A. because 2 is the greatest integer not greater than 2.4, and -3 is the greatest integer not greater than -2.4
B. because 2.4 rounds to 2, and -2.4 rounds to -3
C. because 2.4 is positive, and -2.4 is negative
D. because 2 is the least integer greater than 2.4, and -3 is the least integer greater than -2.4
Answer (2)
The floor function ⌊ x ⌋ returns the greatest integer less than or equal to x .
⌊ 2.4 ⌋ = 2 because 2 is the greatest integer less than or equal to 2.4.
⌊ − 2.4 ⌋ = − 3 because -3 is the greatest integer less than or equal to -2.4.
The correct statement is: 2 is the greatest integer not greater than 2.4, and -3 is the greatest integer not greater than -2.4.
Explanation
Understanding the Problem The problem asks us to explain why the floor of 2.4 is 2, while the floor of -2.4 is -3. The floor function, denoted by ⌊ x ⌋ , returns the greatest integer less than or equal to x. We need to choose the statement that accurately describes this.
Evaluating the Floor of 2.4 Let's evaluate ⌊ 2.4 ⌋ . The greatest integer less than or equal to 2.4 is 2. This is because 2 is an integer, and 2 ≤ 2.4 , and there is no integer greater than 2 that satisfies this condition.
Evaluating the Floor of -2.4 Now let's evaluate ⌊ − 2.4 ⌋ . The greatest integer less than or equal to -2.4 is -3. This is because -3 is an integer, and − 3 ≤ − 2.4 , and there is no integer greater than -3 that satisfies this condition. Note that -2 is greater than -2.4, but -2 is not less than or equal to -2.4.
Choosing the Correct Statement Comparing our results with the given options, the statement that accurately describes the floor function is: "because 2 is the greatest integer not greater than 2.4, and -3 is the greatest integer not greater than -2.4".
Final Answer Therefore, the correct answer is:
because 2 is the greatest integer not greater than 2.4 , and -3 is the greatest integer not greater than -2.4
Examples
The floor function is used in computer science to determine the size of an array or a data structure. For example, if you have an array of size 2.4, you would need to round it down to 2 to allocate the correct amount of memory. Another example is in scheduling tasks. If a task needs to be performed every 2.4 hours, you would use the floor function to determine that the task should be performed every 2 hours.
The floor function ⌊ x ⌋ returns the greatest integer less than or equal to x . For ⌊ 2.4 ⌋ , the greatest integer is 2, and for ⌊ − 2.4 ⌋ , it is -3. Therefore, the correct answer is option A: because 2 is the greatest integer not greater than 2.4, and -3 is the greatest integer not greater than -2.4.
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