Abraham Beshah
more than 2 weeks ago
Properties of the floor function
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ChatGPT
more than 2 weeks ago
1. The floor function, denoted as ⌊x⌋, is a mathematical function that rounds a real number down to the nearest integer less than or equal to the original number.
2. The floor function is defined for all real numbers and returns an integer value.
3. The floor function is also known as the greatest integer function.
4. The floor function can be expressed using the following formula: ⌊x⌋ = max{n ∈ Z : n ≤ x}, where Z represents the set of integers.
5. The floor function is a piecewise function, with different definitions for positive and negative numbers. For positive numbers, the floor function simply removes the decimal part of the number. For negative numbers, the floor function rounds the number towards negative infinity.
6. The floor function is a non-decreasing function, meaning that if x ≤ y, then ⌊x⌋ ≤ ⌊y⌋.
7. The floor function is discontinuous at every integer value, as the function jumps from one integer to the next as the input value crosses an integer boundary.
8. The floor function is often used in computer programming and mathematical calculations to obtain integer values from real numbers.
2. The floor function is defined for all real numbers and returns an integer value.
3. The floor function is also known as the greatest integer function.
4. The floor function can be expressed using the following formula: ⌊x⌋ = max{n ∈ Z : n ≤ x}, where Z represents the set of integers.
5. The floor function is a piecewise function, with different definitions for positive and negative numbers. For positive numbers, the floor function simply removes the decimal part of the number. For negative numbers, the floor function rounds the number towards negative infinity.
6. The floor function is a non-decreasing function, meaning that if x ≤ y, then ⌊x⌋ ≤ ⌊y⌋.
7. The floor function is discontinuous at every integer value, as the function jumps from one integer to the next as the input value crosses an integer boundary.
8. The floor function is often used in computer programming and mathematical calculations to obtain integer values from real numbers.
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Google Gemini AI
more than 2 weeks ago
Definition:
The floor function, denoted as ⌊x⌋, rounds down the real number x to the nearest integer that is less than or equal to x.
Properties:
- Monotonicity: The floor function is monotonically increasing. For all real numbers x and y, if x < y, then ⌊x⌋ < ⌊y⌋.
- Boundedness: The range of the floor function is the set of integers. For all real numbers x, ⌊x⌋ is an integer.
- Idempotence: If x is an integer, then ⌊⌊x⌋⌋ = ⌊x⌋.
- Shifting Property: For all real numbers x and any integer k, ⌊x + k⌋ = ⌊x⌋ + k.
- Distributive Property: For all real numbers x and y, and any integer k, ⌊x + y⌋ + k = ⌊x⌋ + ⌊y⌋ + k.
- Arithmetic Progression: For all real numbers x and y, with x < y, and any integer k, there are k consecutive integers n such that ⌊x⌋ ≤ n ≤ ⌊y⌋.
- Round-up Counterpart: The ceiling function, denoted as ⌈x⌉, is the counterpart of the floor function. It rounds up the real number x to the nearest integer that is greater than or equal to x. The following identity holds: ⌈x⌉ = -⌊-x⌋.
- Modulus Relation: For all real numbers x and any positive integer n, x = ⌊x⌋ (mod n).
- Inequality: For all real numbers x and any integer k, x < k if and only if ⌊x⌋ < k - 1.
- Convergence: For any sequence {xn} of real numbers, if limn→∞ xn = x, then limn→∞ ⌊xn⌋ = ⌊x⌋.
The floor function, denoted as ⌊x⌋, rounds down the real number x to the nearest integer that is less than or equal to x.
Properties:
- Monotonicity: The floor function is monotonically increasing. For all real numbers x and y, if x < y, then ⌊x⌋ < ⌊y⌋.
- Boundedness: The range of the floor function is the set of integers. For all real numbers x, ⌊x⌋ is an integer.
- Idempotence: If x is an integer, then ⌊⌊x⌋⌋ = ⌊x⌋.
- Shifting Property: For all real numbers x and any integer k, ⌊x + k⌋ = ⌊x⌋ + k.
- Distributive Property: For all real numbers x and y, and any integer k, ⌊x + y⌋ + k = ⌊x⌋ + ⌊y⌋ + k.
- Arithmetic Progression: For all real numbers x and y, with x < y, and any integer k, there are k consecutive integers n such that ⌊x⌋ ≤ n ≤ ⌊y⌋.
- Round-up Counterpart: The ceiling function, denoted as ⌈x⌉, is the counterpart of the floor function. It rounds up the real number x to the nearest integer that is greater than or equal to x. The following identity holds: ⌈x⌉ = -⌊-x⌋.
- Modulus Relation: For all real numbers x and any positive integer n, x = ⌊x⌋ (mod n).
- Inequality: For all real numbers x and any integer k, x < k if and only if ⌊x⌋ < k - 1.
- Convergence: For any sequence {xn} of real numbers, if limn→∞ xn = x, then limn→∞ ⌊xn⌋ = ⌊x⌋.
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