Calculating P(0) For P(x) = ⌊60/x⌋ And The Undefined Nature Of Division By Zero

Introduction

The problem at hand involves understanding the floor function and how it applies to a given expression. Specifically, we are asked to evaluate p(0) for the function p(x) = ⌊60/x⌋, where ⌊x⌋ denotes the floor function. The floor function, also known as the greatest integer function, returns the largest integer less than or equal to x. This is a fundamental concept in mathematics, particularly in number theory and real analysis. To address this problem, we will first define the floor function more formally and then discuss the implications of evaluating the given expression at x = 0. We will highlight the mathematical principles involved, including the concept of division by zero, which is crucial in this scenario.

In this article, we will explore the intricacies of the floor function and its application to the expression p(x) = ⌊60/x⌋. We aim to provide a comprehensive understanding of why evaluating p(0) leads to an undefined result. This exploration will involve a detailed examination of the floor function's definition, the implications of dividing by zero, and the behavior of the expression as x approaches zero. By the end of this discussion, readers should have a clear understanding of the mathematical principles involved and the correct approach to solving such problems. This article serves as a guide to navigate the nuances of mathematical expressions and functions, ensuring a solid grasp of these concepts.

Defining the Floor Function

The floor function, denoted by ⌊x⌋, is a mathematical function that takes a real number x as input and returns the greatest integer less than or equal to x. In simpler terms, it rounds the number down to the nearest integer. For example, ⌊3.14⌋ = 3, ⌊-2.7⌋ = -3, and ⌊5⌋ = 5. The floor function is an essential tool in various mathematical contexts, including number theory, real analysis, and computer science. Understanding its properties and behavior is crucial for solving problems involving integers and real numbers.

To fully grasp the floor function, consider its graphical representation. The graph of y = ⌊x⌋ is a step function, where each step has a height of 1 and extends from an integer value to the next integer value, excluding the upper bound. This step-like nature is a direct consequence of the function's definition, which always rounds down to the nearest integer. For instance, between 2 and 3, the floor function will always output 2, regardless of how close x is to 3, until x actually reaches 3. This discrete behavior makes the floor function particularly useful in scenarios where integer values are of primary interest. In the context of this problem, the floor function will play a critical role in determining the integer output of the expression ⌊60/x⌋.

The floor function's properties extend beyond simple rounding. It is a non-decreasing function, meaning that if a ≤ b, then ⌊a⌋ ≤ ⌊b⌋. Additionally, it satisfies several important identities, such as ⌊x + n⌋ = ⌊x⌋ + n for any integer n. These properties make the floor function a versatile tool in mathematical manipulations and proofs. In our specific problem, understanding how the floor function interacts with division is essential. The expression ⌊60/x⌋ combines the floor function with a rational expression, introducing complexities that need careful consideration. The denominator x plays a crucial role, especially when we consider values close to zero, where the behavior of the expression can be quite different from what one might intuitively expect.

Analyzing p(x) = ⌊60/x⌋

Given the function p(x) = ⌊60/x⌋, we need to understand its behavior as x varies. The expression involves dividing 60 by x and then applying the floor function to the result. As x gets larger, the value of 60/x approaches zero, and consequently, ⌊60/x⌋ will also approach zero. Conversely, as x gets smaller (but remains positive), the value of 60/x becomes very large, and ⌊60/x⌋ will also be a large integer. This inverse relationship between x and 60/x is a key characteristic of the function.

To further analyze p(x), let's consider some specific examples. When x = 1, p(1) = ⌊60/1⌋ = 60. When x = 2, p(2) = ⌊60/2⌋ = 30. When x = 4, p(4) = ⌊60/4⌋ = 15. As x increases, p(x) decreases, but it does so in discrete steps due to the floor function. This step-like behavior is a direct consequence of the floor function rounding down to the nearest integer. It's important to note that the floor function introduces discontinuities at points where 60/x is an integer. For example, when 60/x transitions from slightly less than an integer to the integer itself, the floor function's output changes abruptly.

The behavior of p(x) for negative values of x mirrors the positive case but with a sign change. For negative x, 60/x will be negative, and the floor function will round down to the next smallest integer. For example, when x = -1, p(-1) = ⌊60/-1⌋ = -60. When x = -2, p(-2) = ⌊60/-2⌋ = -30. The function remains inversely proportional, but the output values are negative integers. The step-like nature of the floor function persists, creating a series of negative discrete values as x varies. The most critical aspect of analyzing this function, however, arises when we consider x approaching zero, which is the focus of our original problem.

The Issue of Division by Zero

The core of the problem lies in evaluating p(0) = ⌊60/0⌋. Division by zero is undefined in mathematics. It violates fundamental arithmetic principles and leads to mathematical inconsistencies. To understand why, consider the definition of division. Dividing a number a by a number b (where b is not zero) means finding a number c such that a = b × c. However, if b = 0, then the equation becomes a = 0 × c, which simplifies to a = 0. If a is not zero, there is no value of c that can satisfy this equation. If a is zero, then any value of c would satisfy the equation, leading to an indeterminate result.

In the context of our problem, 60/0 is undefined because there is no number that, when multiplied by zero, equals 60. This fundamental principle of arithmetic prevents us from assigning any meaningful value to the expression 60/0. Therefore, attempting to evaluate p(0) = ⌊60/0⌋ results in an undefined expression. The floor function cannot operate on an undefined input, so the entire expression becomes undefined. This highlights the importance of carefully considering the domain of a function and avoiding values that lead to undefined operations.

To further illustrate the issue, consider what happens as x approaches zero from the positive side. As x gets closer and closer to zero, 60/x becomes increasingly large, approaching infinity. However, infinity is not a number; it is a concept representing unbounded growth. Therefore, even as 60/x becomes very large, it remains undefined at x = 0. Similarly, as x approaches zero from the negative side, 60/x becomes increasingly large in the negative direction, approaching negative infinity. Again, this does not provide a defined value for 60/0. The undefined nature of division by zero is a cornerstone of mathematical rigor, and it is essential to recognize and address it when evaluating expressions.

Conclusion: p(0) is Undefined

In conclusion, when we attempt to evaluate p(0) for the function p(x) = ⌊60/x⌋, we encounter the undefined operation of division by zero. The expression 60/0 is not defined in mathematics, and therefore, ⌊60/0⌋ is also undefined. The floor function cannot operate on an undefined input, making p(0) an undefined quantity. This highlights the critical importance of understanding the domain of a function and recognizing when an expression is undefined.

The options provided (A. 8, B. 85, C. 86, D. 9) are all numerical values, and none of them correctly represent the fact that p(0) is undefined. The question, therefore, does not have a valid answer among the given choices. This exercise underscores the significance of mathematical rigor and the need to avoid operations that violate fundamental principles, such as division by zero. It serves as a reminder that mathematical expressions must be evaluated within their defined domains to yield meaningful results. Understanding these principles is essential for solving mathematical problems accurately and avoiding common pitfalls.

Therefore, the correct understanding is that p(0) is undefined due to the division by zero, and none of the provided options are correct.