Problem 34 - Thomas Calculus

Question

What real numbers \(x\) satisfy the equation \(\lfloor x\rfloor=\lceil x\rceil ?\)

Step-by-Step Solution

Answer

All integers satisfy the equation.

Step 1: Understand the Floor and Ceiling Functions

The floor function, denoted by \(\lfloor x \rfloor\), gives the greatest integer less than or equal to \(x\). The ceiling function, denoted by \(\lceil x \rceil\), gives the smallest integer greater than or equal to \(x\).

Step 2: Analyze the Given Equation

We are given the equation \(\lfloor x\rfloor = \lceil x\rceil\). This means that both the greatest integer less than or equal to \(x\) and the smallest integer greater than or equal to \(x\) are equal.

Step 3: Consider the Conditions for Equality

The only way for \(\lfloor x \rfloor\) and \(\lceil x \rceil\) to be equal is if \(x\) itself is an integer. This is because for any non-integer \(x\), \(\lfloor x \rfloor\) and \(\lceil x \rceil\) differ by at least \(1\).

Step 4: Conclusion of the Analysis

Since \(x\) must be an integer for \(\lfloor x \rfloor = \lceil x \rceil\), all real numbers \(x\) that are integers will satisfy this equation. Hence, the set of solutions is all integers.

Key Concepts

Real Numbers

Real numbers are an essential part of mathematics. They include all the numbers you can think of that are not imaginary. This vast group encompasses positive numbers, negative numbers, and zero. Real numbers can be categorized in a few key ways:

• **Rational numbers**: These are numbers that can be expressed as a fraction of two integers, such as 1/2 or -3/4.
• **Irrational numbers**: These numbers cannot be expressed as a fraction of two integers. Examples include numbers like \(\pi\) and \(\sqrt{2}\).
• **Whole numbers and integers**: While all integers are whole numbers, not all whole numbers are integers. Whole numbers include 0, 1, 2, etc., while integers add in negative whole numbers, like -1, -2, etc.

Understanding real numbers is crucial because they are used in many applications like measuring quantities, solving equations, and expressing quantities in real-life scenarios. Real numbers can be represented on a number line, covering everything from the smallest negative infinity to the largest positive infinity.

Integer Solutions

The term integer solution refers to a solution of an equation or inequality that is a whole number, which can be positive, negative, or zero. In the problem at hand, the equation \(\lfloor x \rfloor = \lceil x \rceil\) suggests that we are looking for values of \(x\) where the floor and ceiling function results are the same. By definition, this happens only when \(x\) itself is an integer.

• When \(x\) is an integer, \(\lfloor x \rfloor = x\) and \(\lceil x \rceil = x\), so naturally \(\lfloor x \rfloor = \lceil x \rceil = x\).
• If \(x\) is not an integer, the floor and ceiling functions will differ. For example, for \(x = 2.5\), \(\lfloor 2.5 \rfloor = 2\) and \(\lceil 2.5 \rceil = 3\).

This tells us that the solutions to this type of equation are exclusively the integers themselves. When solving equations like this, identifying whether you're looking for real, integer, or other types of solutions is crucial for getting the correct answer.

Piecewise Functions

Piecewise functions are mathematical functions defined by different formulas or expressions depending on the value or range of the input variable. They are like having a rule book with several rules, and which one you apply depends on certain conditions. Piecewise functions are often used to model scenarios where a function behaves differently in different intervals.

• An example of a piecewise function might be a tax rate: one rate up to a certain income and a different rate above that.
• Another common example could include step functions, which appear like a staircase on a graph.

In relation to our problem involving the floor and ceiling functions, you can see a similar behavior where these functions apply different 'rules' or outputs based on whether the input \(x\) is a whole number, above a whole number, or below it. While not inherently piecewise on their plots, the segment-like behavior of floor or ceiling functions might make one think of a piecewise function. Understanding piecewise functions is crucial, especially in real-world applications where different rules may apply at different times or situations.