Problem 9 - A Gentle Introduction to the Art of Mathematics

Question

Prove that \(\forall x \in \mathbb{R}, x \notin \mathbb{Z} \Longrightarrow\lfloor x\rfloor+\lfloor-x\rfloor=-1\).

Step-by-Step Solution

Answer

For \(x \otin \mathbb{Z} \), \(\floor{x} + \floor{-x} = -1\).

Step 1: Understand the Floor Function

The floor function \(\floor{x}\) of a real number \(x\) is the greatest integer less than or equal to \(x\). For example, \(\floor{2.3} = 2\) and \(\floor{-1.7} = -2\).

Step 2: Analyze \(x\) and \(-x\) for Non-Integer x

Given \(x otin \mathbb{Z}\), \(x\) is not an integer. This means \(x\) can be written as \(n + f\) where \(n\) is an integer and \(0 < f < 1\). Similarly, \(-x = -n - f\).

Step 3: Apply the Floor Function

Apply the floor function to \(x\) and \(-x\). We get \(\floor{x} = \floor{n + f} = n\) because \(n\) is the largest integer ≤ \(n + f\). For \(-x\), we have \(\floor{-x} = \floor{-n - f} = -n - 1\) because \(-n - 1\) is the largest integer ≤ \(-n - f\).

Step 4: Sum the Values

Sum the values obtained in the previous step: \(\floor{x} + \floor{-x} = n + (-n - 1) = -1\).

Step 5: Conclusion

We've shown that \(\floor{x} + \floor{-x} = -1\) for any \(x otin \mathbb{Z}\). This means \(\forall x \in \mathbb{R}, x \otin \mathbb{Z}, \floor{x} + \floor{-x} = -1\).

Key Concepts

Understanding Real Numbers

Real numbers, denoted as \(\backslash mathbb{R}\), include all the numbers that can be found on the number line. This category encompasses:

• Integers like \(-2, 0, 5\)
• Rational numbers such as \(\frac{1}{2}, -7.3, 8\frac{1}{4}\)
• Irrational numbers like \(\backslash sqrt{2}, π\)

Real numbers can be fractions, whole numbers, or decimal points that extend infinitely without repetition. They play a crucial role in mathematics as they include every number type you will generally encounter.

Non-Integer Values

Non-integer values are real numbers that are not whole numbers. These include:

• Fractions and decimals (e.g., \(\frac{3}{4}\) and 0.65)
• Irrational numbers (e.g., \(\backslash sqrt{3}\) and π)

In the context of the floor function, non-integers have specific properties. Given a non-integer value \(x \otin\ \backslash mathbb{Z}\), it can be expressed as \(n + f\), where \(n\) is an integer part and \(f\) is a fractional part (\(0 < f < 1\)). Understanding these properties helps in mathematical tasks like proving the sum of the floor and the floor of the negative of non-integers.

Mathematical Proofs

Mathematical proofs are logical arguments demonstrating that a statement is true. They involve a sequence of steps based on axioms, definitions, and previously established results. Proofs can be:

• Direct: Where the truth is established through straightforward reasoning.
• Indirect: By showing the negation of the statement leads to a contradiction.

In the exercise, we used a direct proof to show that \( \forall x ∈ \backslash mathbb{R}, x \otin\ \backslash mathbb{Z} \backslash Longrightarrow \backslash floor{x} + \backslash floor{-x} = -1 \). This involved breaking down the floor function behavior for non-integers and logically summing the results.