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Chapter 3: Problem 12

Prove that if \(n\) is any even integer, then \(\lfloor n / 2\rfloor=n / 2\).

Short Answer

Expert verified
To prove that \(\lfloor n / 2\rfloor = n / 2\) for any even integer n, we first rewrite the even integer as \(n=2k\) where k is an integer. Then, by substituting \(2k\) for n, we get \(\frac{n}{2} = \frac{2k}{2} = k\). Since k is an integer, the floor function \(\lfloor n/2 \rfloor = \lfloor k \rfloor = k\), thus proving that \(\lfloor n/2 \rfloor = n/2\) holds for any even integer n.

Step by step solution

01

Define even integers and floor function

An even integer is any integer n that can be written as \(n=2k\) where k is an integer. The floor function, denoted as \(\lfloor x \rfloor\), returns the greatest integer that is less than or equal to x.
02

Rewrite the fraction using the definition of even integers

We are given an even integer n, so it can be written as \(n=2k\). To rewrite the fraction \(n/2\), we substitute \(2k\) for n: \[\frac{n}{2}=\frac{2k}{2}\]
03

Simplify the fraction

Simplify the fraction by dividing 2k by 2: \[\frac{2k}{2} = k\]
04

Express the floor division as a floor function

Replace the fraction in the floor function with the simplified fraction as obtained in step 3: \[\lfloor \frac{n}{2} \rfloor = \lfloor k \rfloor\]
05

Evaluate the floor function

Since k is an integer, the greatest integer less than or equal to k is k itself. Therefore, the floor function of k is k: \[\lfloor k \rfloor = k\]
06

Conclude the proof

We have shown that when n is an even integer, the floor function of n divided by 2 is equal to the simplified form k of n divided by 2: \[\lfloor \frac{n}{2} \rfloor = k\]. Since we showed that \(n/2=k\), we can conclude that \(\lfloor n/2 \rfloor = n/2\) for any even integer n.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Floor Function Demystified
The floor function is a mathematical concept that seems simple at first but is incredibly useful in real-world applications. It's represented as \( \lfloor x \rfloor \) and essentially asks the question, "What is the greatest whole number that is less than or equal to \( x \)?"
As an example, consider \( x = 3.7 \). The floor of 3.7 is 3, as 3 is the greatest integer less than 3.7.
Similarly, for negative numbers, if \( x = -2.3 \), then \( \lfloor -2.3 \rfloor = -3 \) because -3 is the greatest integer less than or equal to -2.3.
  • The floor function "rounds down" to the nearest whole number.
  • It is a crucial tool in mathematical proofs and problem-solving, especially when dealing with division and fractions.
Understanding Integer Division
Integer division is another key concept that plays a substantial role in mathematics. When dividing two numbers using integer division, we ignore any remainder and only consider whole numbers in the result.
For instance, dividing 7 by 2 using integer division results in 3, since the remainder is ignored. In a sense, integer division performs a similar function to the floor operation when continuous numbers are divided.
  • Think of integer division as "cutting off" the decimal part of a division.
  • This is particularly useful in algorithms and programming, where only whole numbers are important.
Proof by Simplification: A Step-by-Step Approach
Proof by simplification is a method used to establish the truth of mathematical statements. It's based on simplifying both sides of the equation to demonstrate their equality.
This technique can be seen in the original exercise, where we proved \( \lfloor n/2 \rfloor = n/2 \) for even integers.
  • Start with a known identity or fact that applies to the problem.
  • Simplify the expressions, step by step, until both sides match.
  • Such a proof requires breaking complex expressions into easier or smaller chunks.
In our example, we showed that if \( n = 2k \), then \( n/2 = k \). By verifying \( \lfloor n/2 \rfloor = k \), we directly reached the desired conclusion.
These steps help simplify complex expressions into manageable, understandable components.

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Most popular questions from this chapter

If \(r\) is any rational number and \(s\) is any irrational number, then \(r / s\) is irrational.

Prove those that are true and disprove those that are false.The square root of an irrational number is irrational.

The difference of any two even integers is even.

If \(p\) is a prime number, must \(2^{p}-1\) also be prime? Prove or give a counterexample.

Let \(p_{1}, p_{2}, p_{3}, \ldots\) be a list of all prime numbers in ascending order. Here is a table of the first six:$$ \begin{array}{|c|c|c|c|c|c|} \hline p_{1} & p_{2} & p_{3} & p_{4} & p_{5} & p_{6} \\ \hline 2 & 3 & 5 & 7 & 11 & 13 \\ \hline \end{array} $$a. For each \(i=1,2,3,4,5,6\), let \(N_{i}=p_{1} p_{2} \cdots p_{i}+1\). Calculate \(N_{1}, N_{2}, N_{3}, N_{4}, N_{5}\), and \(N_{6}\). b. For each \(i=1,2,3,4,5,6\), find the smallest prime number \(q_{i}\) such that \(q_{i}\) divides \(N_{i}\).
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