/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Evaluate each, where \(n\) is an... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 3

Evaluate each, where \(n\) is an integer. $$[n+1 / 2]$$

Short Answer

Expert verified
The short answer for evaluating the expression \(\left[ n+\frac{1}{2} \right]\), where \(n\) is an integer, is: \( \left\lfloor n+\frac{1}{2} \right\rfloor = n \).

Step by step solution

01

Identify the floor function

The floor function is represented by square brackets. In this case, the floor function is applied to the expression \(\left[n+\frac{1}{2}\right]\). This means we round down the value inside the brackets to the nearest integer.
02

Apply the floor function to the expression

Now, we need to apply the floor function to the expression. Since \(n\) is an integer, adding \(\frac{1}{2}\) will always result in a number halfway between two integers. For example, if \(n\) is 3, then the expression inside will be \(3.5\). The floor function will round down the number to the nearest integer, so the result will be 3. Therefore, the general formula for the given expression \(\left[n+\frac{1}{2}\right]\) is as follows: \( \left\lfloor n+\frac{1}{2} \right\rfloor = n \)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Rounding Down
Rounding down is a numerical process where we convert a decimal or fractional number to the nearest lower integer value. This is often achieved using the floor function.
  • Imagine you own a number, such as 3.7. By rounding down, you convert it to 3.
  • This process is crucial for calculations in mathematics, ensuring we handle numbers efficiently without overestimating values.

In the context of the given problem, rounding down helps in determining how we manage half-way numbers like \(n + \frac{1}{2}\) by moving them to the lower integer, specifically when dealing with discrete values. Understanding when and how to round down can simplify solving problems in various math fields.
Integer Arithmetic
Integer arithmetic involves performing calculations with numbers that don't have fractional parts. It's quite straightforward because we work with whole numbers.
Some primary operations included are addition, subtraction, multiplication, and division (with care).
  • Adding two integers always results in another integer. For example, 2 + 3 = 5.
  • Multiplying integers is similar; for instance, 4 * 5 = 20.

Integer arithmetic is essential when using the floor function. For example, if we add 0.5 to an integer n, the integer part remains unaffected, which shows how integers operate distinctly from fractions in mathematical expressions.
Discrete Mathematics
Discrete mathematics focuses on structures that are distinct and separated. It contrasts with continuous mathematics involving smooth and flowing values like real numbers.
  • It includes studies on integers, graphs, and logical statements.
  • It's extensively used in computer science, cryptography, and network analysis.

The floor function is a tool frequently used in discrete mathematics because it helps to map continuous data into discrete steps. In the exercise, \[n + \frac{1}{2}\right]\] is a typical example where discrete math enables rounding elements to whole numbers, aiding in decision-making models that require well-defined outcomes.
Mathematical Expressions
Mathematical expressions are combinations of numbers, symbols, and operators showing a value. In mathematics, they give us a clear way to represent problems or scenarios.
  • They can consist of simple terms like multiplication **2 * 3** or more complex functions like **\(\left\lfloor n+\frac{1}{2}\right\rfloor\)**.
  • Expressions must be evaluated, meaning each component is simplified or calculated, such as observing how \(n + \frac{1}{2}\) transforms under a floor function.

Understanding how to construct and evaluate expressions is crucial in solving problems like the one provided. The expression \[n + \frac{1}{2}\right]\] is a fantastic demonstration of using math language to create concise solutions through operations like rounding down.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prove. The set of irrational numbers is uncountable. (Hint: Prove by contradiction.)

Prove. The set \(Q^{+}\) of positive rational numbers is countable.

Let \(A=\left[\begin{array}{ccc}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{ccc}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right]\) . Find each. $$3 A+(-2) B$$

Determine if the functions are bijective. If they are not bijective, explain why. \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\) decimal value of \(x,\) where \(\Sigma=\\{0,1\\}.\)

Evaluate each sum and product, where \(p\) is a prime and \(I=\\{1,2,3,5\\}.\) $$\begin{aligned} &\prod i^{j}\\\ &i, j \in I\\\ &i \leq j \end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.