/*! 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 34 Perform the indicated calculatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 34

Perform the indicated calculations. $$3+1+2+3 \text { in } \mathbb{Z}_{4}$$

Short Answer

Expert verified
The result is 1 in \(\mathbb{Z}_4\).

Step by step solution

01

Review the Operation

The problem asks us to perform the sum \(3 + 1 + 2 + 3\) in the set of integers modulo 4, \(\mathbb{Z}_4\). In modular arithmetic, numbers wrap around after reaching a certain value, which in this case is 4.
02

Compute Each Sum Modulo 4

Begin by calculating each sum sequentially while applying the modulo operation:1) Calculate \((3 + 1) \mod 4\): \(3 + 1 = 4 \equiv 0 \mod 4\)2) Add the next number: \((0 + 2) \mod 4\): \(0 + 2 = 2 \equiv 2 \mod 4\)3) Add the next number: \((2 + 3) \mod 4\): \(2 + 3 = 5 \equiv 1 \mod 4\) so we find the result is 1.

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.

Integers Modulo
To understand modular arithmetic, it's essential to grasp the concept of integers modulo. This is often represented as \( \mathbb{Z}_n \), where \( n \) is the modulus. This set consists of integers from 0 up to \( n-1 \). The key idea here is that numbers "wrap around" when they reach \( n \). This means that after reaching the modulus, we start back at 0.Let's think of it like a clock that only has \( n \) hours. Once it hits the maximum, it resets to the beginning.In the provided solution, you're working in \( \mathbb{Z}_4 \), the integers modulo 4. So, our set of numbers is \( \{0, 1, 2, 3\} \). Whenever calculations surpass 3, we cycle back to 0.
Modulo Operation
The modulo operation is a fundamental part of modular arithmetic. It is often denoted using the "mod" symbol. When you see something like \( a \mod n \), it means you're finding the remainder of dividing \( a \) by \( n \). For instance, when calculating \( 3 + 1 \mod 4 \), the sum of 3 and 1 is 4. But since we are working in mod 4, we take 4 and find what's left after dividing by 4, which is 0.This operation ensures that our results are always within the set of integers \( \{0, 1, 2, 3\} \). It's like a reset button, smoothing out calculations to fit within our defined modular system. You are essentially looking for the leftovers, keeping everything in neat bounds.
Modular Addition
Modular addition is similar to regular addition, but it includes the modulo operation to keep results within a specific range. When you add numbers in a modular sense, you sum them as usual, then immediately apply the modulo operation.For example, in the problem you're tackling, the calculation is \( 3 + 1 + 2 + 3 \) in \( \mathbb{Z}_4 \). Here鈥檚 how each step unfolds:
  • Add \( 3 + 1 \) to get 4, then compute \( 4 \mod 4 \), resulting in 0.
  • Continue with 0, add 2, which gives you 2, and since \( 2 \mod 4 \) is 2, it stays as is.
  • Finally, adding 3 to 2 gives you 5; applying \( 5 \mod 4 \) results in 1.
This keeps calculations firmly within the boundaries of \( \mathbb{Z}_4 \), showing how modular arithmetic keeps everything tidy and predictable, even as numbers rise.

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

Perform the indicated calculations. $$8(6+4+3) \text { in } \mathbb{Z}$$

u and v are binary vectors. Find \(\mathbf{u}+\mathbf{v}\) in each case. $$\mathbf{u}=\left[\begin{array}{l} 0 \\ 1 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$

The rectangle \(A B C D\) has vertices at \(A=(1,2,3)\) \(\mathrm{B}=(3,6,-2),\) and \(\mathrm{C}=(0,5,-4) .\) Determine the coordinates of vertex \(D\).

A \(10 \mathrm{kg}\) block lies on a ramp that is inclined at an angle of \(30^{\circ}\) (Figure 1.80 ). Assuming there is no friction, what force, parallel to the ramp, must be applied to keep the block from sliding down the ramp?

Solve the given equation or indicate that there is no solution. $$2 x=1 \text { in } \mathbb{Z}_{3}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.