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Chapter 1: Problem 9

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

Short Answer

Expert verified
The result is 1 in \( \mathbb{Z}_{3} \).

Step by step solution

01

Simplify the Expression Inside the Parentheses

Start by adding the numbers inside the parentheses: \(2 + 1 + 2\). This gives \(5\).
02

Multiply by 2

Now, take the simplified sum and multiply it by 2: \(2 \times 5 = 10\).
03

Apply Modulo Operation

Since we are working under the modulo \( \mathbb{Z}_{3} \), we take the result from Step 2 and find \(10 \mod 3\). Since 10 divided by 3 gives a remainder of 1, we have \(10 \equiv 1 \mod 3\).

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Key Concepts

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

Modulo Operation
The modulo operation is a mathematical technique often used in various fields of computer science and mathematics. It finds the remainder of the division of one number by another. This operation is universally known as "mod". In simple terms, when you divide a number by another, the modulo operation gives you the remainder:
- For example, if you take 10 modulo 3, you divide 10 by 3, which results in 3 with a remainder of 1. - Therefore, 10 modulo 3 equals 1, written as: \[10 \equiv 1 \mod 3\]
The modulo operation is crucial for calculations in modular arithmetic, especially in cryptography and computer algorithms.
Remainder
When performing division, the remainder is what remains after dividing one number by another. In the context of a modulo operation, the remainder is the result. It can range from zero to one less than the divisor.
- For example, if you divide 7 by 4, you will have 1 as a remainder because 4 goes into 7 once with a leftover of 3. - Therefore, 7 modulo 4 equals 3: \[7 \equiv 3 \mod 4\]
The remainder is a core aspect of understanding modular arithmetic, impacting calculations and how results are derived and interpreted.
Integer Addition
Integer addition is a fundamental arithmetic operation. It involves combining two or more whole numbers to get a sum. Understanding this concept is essential for performing operations within a modular system, especially when numbers are large.
- Adding numbers is straightforward: - For instance, adding 2, 1, and 2 yields 5: \[(2 + 1 + 2 = 5)\]
Working with integers becomes even more interesting in modular systems where the sums are frequently combined with modulo operations.
Arithmetic in Modular Systems
Arithmetic in modular systems involves performing standard arithmetic operations like addition, subtraction, multiplication, and division, followed by a modulo operation. Modular systems are defined by a set number, known as modulus, that defines the upper limit of numbers before they reset back to zero, creating a cyclical system.
  • The primary benefit is simplifying calculations and enabling wrapping of values within a specific range, such as a clock which uses modulo 12.
  • In applications like calendar calculations and cryptography, modular arithmetic is invaluable.
Modular arithmetic provides an alternative way of performing arithmetic where numbers "wrap around" after reaching a certain value, ensuring consistency and predictability in systems.

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Most popular questions from this chapter

Write the equation of the plane passing through \(P\) with normal vector \(\mathbf{n}\) in (a) normal form and (b) general form. $$P=(-3,5,1), \mathbf{n}=\left[\begin{array}{r} 1 \\ -1 \\ 5 \end{array}\right]$$

\(\mathbf{u}\) and \(\mathbf{v}\) are binary vectors. Find \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u} \cdot \mathbf{v}\) in each case $$\mathbf{u}=[1,1,0,1,0], \mathbf{v}=[0,1,1,1,0]$$

Show that the plane and line with the given equations intersect, and then find the acute angle of intersection between them. The plane given by \(x+y+2 z=0\) and the line given by \(x=2+t\) \\[ \begin{array}{l} y=1-2 t \\ z=3+t \end{array} \\]

(a) Prove that if a transposition error is made in the fourth and fifth entries of the ISBN \([0,6,7,9,7,6,\) 2,9,0,6]\(,\) the error will be detected. (b) Prove that if a transposition error is made in any two adjacent entries of the ISBN in part (a), the error will be detected. (c) Prove, in general, that the ISBN code will always detect a transposition error involving two adjacent entries.

A parity check code vector \(\mathbf{v}\) is given. Determine whether a single error could have occurred in the transmission of \(\mathbf{v}\) $$\mathbf{v}=[1,1,0,1,0,1,1,1]$$
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