/*! 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 89 Which property of real numbers j... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 89

Which property of real numbers justifies each statement? $$(x+y)+z=z+(x+y)$$

Short Answer

Expert verified
The property of real numbers that justifies the statement is the Commutative Property of Addition.

Step by step solution

01

Understand the statement

First, we need to understand the statement. The statement is given by \((x+y)+z=z+(x+y)\), which involves only the operation of addition.
02

Identify the property

Now, we need to identify the property of real numbers that justifies this statement. In this statement, the number 'z' is getting added to both sides and maintaining the equality. This fact aligns with the **commutative property of addition**. According to this property, reversing the order or changing the arrangement of the numbers does not change the result of the addition.
03

Verifying the property

Finally, it's important to confirm whether our identified property justifies the statement. So, using the commutative property of addition, we can rearrange the left side of the equation \(z+(x+y)\), where 'x+y' is rearranged as 'y+x', which will also equal to 'z+(x+y)'. Thus, the identified property justifies the provided statement perfectly.

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.

Real Numbers
Real numbers are a fundamental concept in mathematics that encompasses different sets of numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Simply speaking, real numbers include all the numbers on the number line. They are versatile and essential in both everyday contexts and higher-level mathematics.

Real numbers can be:
  • Positive (e.g., 1, 2, 3.5)
  • Negative (e.g., -1, -2, -3.5)
  • Zero
  • Fractions (e.g., 1/2)
  • Irrational numbers like 蟺 (pi) or 鈭�2 that cannot be expressed as a simple fraction
Understanding real numbers is crucial when exploring different mathematical operations, including addition. This is because rules that apply to real numbers form the foundation for performing these operations effectively.
Addition
Addition is one of the basic arithmetic operations and is represented by the plus symbol (+). When you add real numbers, you combine their values to get a sum. The process of addition is intuitive and relatable, being used in everyday life for tasks like calculating totals or balancing checkbooks.

Key points about addition include:
  • Adding two positive numbers results in a larger number.
  • Adding two negative numbers gives a more negative number.
  • Adding a positive and a negative number involves finding the difference and taking the sign of the larger absolute value.
  • Adding zero to a number doesn't change its value.
Addition operations with real numbers are straightforward, but they rely on understanding properties like the commutative and associative properties to manipulate numbers effectively.
Properties of Addition
The properties of addition define the rules and behaviors of the addition operation when applied to real numbers. Comprehending these properties can significantly enhance your ability to perform mental math and solve complex equations.

The primary properties of addition are:
  • Commutative Property: This property states that the order in which two numbers are added does not affect the sum, i.e., \(a + b = b + a\).
  • Associative Property: This property indicates that no matter how numbers are grouped during addition, the result is the same, i.e., \((a + b) + c = a + (b + c)\).
  • Identity Property: This property tells us that adding zero to any number will not change the value of that number, i.e., \(a + 0 = a\).
Familiarity with these properties allows for flexibility in arranging numbers to make calculations easier, as demonstrated in the original exercise with the commutative property helping to justify the statement \((x+y)+z=z+(x+y)\).

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

A high-efficiency home heating system can be installed for \(\$ 4,170,\) with an average monthly heating bill of \(\$ 57.50 .\) A regular furnace can be installed for \(\$ 1,730,\) but monthly heating bills average \(\$ 107.75 .\) After three years, which system has cost more altogether?

Let \(x=8, y=4,\) and \(z=2 .\) Write each phrase as an algebraic expression, and evaluate it. Assume that no denominators are \(0 .\) 7 less than the sum of \(x\) and \(y\)

Write each sentence as a mathematical expression. Thirty-seven is greater than the product of multiplying three and four.

Describe how you would find the common denominator of two fractions.

Write each sentence as a mathematical expression. Eight is less than or equal to eight.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

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.