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Chapter 1: Problem 84

Which property of real numbers justifies each statement? $$(3+x)+y=3+(x+y)$$

Short Answer

Expert verified
The given statement \((3+x)+y=3+(x+y)\) is justified by the Associative Property of Addition.

Step by step solution

01

Consider The Statement

Consider the given statement \((3+x)+y=3+(x+y)\). We have to figure out which axiom of real numbers justifies this statement.
02

Analyze The Expression

Analyzing the expression, we can see that the sum of three terms, \(3\), \(x\), and \(y\), is the same regardless of how the sum is grouped. The only thing that changes is that the parentheses move; in other words, how the terms are grouped before summing.
03

Identify The Property

The property that allows us to regroup numbers in a sum is called the Associative Property of Addition. Thus, we can conclude that the given statement is justified by the Associative Property of Addition.

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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, encompassing all the numbers one can think of along a continuous line. This includes:
  • Natural numbers: These are the counting numbers like 1, 2, 3, and so on.
  • Whole numbers: These include all natural numbers plus zero.
  • Integers: This category extends to negative numbers, so it includes natural numbers, zero, and negative numbers like -1, -2, -3.
  • Rational numbers: Numbers that can be expressed as a fraction or ratio of two integers, like 1/2 or 3.5.
  • Irrational numbers: These are numbers that cannot be written as a simple fraction, such as \(\pi\) and \(\sqrt{2}\).
  • Real numbers also include zero, as it is part of both the whole numbers and integers.
Understanding real numbers is crucial because they provide a way to measure quantities and model real-world problems. They form the basis for many mathematical operations and concepts, including those related to algebra, calculus, and beyond.
addition
Addition is one of the most basic operations in mathematics, allowing us to combine numbers into a single sum. It is signified by the '+' symbol and adheres to various mathematical properties including:
  • Commutative Property: This states that the order in which two numbers are added does not change the sum. For example, \(a + b = b + a\).
  • Associative Property: This teaches that the way in which numbers are grouped when adding does not affect the sum. An example of this property is given in the original exercise: \((3 + x) + y = 3 + (x + y)\).
  • Identity Property: According to this property, any number plus zero will result in the same number, like \(a + 0 = a\).
Addition not only helps in basic arithmetic but also sets a foundation for understanding more complex mathematical processes like integration in calculus and vector addition in physics.
mathematical axioms
Mathematical axioms are foundational principles or statements that are accepted as true without proof. They serve as a starting point for further reasoning and logical deductions in mathematics. Axioms are essential because they:
  • Provide the basic rules for logical reasoning.
  • Help define structures such as groups, rings, and fields in algebra.
  • Facilitate the derivation of theorems and other mathematical truths.
One important axiom related to the original exercise is the Associative Property of Addition, which is part of the broader set of axioms defining real numbers. This property justifies rearranging the parentheses in the sum \((3 + x) + y = 3 + (x + y)\), ensuring the sum remains unchanged regardless of grouping. Thus, axioms provide mathematicians a common framework to build and extend rigorous mathematical theories.

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