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Chapter 1: Problem 41

Determine which of the fundamental laws of algebra is demonstrated. $$3+(5+9)=(3+5)+9$$

Short Answer

Expert verified
The Associative Property of Addition is demonstrated.

Step by step solution

01

Identify the Given Equation Structure

Observe that the equation is comparing two ways of adding the numbers 3, 5, and 9, where the numbers are grouped differently. On one side, we have the addition of 5 and 9 first, and then 3 is added. On the other side, 3 and 5 are added first, and the result is added to 9.
02

Recognize the Associative Property of Addition

The associative property of addition states that the grouping of numbers does not change their sum. In mathematical terms, this is expressed as \((a+b)+c = a+(b+c)\). In the given exercise, the numbers are grouped as \(3+(5+9)\) on one side and \((3+5)+9\) on the other side, which aligns with 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.

Laws of Algebra
The Laws of Algebra are essential rules that help us manipulate and simplify algebraic expressions and equations. They provide a foundation for understanding how numbers and variables interact. Some of the primary Laws of Algebra include:
  • Commutative Law: This law applies to both addition and multiplication. It states that the order of numbers does not affect the result. For example, for addition: \(a + b = b + a\).
  • Associative Law: This law concerns the grouping of numbers. It indicates that how numbers are grouped in operations like addition or multiplication does not change their result. This is the law demonstrated in the given exercise with addition.
  • Distributive Law: This involves both addition and multiplication, stating that \(a(b + c) = ab + ac\).
Understanding these laws simplifies solving algebraic problems and helps in identifying relationships within an equation.
Addition
Addition is one of the most fundamental operations in mathematics. It combines two or more numbers to yield a sum. The addition operation is denoted by the plus sign \(+\), and the numbers involved are called addends. Here are some key points about addition:
  • Simplicity: It is the most basic arithmetic operation, taught from a very young age.
  • Order and Grouping: As illustrated by the Associative Property, the order or grouping of addends does not affect their sum.
  • Identity Element: Adding zero to any number leaves the number unchanged, known as the identity property of addition \(a + 0 = a\).
In our daily life, addition helps calculate totals, balance budgets, and more, making it an indispensable part of numeracy.
Mathematical Properties
Mathematical Properties are rules that apply to numbers and arithmetic operations, forming the basic building blocks for more complex mathematical concepts. These properties ensure consistency and make calculations predictable. Some properties related to the exercise are:
  • Associative Property: As mentioned earlier, it's crucial for addition and multiplication, showing that the grouping of operands doesn't change the result. This property was demonstrated in the exercise with \(3+(5+9)\) and \((3+5)+9\).
  • Commutative Property: Whether you're adding or multiplying numbers, changing the order doesn't change the result.
  • Distributive Property: An essential property that blends addition with multiplication, useful for expanding expressions.
  • Identity Property: The result of arithmetic operations involving the neutral element (0 for addition and 1 for multiplication) remains unchanged.
Collectively, these properties create a structured approach to tackle mathematical problems, making them more manageable and understandable.

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