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Chapter 5: Problem 83

State the associative property of addition and give an example.

Short Answer

Expert verified
The associative property states that the grouping of numbers does not affect the sum, i.e., \(a + (b + c) = (a + b) + c\). An example is \(2 + (3 + 4) = (2 + 3) + 4\). Both sides result in a sum of 9, confirming the associative property.

Step by step solution

01

Define the Associative Property of Addition

The associative property of addition states that when three or more numbers are added, the sum remains the same regardless of how the numbers are grouped. This can be written mathematically as: For any numbers a, b, and c, it holds that \(a + (b + c) = (a + b) + c\).
02

Provide an Example

Let's take the numbers 2, 3 and 4. According to the associative property of addition, the grouping should not affect the sum. So, \(2 + (3 + 4) = (2 + 3) + 4\). Simplifying both sides, we get \(2 + 7 = 5 + 4\), thus \(9 = 9\). This confirms that the associative property of addition is valid.

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Key Concepts

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

Addition
Addition is one of the basic operations in arithmetic. It involves combining two or more numbers to find their total or sum. For example, when you add 2 and 3, you are finding out how many units you have in total when you put them together. The process of addition is straightforward and can be summarized in the following steps:
  • Identify the numbers you want to add (known as addends).
  • Combine these addends to produce a total or sum.
In mathematical notation, addition is represented by the plus symbol (+), for example, 2 + 3 = 5.
Understanding addition is fundamental, as it forms the basis for more complex operations like multiplication and the manipulations of algebraic expressions.
Mathematical Properties
Mathematical properties act like rules that help us understand and solve math problems efficiently. The associative property is one of these properties and specifically applies to addition and multiplication. It tells us that when adding three or more numbers, the grouping of these numbers does not change their sum. This concept can be observed in the associative property of addition, written as \(a + (b + c) = (a + b) + c\).
Some key advantages of understanding mathematical properties include:
  • Enabling simplifications in calculations.
  • Providing a foundation for solving equations.
Other mathematical properties include the commutative property and the distributive property, each supporting different aspects of arithmetic operations.
Number Theory
Number theory is a branch of mathematics that focuses on the properties and relationships of numbers, especially integers. Within number theory, we often explore concepts like divisibility, prime numbers, and different number sets such as whole numbers, integers, and rational numbers. It serves as a theoretical foundation for understanding more specific properties like the associative property of addition.
This area of math helps in discovering:
  • Patterns in numbers.
  • How numbers interact with one another.
By studying number theory, we can understand why properties like associative hold true and appreciate how numbers behave in various contexts.
Arithmetic Operations
Arithmetic operations are the building blocks of all mathematical calculations. They include four primary operations: addition, subtraction, multiplication, and division. Each operation has its own rules and properties, with addition being one of the most fundamental. The associative property, for instance, directly relates to addition and highlights how different arrangements of numbers yield the same result.
Exploring arithmetic operations involves:
  • Understanding how each operation manipulates numbers.
  • Recognizing the importance of operation order (especially in complex equations).
Mastering these operations empowers students to solve everyday math problems and lays the groundwork for advanced mathematical learning.

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Most popular questions from this chapter

State the distributive property of multiplication over addition and give an example.

The sum, \(S_{n}\), of the first \(n\) terms of an arithmetic sequence is given by$$S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right),$$in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by$$S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r},$$in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(4,-12,36,-108, \ldots\)

Use properties of exponents to simplify each expression. First express the answer in exponential form. Then evaluate the expression. \(\frac{4^{7}}{4^{5}}\)

Write the first six terms of the arithmetic sequence with the first term, \(a_{1}\), and common difference, \(d\). \(a_{1}=-7, d=4\)

Express each number in scientific notation. 3600
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