Problem 13 - Precalculus: Mathematics for Calculus

Question

Properties of Real Numbers State the property of real numbers being used. $$(x+2 y)+3 z=x+(2 y+3 z)$$

Step-by-Step Solution

Answer

Associative Property of Addition.

Step 1: Associative Property Introduction

In this problem, we are dealing with an equation that has been rewritten in a different configuration. Our task is to identify which property of real numbers allows for this change in configuration.

Step 2: Rewrite the Expressions

The original expression is \((x + 2y) + 3z\) and it has been rewritten as \(x + (2y + 3z)\). Notice how the grouping of the terms has changed.

Step 3: Identify the Property

Since the problem involves re-grouping terms without altering their order or the operation of addition, this identifies the use of the Associative Property of Addition, which states that the grouping of numbers does not affect their sum: \((a + b) + c = a + (b + c)\).

Key Concepts

Properties of Real Numbers

The properties of real numbers are crucial rules that help in simplifying and solving math problems. These properties include the commutative, associative, distributive, identity, and inverse properties. They describe how we can add, subtract, multiply, and divide numbers, ensuring operations' consistency.

The associative property is particularly useful. It's all about grouping numbers in addition or multiplication tasks. Even though the numbers are grouped differently, the result doesn't change. This can simplify solving complex equations. Understanding properties helps in formulating strategies for smart calculation.

For example, combining like terms in an equation often uses the associative property. Recognizing and applying these properties allows for greater flexibility and efficiency in problem-solving.

Here are a few functions of the properties of real numbers:

• They ensure computations are consistent and predictable.
• They often simplify expressions and equations.
• They provide methods to check correctness of your work.

Associative Property of Addition

The associative property of addition is a fundamental aspect of real numbers. It states that when adding three or more numbers, the way in which the numbers are grouped does not affect the sum. In simpler terms, it doesn't matter how you place the parentheses when adding numbers.

For example, given three numbers, a, b, and c, the associative property tells us:

• \( (a + b) + c = a + (b + c) \).

This property is highlighted in the example equation

• \((x + 2y) + 3z = x + (2y + 3z)\).

Here, the sum remains the same despite the regrouping of terms.

This property is very handy when dealing with complex algebraic expressions. By reordering or regrouping terms, we can simplify problems or make computations more manageable. So, whenever you see numbers being rearranged without changing the operation, the associative property is likely at play.

Real Numbers

Real numbers encompass a wide range of numbers used in everyday math. They include both rational numbers (which can be expressed as fractions) and irrational numbers (which cannot be expressed as simple fractions). This category of numbers is central to most basic arithmetic operations.

Rational numbers are numbers like 3, -5, or 7/2, which can either be whole numbers, decimals, or fractions. Meanwhile, irrational numbers include numbers like \( \sqrt{2} \) or \( \pi \), which cannot be accurately expressed as fractions and have an infinite, non-repeating decimal representation.

Most importantly, real numbers adhere to specific mathematical properties, including the associative property outlined above. Understanding real numbers is essential for grasping further mathematical concepts, as they form the basis of our understanding of mathematical reality.

Here’s why real numbers are important:

• They form the foundation for most math lessons in schools and colleges.
• They allow for accurate representation and solution of tasks in algebra.
• They enable various scientific and real-world applications that rely on mathematical models.