Problem 9 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

This equation uses the associative property of addition.

Step 1: Identify the Operation

The expression involves addition of three terms: \((x + 2y)\) and \(3z\) on the left-hand side, rearranged as \(x\) and \((2y + 3z)\) on the right-hand side.

Step 2: Recognize the Structure

Observe that the grouping of terms in the addition operation changes between the left-hand side and the right-hand side. On the left, \((x + 2y) + 3z\) is grouped, while on the right, \(x + (2y + 3z)\) is grouped.

Step 3: Apply the Associative Property

The property being applied here is the associative property of addition, which states that the sum remains the same regardless of how the numbers are grouped. Formally, it can be expressed as: \((a + b) + c = a + (b + c)\).

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics. They include all the numbers we usually work with:

• Positive and negative numbers
• Whole numbers and fractions
• Integers and decimals

This broad category encompasses rational and irrational numbers. Real numbers can be represented on a number line, which provides a visual way to understand their relationships.
This makes them easy to add, subtract, multiply, and divide using standard arithmetic operations. When dealing with properties like the associative property, real numbers play a crucial role because these properties hold true for all real numbers.

Addition Operation

The addition operation is one of the basic arithmetic operations used to combine numbers. It's represented by the plus symbol "+". Let's say you have two numbers, such as 3 and 5. Adding them gives you 8, written as:

• 3 + 5 = 8

When you apply addition to more than two numbers, the process remains simple. For instance, 2 + 3 + 4 equals 9.
Addition is commutative, which means that the order of numbers doesn't affect the result. For example, 4 + 2 is the same as 2 + 4. When using addition, remember that you can group the numbers differently without changing the sum, thanks to the associative property.

Grouping Terms

Grouping terms refers to how you organize numbers in mathematic operations like addition. In the expression each term represents a number or a combination of numbers like
Note that in the expression (x + 2y) + 3z = x + (2y + 3z), the terms are grouped differently.

• Left side: (x + 2y) is grouped
• Right side: (2y + 3z) is grouped

Grouping doesn’t affect the final result in addition due to the associative property. This ability to regroup terms without changing the outcome is vital when simplifying expressions.

Mathematical Properties

Mathematical properties are rules that numbers follow during operations. The associative property is one of these important rules, particularly for addition. It states that how you group numbers does not affect their sum. Mathematically, this is shown as:

• \((a + b) + c = a + (b + c)\)

This property allows us to change the grouping of numbers to make calculations easier. Suppose you have (3 + 5) + 2. You can regroup it as 3 + (5 + 2), and the sum remains the same, which is 10.
Recognizing and using mathematical properties, like the associative property, enhances the flexibility and efficiency of solving math problems.