Problem 100 - Elementary Algebra Within Reach

Question

In Exercises 97-100, identify the property of real numbers illustrated by the statement. $$ 7+(8+5)=(7+8)+5 $$

Step-by-Step Solution

Answer

The given equation demonstrates the associative property of real numbers.

Step 1: Identify the structure of the equation

On the left side, 8+5 is within the parentheses, and they are added first due to the order of operations. This sum is then combined with 7. On the right side, however, 7+8 is added first, and then added to 5.

Step 2: Confirm the associative property

The associative property states that when adding (or multiplying), the way the numbers are grouped doesn't change the result. To confirm that the equation follows this property, it's necessary to perform the operations on both sides of the equation and ensure they equal the same value.

Step 3: Perform the operations

Perform the addition from Step 1. On the left side, adding 8+5 gives 13. When 7 is added to that value, we have 20. On the right side, adding 7+8 results in 15, and adding 5 to that value also gives 20. Hence, the result is the same regardless of how the numbers are grouped.

Key Concepts

Real Numbers

Real numbers encompass all the numbers that you can find on the number line. They include:

• Rational numbers like fractions and whole numbers, e.g., 3, -1, 1/2
• Irrational numbers such as roots and pi, e.g., \( \sqrt{2} \), \( \pi \)

Real numbers are essential in mathematics because they represent an unending continuum of values that can describe various quantities and measures.
Moreover, every number used in calculations like addition or multiplication in elementary algebra belongs to this group. Understanding real numbers is fundamental because they are the building blocks of more complex math concepts.

Order of Operations

When performing any calculations in algebra or arithmetic, the order in which operations are executed is critical. This sequence can be remembered by the acronym PEMDAS:

• Parentheses
• Exponents (powers and roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In problems like the one from the exercise, this order dictates how we evaluate expressions. For instance:
On the left side of the equation \(7+(8+5)\), parentheses define that 8 plus 5 must be calculated first. Similarly, on the right side with \((7+8)+5\), the addition inside the parentheses is prioritized.

Addition Properties

The addition properties are rules that help us simplify and understand addition operations in mathematics. The two main properties are:

• Commutative property: The order of numbers doesn't affect the sum. For example, \(a + b = b + a\).
• Associative property: The grouping of numbers doesn't change the sum, e.g., \((a + b) + c = a + (b + c)\).

In the exercise provided, the associative property of addition is the key concept. It demonstrates that changing the grouping of numbers, as shown in \(7+(8+5) = (7+8)+5\), does not impact the final result of the addition, confirming the robustness of this property.

Elementary Algebra

Elementary algebra involves the basic principles and rules that form the foundation of algebra. This includes operations with numbers and variables, understanding expressions, and solving equations.
In this context, algebra simplifies and generalizes arithmetic operations. By applying concepts such as:

• Using variables to represent numbers
• Organizing expressions with operations and properties
• Solving equations step-by-step

Students learn to manipulate equations like \(7+(8+5) = (7+8)+5\). The principles of elementary algebra provide a systematic approach to dealing with mathematical problems, ensuring consistency and clarity in solution methods.