Problem 97 - Elementary Algebra Within Reach

Question

In Exercises 97-100, identify the property of real numbers illustrated by the statement. $$ 3(4)=4(3) $$

Step-by-Step Solution

Answer

The property of real numbers illustrated by the statement \(3(4)=4(3)\) is the commutative property of multiplication.

Step 1: Analyze the Operation Used

In the statement \(3(4)=4(3)\), the operation used is multiplication. It's noticed that the order of the numbers being multiplied is switched, but the result remains the same. This suggests that the commutative property might be the one being illustrated.

Step 2: Identify the Property

The commutative property states that the order in which numbers are added or multiplied does not change the result. This is exactly what's happening in the given statement. Therefore, the property illustrated by the statement \(3(4)=4(3)\) is the commutative property of multiplication.

Key Concepts

Real Numbers

Real numbers include all the numbers we typically use in everyday arithmetic: positive and negative whole numbers, fractions, and mathematical constants like \(\pi\). They are called 'real' because they can be found on the number line, representing all possible magnitudes, including both rational and irrational numbers. Understanding real numbers is fundamental in algebra, as they represent possible values of variables and constants in equations.

When we speak of real numbers in the context of multiplication, we imply that these numbers behave predictably. Properties such as the commutative property apply to them, which makes algebraic manipulations consistent and reliable. It’s important to comprehend that the commutative property confirming \(3(4)=4(3)\) holds not just for whole numbers, but for all members of the real number set.

Properties of Numbers

Properties of numbers, such as commutative, associative, distributive, identity, and inverse, serve as foundational rules for operations in mathematics. In elementary algebra, being aware of these properties allows us to manipulate and solve equations with understanding and ease.

For instance, the commutative property for both addition and multiplication underpins the flexibility in rearranging terms and factors without altering the outcome. This property is crucial when simplifying expressions or solving equations.

The Commutative Property of Multiplication

Specifically, it states that changing the order of factors does not affect the product: \(a \times b = b \times a\). In our exercise \(3(4)=4(3)\), this property is exactly what's being demonstrated.

Elementary Algebra

Elementary algebra is an area of mathematics that introduces the use of symbols and letters to represent numbers and quantities in formulas and equations. This abstraction allows for generalizations of arithmetic to be made, beyond simple computation, and into solving for unknown values.

An understanding of the properties of numbers is essential in this area because it helps decipher how to manipulate these symbolic expressions effectively. The commutative property of multiplication is a basic yet powerful tool in algebra that aids in reorganizing and simplifying algebraic expressions and equations. Recognizing this property, students can feel confident in solving an array of problems without worrying about altering number sequences when multiplying.

Multiplication

Multiplication is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction, and division. The multiplication of two numbers results in a product, and it can be thought of as repeated addition. For example, \(4 \times 3\) represents adding the number 4 to itself 3 times.

The commutative property is one of the key characteristics of multiplication of real numbers. It allows for flexible computational strategies and can be observed when multiplying numbers in either order; you'll obtain the same result. In the exercise \(3(4)=4(3)\), this property simplifies computation and assures us that no matter which way we multiply these numbers, we're guaranteed to get the same answer, highlighting one of the reassuring consistencies within mathematics.