Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ (2 x+y)+z=z+(2 x+y) $$

Step-by-Step Solution

Answer

The real number property that justifies the statement \((2 x+y)+z=z+(2 x+y)\) is the Commutative Property of Addition.

Step 1: Identify the real number property

The given statement involves addition of real numbers and their order. The property that deals with this situation is the Commutative Property of Addition, which states that the order in which we add real numbers does not matter. In other words, for all real numbers a and b: \( a+b = b+a \)

Step 2: Recognize the components of the expression

In our given statement, the two real numbers being added are: a = (2x + y) b = z

Step 3: Apply the Commutative Property of Addition

Using the Commutative Property of Addition for the real numbers a and b: (2x + y) + z = z + (2x + y) Hence, the real number property that justifies the given statement is the Commutative Property of Addition.

Key Concepts

Real Numbers

The concept of real numbers is foundational in mathematics, often serving as the building blocks for various mathematical theories and equations. 'Real numbers' include both rational numbers (like 1, 1/2, and 0.333…) and irrational numbers (like \( \sqrt{2} \) or \( \pi \)). They can be thought of as any point on an infinitely long number line.

Understanding real numbers is crucial because they are used to measure continuous quantities in the real world, such as distance, temperature, and time. When it comes to mathematical operations such as addition, real numbers adhere to certain rules or properties which allow for consistent and reliable calculations. These properties, like the Commutative Property of Addition demonstrated in the given exercise, provide the structure needed to manipulate and work with numbers efficiently in mathematics.

Associative Property

The associative property is another essential mathematical concept, particularly with respect to addition and multiplication. This property tells us that the way in which numbers are grouped when adding or multiplying does not affect the final result. For instance, when adding real numbers, whether we add \( a \) and \( b \) first, and then add \( c \) (\( (a+b)+c \) ), or if we add \( b \) and \( c \) first, and then add this sum to \( a \) (\( a+(b+c) \)), the outcome will be the same.

Mathematically, for any real numbers \( a \) , \( b \) , and \( c \) , the associative property of addition is expressed as:
\( (a+b)+c = a+(b+c) \). The same holds true for multiplication. This property is immensely beneficial when solving complex expressions as it allows us to rearrange and group numbers in a way that simplifies the computation.

Mathematical Properties

Mathematical properties are the rules that govern the operations we perform with numbers. They are essential in ensuring the consistency and logic within mathematics, making problem-solving more systematic and predictable.

Aside from the commutative and associative properties, there are several other properties worth noting:

• Identity Properties: The identity properties state that adding 0 to any number doesn't change the number (Additive Identity), and multiplying any number by 1 does not change the number (Multiplicative Identity).
• Inverse Properties: These properties tell us that for every nonzero number, there is a reciprocal for multiplication (Multiplicative Inverse) and an opposite for addition (Additive Inverse), which will return the number back to its identity value.
• Distributive Property: The distributive property connects multiplication and addition, letting us multiply a sum by a number equivalently by multiplying each addend individually by that number and then adding the results.

These properties not only facilitate computation but also help in proving theorems and solving algebraic expressions. They're fundamental tools that are applied across various fields of mathematics and play a significant role in the study of arithmetic, algebra, and beyond.