Problem 6 - Intermediate Algebra

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$ (-1)(48)=-48 $$

Step-by-Step Solution

Answer

Multiplication Property of -1.

Step 1: Understanding the Problem

We are given the expression \((-1)(48) = -48\) and need to identify which mathematical property justifies this statement. To do this, we need to analyze the operation being performed: multiplication with a negative number.

Step 2: Identifying the Operation

The expression involves multiplication of a number by \(-1\). In mathematics, multiplying any number by \(-1\) inverts the sign of the number, making a positive number negative and a negative number positive.

Step 3: Recognizing the Property

The specific rule governing the multiplication by \(-1\) is part of the properties of real numbers. It states that multiplying any number by \(-1\) will result in the additive inverse (or opposite) of the number.

Step 4: Reaching Conclusion

The property used in the given expression \((-1)(48) = -48\) is the "Multiplication Property of -1", which states that multiplying a number by \(-1\) results in its negative.

Key Concepts

Multiplication Property of -1

The Multiplication Property of -1 is a unique and intriguing feature of real numbers. It simply states that when you multiply any number by \(-1\), you obtain a new number that is the additive inverse, or the opposite, of the original. This is why when you take the example of 48 and multiply it by \(-1\), the result is \(-48\).

• Positive numbers become negative.
• Negative numbers become positive.
• Zero remains zero, since the inverse of zero is zero.

This property is particularly useful in solving equations and simplifying expressions, as it allows for quick sign changes without altering the magnitude of a number. Imagine having a pile of 48 apples, and using the multiplication by \(-1\) like a magic wand that changes your pile of apples to take away 48 apples instead! It's a simple yet powerful way to understand how signs change in arithmetic.

Additive Inverse

The concept of an Additive Inverse is central to understanding negative numbers and their role in arithmetic. Essentially, the additive inverse of a number is what you add to that number to get zero. Every real number has an additive inverse.

• For a positive number like 48, the additive inverse is \(-48\).
• For a negative number like \(-7\), the additive inverse is \(7\).
• The additive inverse of zero is again zero since \(0 + 0 = 0\).

This idea is the basis for subtraction, which can be viewed as adding the additive inverse. For instance, if you were to delve into subtracting \(8 - 3\), it is the same as adding \(8 + (-3)\). Here, \(-3\) is the additive inverse of \(3\), helping you reach the result of \(5\). Recognizing and using additive inverses effectively can simplify the manipulation and balancing of equations.

Commutative Property of Addition

The Commutative Property of Addition is one of the fundamental characteristics of real numbers that makes arithmetic intuitive and flexible. This property asserts that the order in which you add two numbers does not affect their sum. In simpler terms:

• \(3 + 5 = 5 + 3\)
• Both result is \(8\).
• This property holds true for all real numbers, whether positive, negative, or zero.

Think of it as relating to physical objects, like stacking books. Whether you place Book A on top of Book B or the other way around, the total height stays the same. This property fosters a more efficient computation, such as quick mental math, and helps in reordering terms in algebraic expressions to simplify them. Recognizing this property also aids in developing strategies for checking your work, especially in lengthy mathematical computations where reordering can lead to quicker verification of answers.