Problem 123 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain how to multiply two real numbers. Provide examples with your explanation.

Step-by-Step Solution

Answer

Negative three times five equals negative fifteen, and negative two times negative three equals six.

Step 1: Realize the Nature of Numbers

Understand that real numbers include both positive, negative numbers and zero. When two positive numbers are multiplied, the result is positive. Likewise, when two negative numbers are multiplied, the result is also positive since the minuses cancel out. However, when a positive and a negative are multiplied, the result is a negative.

Step 2: Perform Basic Multiplication

Simply multiply the two numbers as if they were both positive. For example, if you are multiplying -3 and 5, you just multiply 3 and 5 to get 15.

Step 3: Assign the Sign to the Result

After performing the multiplication, assign the correct sign to the result. The sign is determined by the rules discussed in step 1. In this case, since we multiplied -3 (negative) with 5 (positive), the result is -15.

Step 4: Example

Let's take an example of two negative numbers, say -2 and -3. As per normal multiplication, 2*3 would yield 6. But since both numbers were negative and the rule states the product of two negatives is a positive, the result is 6.

Key Concepts

Properties of Real Numbers

The concept of real numbers is foundational in mathematics, especially when it comes to the rules that govern arithmetic operations. Real numbers can be positive, negative, or zero and include all the numbers on the number line, such as whole numbers, fractions, irrational numbers, and more. Understanding their properties is essential for successful multiplication and other operations.

One of the fundamental properties of real numbers is the commutative property which tells us that the order in which we multiply numbers does not affect the result, for example, \( 3 \times 5 = 5 \times 3 \). Similarly, the associative property indicates that grouping does not change the product, as seen in \( (2 \times 3) \times 4 = 2 \times (3 \times 4) \).

Another property to consider is the distributive property, which helps us multiply a number by a sum or difference within a set of parentheses, such as in the expression \( 2 \times (3 + 4) = (2 \times 3) + (2 \times 4) \). This is particularly useful when working with algebraic expressions or larger numbers. These properties ensure consistency and order within the realm of mathematics, streamlining computations across diverse problems.

Positive and Negative Number Multiplication

When multiplying positive and negative numbers, the rules are straightforward, yet essential for deriving the accurate sign of the answer. The multiplication of two positive numbers, such as \( 4 \times 3 \), results in a positive product, \( +12 \). The same positive outcome applies when multiplying two negative numbers because of a key rule: the signs cancel out, for example, \( (-2) \times (-6) = +12 \).

However, when you multiply a positive number with a negative one, the product is always negative, such as in \( 7 \times (-3) = -21 \), reflecting the contrasting nature of their signs. It's helpful to remember this as 'different signs result in a negative product' while 'same signs result in a positive product'. These rules are not only essential for basic arithmetic but also for understanding more complex algebraic expressions where the correct sign can determine the direction of an inequality or the side of the axis a function may lie on.

Basic Arithmetic Operations

The ability to perform basic arithmetic operations, which include addition, subtraction, multiplication, and division, is like holding the keys to the castle of mathematics. Multiplication, one of these fundamental operations, involves combining equal groups or adding a number to itself repeatedly. For instance, \( 4 \times 5 \) is the same as adding 4 to itself 5 times (\( 4 + 4 + 4 + 4 + 4 \) which equals 20).

Mastering multiplication requires familiarity with times tables and the associative and commutative properties to simplify calculations. When working with real numbers, it's useful to keep these properties in mind and apply them to solve problems more efficiently. Always start by simplifying the numbers if possible, before adhering to the correct procedure for positive and negative multiplication. Whether we are calculating the area of a rectangle or scaling a recipe for a larger crowd, multiplication is one of the basic operations that permeate everyday calculations and deeper mathematical concepts alike.