Problem 121 - Beginning and Intermediate Algebra

Question

Explain why the product of an even number of negative numbers is a positive number.

Step-by-Step Solution

Answer

The product is positive because negative numbers multiplied in pairs yield positive results, and an even number of pairs results in a positive product.

Step 1: Understand Negative Number Multiplication

When you multiply two negative numbers, the result is positive. This is because a "negative times a negative equals a positive" following the rule of signs.

Step 2: Pair the Negative Numbers

An even number of negative numbers can be paired up into groups of two. For example, if you have 4 negative numbers: \(-a, -b, -c, -d\), pair them up as \((-a) \times (-b)\) and \((-c) \times (-d)\).

Step 3: Multiply Within Each Pair

Each pair of negative numbers produces a positive result. For instance, \((-a) \times (-b) = ab\) and \((-c) \times (-d) = cd\), both of which are positive results.

Step 4: Multiply the Positive Results

Finally, multiply the results of the positive numbers obtained from the pairs. Since all products are positive, their multiplication will also result in a positive number, i.e., \(ab \times cd\).

Key Concepts

Even Number of Negative Numbers

When dealing with negative numbers, it's interesting to find that multiplying an even number of negative numbers results in a positive outcome. Let's break it down so it's easy to grasp. Imagine you have four negative numbers:

• These can be represented as i\(-a, -b, -c, -d\).
  When we speak of an even number, we're referring to counts like 2, 4, 6, etc., which always can be divided into equal pairs.
• In this case, with four numbers: each pair will have exactly two negative numbers, such asi\((-a) \times (-b)\) and \((-c) \times (-d)\).

The importance of pairing these negative numbers is pivotal because each of these pairs will result in a positive product, as will be explained in the following sections.

Rule of Signs

The multiplication of numbers, positive or negative, adheres to what we call the "rule of signs." It is a fundamental principle in mathematics, particularly when dealing with integers:

• If you multiply two positive numbers, the product is positive.
• If you multiply a positive number by a negative number, the product is negative.
• If you multiply two negative numbers, the product becomes positive.

This transformation of negatives to a positive is crucial when calculating products of negative numbers.
By applying the rule of signs, when two negatives are multiplied together, each pair leads to a transformation that turns potentially negative products into positive ones.
For example, \[(-a) \times (-b) = ab\], reflects this principle in action.

Positive Product from Negative Pairs

After understanding that pairs of negative numbers yield positive products when multiplied, we can tackle the final step: what occurs when you multiply these positive results?
Once you have formed pairs and multiplied them individually, each pair results in a positive product. For instance,

• The pairs \((-a) \times (-b) \quad \text{and} \quad (-c) \times (-d)\) will each yield positive results: \[ ab \quad \text{and} \quad cd \].
• Multiplying these positive outcomes, \[ ab \times cd \], maintains the positivity.

This process illustrates why an even number of negative numbers, after being multiplied, will yield a positive product. The step-by-step transformation from negative to positive is key to this phenomenon. Once you understand it, multiplying negative numbers becomes clear and predictable.