Problem 3 - Fundamentals of Mathematics

Question

Find the following products. $$ (-6)(-5) $$

Step-by-Step Solution

Answer

The product of (-6) and (-5) is 30.

Step 1: Understand the Problem

We are given the task to find the product of two numbers: (-6) and (-5). The problem involves multiplying two negative integers, which is essential to understand before proceeding.

Step 2: Identify the Rule for Negative Multiplication

When multiplying two negative integers, the resulting product is always positive. This is an important property that we'll use to solve the problem.

Step 3: Calculate the Product

To find the product of (-6) and (-5), we multiply the absolute values of the two numbers: \[ 6 \times 5 = 30 \]Since both numbers are negative, according to the rule, the result will be positive. Thus, the product of (-6) and (-5) is 30.

Key Concepts

Negative Numbers

Negative numbers are a fundamental part of mathematics. They represent values less than zero and are often used to indicate loss, debt, or a decrease. Negative numbers are found to the left of zero on the number line. In our example, both (-6) and (-5) are negative. Imagine you owe someone $6 and you also owe another person $5, both amounts can be represented as negative numbers.
Understanding negative numbers is crucial in many real-world situations. They allow us to handle scenarios with deficit or debt accurately. For example,

• Temperatures below zero are measured in negative numbers.
• Bank overdrafts or deficits in finances are expressed as negative amounts.

Negative numbers follow particular rules when added, subtracted, or multiplied, which leads us to their behavior in different operations.

Multiplication Rules

Multiplication rules for positive and negative numbers help determine the product’s sign. Here’s what you need to know:

• When two positive numbers are multiplied, the result is positive.
• When a positive number is multiplied by a negative number, the result is negative.
• When two negative numbers are multiplied, like (-6) and (-5), the result is positive.

This happens due to the property that a negative times a negative results in a positive, which may seem counterintuitive initially. A simple way to think about it is cancelation of negative signs, leading to a positive number. In our case, the product of (-6) and (-5) turns out to be +30, as multiplying their absolute values (ignoring signs initially) gives 30.

Mathematical Properties

Understanding the multiplication of integers involves recognizing mathematical properties such as the commutative and associative properties. These properties make calculations more efficient and straightforward.
The **commutative property** states that the order of multiplication does not affect the outcome. For instance, (-6) multiplied by (-5) will yield the same result as (-5) multiplied by (-6). Hence, \((-6) \times (-5) = (-5) \times (-6)\).
The **associative property** allows for grouping numbers in a multiplication problem, facilitating easier computation: \((a \times b) \times c = a \times (b \times c)\). This property, while not directly applied in our problem, ensures that complex calculations can be tackled by grouping smaller parts.
By applying these rules and properties, mathematics becomes a more consistent and logical language, especially when dealing with both positive and negative integers.