Problem 6 - Fundamentals of Mathematics

Question

Find the following products. $$ (-7) 7 $$

Step-by-Step Solution

Answer

The product is -49.

Step 1: Identify the Numbers

We are given the numbers \(-7\) and \(7\). The task is to find their product, which is their result when multiplied together.

Step 2: Determine the Sign of the Product

When multiplying two numbers, if one number is negative and the other is positive, the product is negative. In this case, \(-7\) is negative and \(7\) is positive, so the product will be negative.

Step 3: Multiply the Absolute Values

Ignore the signs and multiply the absolute values of \(-7\) and \(7\). The absolute value of \(-7\) is \(7\), and the absolute value of \(7\) is \(7\). Calculate \(7 \times 7 = 49\).

Step 4: Apply the Sign to the Product

Incorporate the sign determined in Step 2. Since the product is negative, the final result is \(-49\).

Key Concepts

Negative Numbers

When working with negative numbers, it's essential to understand their role in multiplication. Negative numbers are numbers that are less than zero, often represented with a minus sign \(-\). They are situated to the left of zero on the number line. For example, in the number \(-7\), the minus sign indicates that 7 units are below zero.
When multiplying numbers, the sign plays a critical role in determining the outcome of the product. Here's a simple rule to follow:

• When both numbers have the same sign (either both positive or both negative), the product is positive.
• If the numbers have different signs (one positive and one negative), the product is negative.

This rule ensures that your calculations maintain consistency and accuracy.

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, disregarding its sign. It is always a non-negative number. For example, the absolute value of \(-7\) is 7, represented as \(|-7| = 7\). Similarly, the absolute value of 7 is simply \(|7| = 7\).
Absolute values are helpful, especially in multiplication, because they allow us to focus only on the magnitude of numbers without worrying about their signs. Once the multiplication of absolute values is complete, you can then apply the appropriate sign according to the rules for negative numbers.

Product of Numbers

The product of numbers refers to the result obtained by multiplying two or more numbers. Understanding how to determine the product becomes straightforward with practice. Let's walk through an example:
Consider multiplying \(-7\) and \(+7\). Follow these steps:

• Identify the absolute values: Both numbers have an absolute value of 7.
• Multiply the absolute values: Calculate \(7 \times 7 = 49\).
• Apply the sign rule: Since one number is negative and the other is positive, the product is negative.

This method ensures that you grasp not only how to find the product but also the underlying principles governing the signs of the result.