Problem 4 - Prealgebra

Question

Find each of the following products. (Multiply.) $$4(-8)$$

Step-by-Step Solution

Answer

The product is -32.

Step 1: Understanding Multiplication with Negative Numbers

In this problem, we need to multiply two numbers: 4 and -8. The rules of multiplication say that multiplying a positive number by a negative number results in a negative number.

Step 2: Performing the Multiplication

Now, we multiply the absolute values of the numbers, 4 and 8, ignoring the signs for this step. The product of 4 and 8 is 32.

Step 3: Assigning the Correct Sign

Since the multiplication involves a positive number and a negative number, the result of this multiplication will be negative. Therefore, the product is -32.

Key Concepts

Negative Number Multiplication

When you first encounter the concept of multiplying negative numbers, it might seem a bit confusing. However, the rule is rather straightforward. Whenever you multiply a negative number by a positive number, the result is always a negative number. This is because the positive and negative signs essentially "cancel out" to some extent. For example, when you multiply 4 by -8, since one of these numbers is positive (4) and the other is negative (-8), the product becomes negative, so you end up with -32.
The rules are easy to remember once you get the hang of it:

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

These rules are consistent and can be applied to all real numbers, making the process of multiplication much easier to understand.

Absolute Value

Before diving into multiplication, it's essential to grasp what absolute value means. The absolute value of a number is its distance from zero on the number line, without regard to a positive or negative sign.
In simpler terms, the absolute value of a number is always positive. Whether you are dealing with 4, -8, or 8, the absolute value ignores the sign and focuses purely on the magnitude. Thus:

• The absolute value of 4 is 4.
• The absolute value of -8 is 8.
• The absolute value of 8 is 8.

In our example of multiplying 4 by -8, ignoring the signs initially allows us to compute the product of the absolute values, 4 and 8, giving us 32. After determining the absolute value, the final step involves applying the rules of negative number multiplication to ascertain the correct sign of the product.

Integers Multiplication

Multiplying integers follows specific rules that relate closely to signs and absolute values. An integer is any whole number, positive or negative, including zero. Understanding how multiplication works with integers is fundamental because most calculations in basic arithmetic involve integers.
Let's break down integer multiplication:

• If both integers are positive, simply multiply their absolute values and the result remains positive.
• If both are negative, multiply their absolute values; however, the result will be positive because two negatives make a positive.
• If one integer is positive and the other is negative, multiply their absolute values, but the result is negative.

Remember, the key to mastering integer multiplication lies in properly applying these sign rules. For instance, in multiplying 4 by -8, since only one of these integers is negative, the result naturally becomes negative, i.e., -32.