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Chapter 1: Problem 23

Multiply. See Example 1. $$ 12(-5) $$

Short Answer

Expert verified
The product of 12 and -5 is -60.

Step by step solution

01

Identify the numbers

We need to multiply the numbers 12 and -5. Here, 12 is a positive number and -5 is a negative number.
02

Multiply the absolute values

First, ignore the signs and simply multiply the absolute values of the two numbers: 12 and 5. So, calculate \(12 \times 5 = 60\).
03

Determine the sign of the product

Since you are multiplying a positive number with a negative number, the product will be negative. Therefore, attach a negative sign to the result obtained in the previous step.
04

Write the final result

Combine the results from the previous steps to get the final answer: \(-60\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Positive and Negative Integers
When we talk about integers, we are referring to whole numbers that can be either positive, negative, or zero. Positive integers are numbers greater than zero, like 1, 2, and 12 in our example. These numbers are situated to the right of zero on a number line. Negative integers, on the other hand, are numbers less than zero, like -1, -2, and -5. They appear to the left of zero on the number line.

Recognizing whether a number is positive or negative is crucial when performing operations like multiplication. The sign of an integer affects the sign of the result, so always note the signs before starting to multiply.
The Concept of Absolute Value
The absolute value of a number is a measure of how far that number is from zero on the number line, regardless of its direction (meaning its sign). It's always a positive value or zero. For example, the absolute value of -5 is 5, written as \(|-5| = 5\).

When multiplying numbers, we often use absolute values to simplify the process by temporarily ignoring the signs. In the exercise, you multiply the absolute values of 12 and 5 to get 60. This step focuses solely on the numerical part without considering whether the numbers are positive or negative. Once the multiplication is complete, you'll determine the final sign of the product.
Applying Multiplication Rules with Negative Numbers
Multiplying with negative numbers follows certain rules that are vital to understand. Here are some basic rules for multiplying integers:
  • Positive 脳 Positive = Positive
  • Negative 脳 Negative = Positive
  • Positive 脳 Negative = Negative
  • Negative 脳 Positive = Negative
In this context, when multiplying a positive integer by a negative integer, the result will always be negative. This is because the absolute values are multiplied first and the negative sign from one of the integers affects the product's sign.

In the given exercise, multiplying 12 (a positive integer) by -5 (a negative integer) results in -60. This negative result aligns with the rule that a positive number multiplied by a negative number yields a negative product.

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Most popular questions from this chapter

Write each fraction as a decimal. If the result is a repeating decimal, use an overbar. $$ \frac{1}{30} $$

Let POS stand for a positive number and NEG stand for a negative number. Determine the sign of each result, if possible. a. \(\mathrm{POS} \cdot \mathrm{NEG}\) b. \(\mathrm{POS}+\mathrm{NEG}\) c. \(\mathrm{POS}-\mathrm{NEG}\) d. \(\frac{\mathrm{POS}}{\mathrm{NEG}}\)

Add. $$ -6.25+8.5 $$

Write \(-3-(-5)\) as addition of the opposite.

Perform the operations. $$ (-2)(-2)(-2)(-2) $$
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