Problem 9 - Algebra for College Students

Question

Perform the following operations with real numbers. $$(-9)(-12)$$

Step-by-Step Solution

Answer

108

Step 1: Understanding Multiplication of Negative Numbers

When multiplying two negative numbers, the product is positive. This is because multiplying a negative by a negative yields a positive, just like how the negative sign of one number cancels out the negative sign of the other.

Step 2: Identify the Numbers to Multiply

We are given the numbers (-9) and (-12). We need to multiply these two negative numbers together.

Step 3: Multiply the Absolute Values

Ignore the negative signs and multiply the absolute values of the numbers. So, multiply 9 by 12:\(9 \times 12 = 108\)

Step 4: Determine the Sign of the Product

Since both numbers were negative, the rule mentioned in Step 1 applies, and the product will be positive. Thus, the multiplication of (-9) and (-12) gives a positive result.

Step 5: Combine Results

Combine the results from the previous steps to find the final answer. The product of (-9) and (-12) is \(108\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign in front. In mathematics, they are used to describe values below a defined point, like sea level temperatures or debts. Negative numbers have unique properties, especially when involved in operations like addition, subtraction, and multiplication.
When multiplying, negative numbers follow specific rules that are essential to grasp. For example, multiplying a negative number by a positive number results in a negative product. Meanwhile, multiplying two negative numbers together flips the signs, leading to a positive result. Understanding these properties is crucial for solving problems involving negative numbers.

Grasping Absolute Value

The absolute value of a number is the distance between the number and zero on a number line, without considering the direction. It’s denoted by vertical bars around the number, such as \(|-9|\), meaning 'the absolute value of -9'.
Absolute value strips away the negative sign, turning negative numbers into their positive counterparts. For example, the absolute value of -9 is 9, just as \(|-12| = 12\).
This concept is particularly useful in multiplication, as you'll often multiply the absolute values first before applying rules about negative numbers to determine the sign of the final product.

The Concept of a Positive Product

Getting a positive product isn't tricky if you understand the rules. When you multiply two negative numbers, the result is always positive. This might seem unintuitive, but one way to picture it is by thinking about reversing direction twice, which brings you back to the start.
For example, multiplying \((-9)\) with \((-12)\) gives a positive product. First, we take the absolute values: \(9 \times 12 = 108\). Then, because two negatives make a positive when multiplied, the product is positive 108.
The rule also applies when multiplying even counts of negative numbers; any even number of negatives results in a positive, while an odd number of negatives results in negative.

Basic Multiplication Rules

Multiplication rules form the foundation of arithmetic operations with real numbers. Basic rules include:

• Multiplying two positive numbers yields a positive result. For example, 3 multiplied by 4 results in 12.
• Multiplying a positive number by a negative number yields a negative result. This is seen in \(3 \times (-4) = -12\).
• Multiplying two negative numbers gives a positive result, as demonstrated by \((-3) \times (-4) = 12\).

By knowing these rules, solving multiplication problems becomes more straightforward. These principles not only make calculations with real numbers more predictable but also aid in complex problem-solving involving algebra and geometry. The rule of signs is a vital component in understanding these operations and is used universally across various mathematical disciplines.