Problem 23 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ (-2)(6) $$

Step-by-Step Solution

Answer

The product is -12.

Step 1: Identify the Factors

The problem requires us to find the product of two numbers: \[-2\] and \[6\].We need to multiply these two numbers to find the solution.

Step 2: Multiply the Numbers

To multiply \[-2\] and \[6\], multiply their absolute values first, which is \[2 \times 6 = 12\].Since one of the numbers is negative, the product will also be negative.

Step 3: Apply the Sign

After obtaining the absolute product, add a negative sign because one of the original numbers is negative:\[-(12) = -12\].So, the product of \[-2\] and \[6\] is \[-12\].

Step 4: Verify with a Calculator

Use a calculator to multiply \[-2\] and \[6\] again:- Enter \[-2\] in the calculator.- Press the multiplication sign.- Enter \[6\] and press equals.- The calculator will confirm that the product is indeed \[-12\].

Key Concepts

Negative Numbers

Negative numbers are numbers with a value less than zero, often represented with a "-" sign. They are crucial in mathematics because they extend the traditional number line, allowing us to represent values less than zero.

Understanding negative numbers is essential for operations like subtraction, temperature readings below zero, and financial debts. When performing operations with negative numbers, the sign of a number plays a crucial role. For instance, multiplying two negative numbers results in a positive product, while multiplying a negative and a positive number—like in the exercise above—yields a negative result. Whenever you encounter an operation involving negative numbers, pay close attention to the signs involved to ensure accurate results.

Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. It is always represented as a non-negative value. The absolute value is especially helpful in making complex calculations simpler by stripping away the signs of the numbers involved.

For the exercise \((-2)(6)\), we first examine the absolute values of \(-2\) and \(6\), which are \(2\) and \(6\), respectively. By converting them into positive numbers, we simplify the multiplication process: \(2\times 6 = 12\). Once calculated, we need to consider the original signs to determine the final outcome. When dealing with applications where direction is insignificant, absolute value simplifies comparisons and calculations.

Integer Operations

Integer operations encompass the basic arithmetic operations—addition, subtraction, multiplication, and division—applied to whole numbers, both positive and negative. Each of these operations with integers follows specific rules, especially when it comes to the signs of the numbers involved.

For multiplication, like in the exercise above, the following rules apply:

• Positive times positive equals positive.
• Negative times negative equals positive.
• Positive times negative equals negative.
• Negative times positive equals negative.

Learning these rules helps you predict outcomes without always using a calculator. For instance, multiplying \(-2\) by \(6\) results in \(-12\) due to the third rule. Being comfortable with these operations enhances your ability to solve math problems efficiently and accurately.