Problem 12 - Algebra for College Students

Question

Perform the following operations with real numbers. $$(-17)(4)$$

Step-by-Step Solution

Answer

The result of (-17)(4) is -68.

Step 1: Understanding the Problem

We need to multiply two real numbers: -17 and 4. To do this, we apply the rules for multiplying a negative number by a positive number.

Step 2: Applying the Multiplication Rule

When multiplying a negative number by a positive number, the result is always negative. In this case, we will multiply the absolute values of the numbers first, and then apply the sign.

Step 3: Calculating the Product

Multiply the absolute values: 17 and 4. Perform the multiplication: \[17 \times 4 = 68\] Since one of the numbers was negative, the product will also be negative, giving us -68.

Step 4: Conclusion

We have determined that the product of -17 and 4 is -68.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero, typically represented with a minus sign (-). These numbers can be found on the left side of the number line. They are used to represent various situations like below-zero temperatures or financial debts.

• Negative numbers follow different arithmetic rules compared to positive numbers.
• When multiplied with positive numbers, the result is negative.
• Negative signs are an important aspect of various real-life equations and calculations.

Learning to work with negative numbers is essential for understanding more complex mathematical operations like multiplication, division, and beyond.

Exploring Positive Numbers

Positive numbers include all numbers greater than zero. They are represented without a sign or with a plus sign (+). These numbers appear on the right side of the number line. Positive numbers often represent quantities such as the number of items, natural counts, or financial gains.

• They follow the basic arithmetic rules, making them easier for calculations.
• Multiplying two positive numbers yields a positive product.
• When combined with negative numbers in operations, they can help in determining direction or result sign.

Understanding positive numbers helps in setting a foundation for all basic arithmetic and algebraic operations.

Defining Absolute Value

The absolute value of a number is its distance from zero on a number line, ignoring its sign. It is always a non-negative number. The absolute value is denoted by vertical bars surrounding the number, like this: \( |a| \).

• For example, the absolute value of -17 is 17.
• Similarly, the absolute value of 4 is 4.
• Absolute value helps in simplifying many mathematical operations involving negative numbers.

Absolute value is fundamental when determining the magnitude of numbers, which is especially useful in problems involving distances or measurements.

Applying Multiplication Rules

When dealing with multiplication in mathematics, certain rules govern the product's sign. Specifically:

• If both numbers are positive, the product is positive.
• If both numbers are negative, the product is positive because the negative signs cancel each other out.
• If one number is positive and the other is negative, the product is negative.

In the problem \((-17)(4)\), the multiplication involves a negative and a positive number, resulting in a negative product. By multiplying the absolute values: \[17 \times 4 = 68\]And applying the multiplication rule, the final product is -68. These rules simplify solving such problems and help in predicting the sign of products quickly during calculations.