Problem 16 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The result is \(-68\).

Step 1: Identify the Operation

The exercise asks for the multiplication of two real numbers: \(-17\) and \(4\).

Step 2: Understand the Signs

Check the signs of both numbers. \(-17\) is negative, and \(4\) is positive.

Step 3: Multiply the Numerical Values

Ignore the signs temporarily and multiply the absolute values: Multiply \(17\) by \(4\) to get \(68\).

Step 4: Apply the Sign Rule

According to the multiplication rule of integers, a negative number multiplied by a positive number results in a negative product.Hence, \((-17)(4) = -68\).

Key Concepts

Multiplication of Real Numbers

When we talk about the multiplication of real numbers, we are referring to the process of taking two numbers and finding their product. Real numbers include all the numbers on the number line, such as whole numbers, fractions, irrational numbers, and decimals. Understanding how to multiply these numbers is crucial in mathematics.

The key steps to multiply real numbers are:

• Identify the numbers being multiplied.
• Understand the signs of those numbers.
• Multiply their absolute values (the numbers without any signs).
• Lastly, apply the appropriate sign to the result, based on the sign rule for multiplication of real numbers.

This simple guide will help you accurately determine the product of any two real numbers, ensuring you comprehend both positive and negative products.

Integer Multiplication

Integer multiplication forms the backbone of understanding how to handle real numbers. Integers are simply whole numbers which can be positive, negative, or zero. Multiplying integers involves calculating the product of numbers without considering their decimal parts.

Let's focus on an integer multiplication example with \(-17\times 4\):

• First, determine the absolute value of each integer. For \(-17\text{,}\) the absolute value is\(17\), and for\(4\), it remains\(4\).
• Next, perform the multiplication of these absolute values:\(17\times 4=68\).
• Then, apply the sign rules, as multiplying a negative integer with a positive integer gives a negative product. This results in\(-68\).

By understanding integer multiplication, it becomes much easier to address more complex problems that include a wider variety of real numbers.

Negative and Positive Numbers

One of the trickiest parts of multiplication of real numbers is managing the signs when dealing with negative and positive numbers. Signs affect the outcome of any multiplication problem significantly.

Here are some crucial points about handling negative and positive numbers in multiplication:

• Multiplying two positive numbers always results in a positive product.
• Multiplying two negative numbers also gives a positive product, since "a negative times a negative equals a positive."
• When one number is negative and the other is positive, the product will be negative. This is because the negative number's influence reverses the positive product.

Understanding these sign rules ensures you can correctly determine the outcome of multiplying real numbers with different signs. It simplifies solving equations and doing calculations involving negative and positive numbers.