Problem 123 - Introductory Algebra for College Students

Question

Explain how to multiply two real numbers. Provide examples with your explanation.

Step-by-Step Solution

Answer

To multiply two real numbers, you have to follow a basic principle of repeated addition. If both numbers are positive, the result is positive. If one number is negative, the result is negative. And if both numbers are negative, their negative signs cancel each other, and the result is positive. For example \(4 x 3 = 12\), \(4 x -3 = -12\), and \(-5 x -2 = 10\).

Step 1: Understanding Multiplication

Multiplication of two real numbers can be thought of as repeated addition.

Step 2: Multiplication of Positive Real Numbers

If given two positive numbers, for example 4 and 3. You could write 4 as \(4 = 1 + 1 + 1 + 1\) and then add 4 three times, \(4+4+4 = 12\). This is the same as multiplying 4 by 3, \(4 x 3 = 12\). The result is a positive number.

Step 3: Multiplication of a Positive and a Negative Real Number

If we are given a positive and a negative number, such as 4 and -3, the process remains the same as before, but now the final result is a negative number. So \(4 x -3 = -12\).

Step 4: Multiplication of Negative Real Numbers

When you multiply two negatives, the result is always a positive. For instance, if we have -5 and -2, it's the same as \(-5 x -2 = 10\). The two negative signs cancel each other out.