Problem 19 - Prealgebra

Question

Find each of the following products. (Multiply.) $$-1(-2)(-3)$$

Step-by-Step Solution

Answer

The product of the numbers is -6.

Step 1: Multiply the First Two Numbers

First, let's multiply the first two numbers: \(-1 \times (-2)\).Multiplying two negative numbers gives a positive result, so:\[-1 \times (-2) = 2\].

Step 2: Multiply the Result with the Third Number

Now, use the result from Step 1 and multiply it by the third number. We need to multiply 2 by (-3):\[2 \times (-3)\].Multiplying a positive and a negative number results in a negative number, so:\[2 \times (-3) = -6\].

Key Concepts

Understanding Negative Numbers

Negative numbers are an essential part of mathematics, and understanding them can help you perform a variety of calculations. Negative numbers are less than zero and are indicated by a minus sign (-). They are often used to represent values below a defined point, such as temperatures below freezing or financial losses. For instance:

• -5 degrees Celsius means 5 degrees below the freezing point.
• -10 dollars could represent a debt of 10 dollars.

Negative numbers have some unique properties, particularly when it comes to multiplication. When two negative numbers are multiplied together, the result is positive. This might seem counterintuitive, but it ensures consistency in mathematical rules. Visualize this by considering the equation \[(-1) imes (-1) = 1\]It shows how the multiplication of two negatives leads to a positive outcome.

Comprehending Positive Numbers

Positive numbers are familiar to most people, as they represent quantifiable amounts that are greater than zero. Positive numbers do not have a sign or are marked with a plus sign (+). Everyday usage includes:

• The number 10 can be thought of as: just 10, or +10.
• 75 degrees Fahrenheit indicates 75 degrees above a baseline, such as freezing point.

Positive numbers differ from negative numbers in terms of calculations and effects. When multiplying a positive number by another positive number, the result is always positive. For instance, multiplying 3 by 2 equals 6:\[3 imes 2 = 6\]They represent tangible quantities and usually characterize gains or heights. Understanding how positive numbers function in operations alongside other numbers, such as negative numbers, is fundamental to mastering arithmetic.

Mastering Integer Multiplication Rules

Integer multiplication rules are the stepping stones to understanding how different types of numbers interact. The key rules for multiplying integers clarify the results when you mix positive and negative numbers in operations.Here’s a quick rundown of these rules:

• Product of two positive integers is positive. E.g., \(2 \times 3 = 6\)
• Product of two negative integers is positive. E.g., \((-2) \times (-3) = 6\)
• Product of one positive and one negative integer is negative. E.g., \(2 \times (-3) = -6\)

To visualize this with our example, by multiplying the negative numbers in the exercise, the first two steps convert a seemingly negative scenario into something positive, and the subsequent step ensures that one negative factor results in a negative product. This methodology keeps calculations consistent across various scenarios.