Problem 8 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$(-11)(-1)$$

Step-by-Step Solution

Answer

The answer is 11.

Step 1: Identify the numbers

The numbers presented are -11 and -1, which are both negative.

Step 2: Apply the rule of multiplication for negative numbers

The rule is: A negative times a negative gives a positive. So (-11) times (-1) gives a positive number.

Step 3: Multiply the numbers

Multiply the absolute values of the numbers, which are 11 and 1. The result is 11.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are typically represented with a minus sign before the number. For instance, in our example, \(-11\) and \(-1\) are negative numbers. This means they are both below zero on the number line.
Negative numbers are found in many real-world situations, like temperatures below freezing or debts in finance.
They are crucial in mathematics as they allow for more complex computations.

• Negative numbers provide insight into directional changes. Going below zero on a number line or graphs may indicate downward or backward movement.
• Operations with negative numbers often involve rules different from positive numbers. Understanding these rules is essential for solving equations and inequalities.

Multiplication Rule

In mathematics, the rule of multiplication for negative numbers is straightforward. When you multiply two negative numbers, the result is positive.
Here's how it works:

• A negative number times a negative number always results in a positive number. For example, \((-11) \times (-1) = 11\).
• If you multiply a positive number by a negative number, the result is negative. For example, \(5 \times (-3) = -15\).

The reason behind this rule is rooted in math's consistency with addition and subtraction patterns. If you think of multiplication as repeated addition, negative times negative can mean 'subtract the negative,' which results in a positive. This rule might seem abstract at first, but over time, with practice, it becomes intuitive.

Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. Think of it as stripping away any negative sign to find a number's magnitude.
In the problem \((-11) \times (-1)\), we took the absolute values of \(-11\) and \(-1\). These are simply \(11\) and \(1\) respectively.
This is what we actually multiply during the calculation. Absolute value is always non-negative. It essentially tells us the size of a number without regard to its sign.

• The absolute value of a negative number is positive. For example, \|-4| = 4\.
• The absolute value of a positive number is the number itself. For example, \|7| = 7\.

Understanding absolute value is crucial, especially when dealing with real-life scenarios where magnitude matters and direction doesn't play a role.