Problem 107 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the next section. In each exercise, a subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. $$-8-(-13)=-8+13=?$$

Step-by-Step Solution

Answer

The sum indicated by the question mark is 5.

Step 1: Understanding addition of negative and positive numbers

In this step, an understanding of how negative and positive numbers add up is significant. Generally: \[ negative + positive = positive - |negative| \] where || denotes the absolute value. In this case, \(-8 + 13\) is the same as \(13 - 8\).

Step 2: Performing the operation

Now, subtract 8 from 13, which gives us 5.

Step 3: Writing the Final Answer

So, \(-8 + 13 = 5\). So the sum indicated by the question mark is 5.

Key Concepts

Negative Numbers

Negative numbers are values less than zero. They are typically represented with a minus sign (-) preceding the number, like -8 or -13.
This minus sign indicates the direction or sign on the number line, showing that it is left of zero.

• They portray quantities such as debts, temperatures below freezing, or any decrease in a value.
• Adding a negative number is often interpreted as subtracting its absolute value.
• For example, when you see \(-8 + (-13)\), it can be thought of as \-8 - 13\. Here, you move 13 units to the left of -8 on the number line.
• To add a negative and a positive number, simply subtract the absolute value of the negative number from the positive and give the result the sign of the greater absolute value.

Understanding how negative numbers interact in mathematical operations sets a foundation for dealing with real-world scenarios and more advanced math concepts.

Positive Numbers

Positive numbers are values greater than zero. They are usually represented without any sign, like 2, 13, or 100, although they can also be indicated with a plus sign (+), such as +13.
On the number line, they are to the right of zero.

• Positive numbers represent quantities such as wealth, temperatures above freezing, or any increase in a value.
• Adding or subtracting positive numbers follows the normal arithmetic pattern. For instance, simply add the absolute values of the numbers.
• In the expression \(-8 + 13\), since 13 is positive, the result moves 13 units to the right starting from -8, ending up at 5.

Recognizing and utilizing positive numbers is essential for all basic calculations in mathematics, as they represent the natural counting numbers everyone is familiar with.

Absolute Value

Absolute value, denoted by vertical bars \(|x|\), represents the distance of a number from zero on the number line, regardless of direction. For example:

• The absolute value of -8 is \(|-8| = 8\).
• The absolute value of 13 is \(|13| = 13\).

It is always a non-negative number.
This concept simplifies the addition of negative numbers by converting them into a form where only distances are considered, not direction. When you see an expression like \(-8 + 13\), think of it as knowing how far -8 is from zero and how far 13 is from zero, then determining the net distance based on the signs.

• For example, \(-8 + 13\) means traveling more distance in the positive direction than in the negative, which results in a net distance of 5, hence the computation: \-8 + 13 = 5\.

Understanding absolute value is crucial as it appears in various advanced mathematical contexts, including functions and complex numbers.