Problem 106 - 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

-21

Step 1: Addition of Numbers

Begin by treating the operation as an addition between the two numbers by ignoring the negative signs for now. If we ignore the negative signs, we have 8 + 13 which equals 21.

Step 2: Consider the Sign

Because both of the numbers we are adding are negative, the result will also be negative. So, our result is not simply 21, but -21.

Key Concepts

Integer Addition

Integer addition is a simple arithmetic operation, where we combine two or more whole numbers to get a total sum. It's important to remember that integers can be both positive and negative. When adding integers, there are a few basic rules to keep in mind:

• If both integers have the same sign, simply add their absolute values and keep the sign.
• If the integers have different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

When you add positive integers, you are simply increasing the total. For example, adding 5 and 3 gives 8. Easy, right? However, when negative integers come into play, things might seem tricky at first. Negative numbers follow their own rules, but with practice, you will get used to them. Subtraction, as seen in the provided exercise, is closely related to addition in that subtraction can be seen as adding the opposite.

Negative Numbers

Negative numbers represent quantities less than zero, often used to express debt or temperature below zero. These numbers can be initially challenging, but understanding them is essential for mastering arithmetic. When working with negative numbers:

• The greater the negative number, the less its 'value'. For instance, -13 is less than -8.
• Adding a negative number is akin to subtracting the positive equivalent. For instance, adding -5 is the same as subtracting 5.

In our exercise, the subtraction of integers transforms into adding opposite values. Here's a deeper look: for the calculation \(-8 - 13\) or \(-8 + (-13)\), we're actually adding \(-8\) and \(-13\). This context helps us identify the integers based on their direction on the number line. By practicing the movement along the number line, you can visualize operations involving negative numbers better.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division, which form the basis of mathematics. These operations follow a conventional sequence, which is crucial while performing calculations.In the provided problem, subtraction as the addition of opposites is a clever way to understand these operations. Instead of subtracting directly, we switch thinking to addition to simplify calculations.For example, instead of using the direct subtraction \(-8 - 13\), we reframe it as \(-8 + (-13)\). This allows us to simply focus on adding two negative values:

• The effect of subtraction is mirrored by moving further left on the number line.
• This is useful when dealing with negative numbers, as the subtraction problem becomes clearer when viewed as addition of negatives.

Overall, mastering these operations requires practice and patience, but understanding the principles behind them can make learning feel intuitive and more manageable.