Problem 30 - Algebra for College Students

Question

Perform the following operations with real numbers. $$-23-38$$

Step-by-Step Solution

Answer

-61

Step 1: Understanding the Problem

The exercise requires you to perform a subtraction of two real numbers: \(-23 - 38\). Here, you're essentially combining the effects of negative numbers on the number line. Let's break it down step-by-step.

Step 2: Identify the Numbers

Observe the numbers involved: \(-23\) and \(-38\). We know that subtracting \(38\) from \(-23\) can be interpreted as moving \(-38\) units further negatively from \(-23\).

Step 3: Perform the Operation

To perform the calculation of \(-23 - 38\), simply add the negative integers:\( (-23) + (-38) = -61\).This is because subtracting \(38\) is the same as adding \(-38\).

Step 4: Write the Final Answer

Now that we have performed the operation, the final result is \(-61\) which is the sum of the original subtraction problem.

Key Concepts

Number Line

Understanding the number line is key when dealing with real numbers, especially negative ones. Imagine it as a horizontal line with zero in the center, positive numbers to the right, and negative numbers to the left. Each point on this line corresponds to a real number.
When you move right on the number line, you're heading towards positive infinity. Moving left takes you towards negative infinity. This is important when performing operations with integers.

• The distance between numbers on the number line is crucial for determining their sum or difference.
• Subtracting a positive number means moving left, while subtracting a negative number involves moving right.

Using the number line, you can easily visualize why (-23 - 38) simply moves you further into the negative toward -61.

Negative Numbers

Negative numbers are numbers less than zero, found on the left side of the number line. They represent values below zero, such as a debt or a temperature below freezing. Understanding how to operate with these numbers is crucial in arithmetic.

• Negative numbers are denoted with a minus sign, for example, -1, -23, and -38.
• They are considered opposites to positive numbers, so 3 and -3 have identical magnitudes but opposite signs.

When dealing with negative numbers:

• Subtracting a positive number results in a movement further to the left on the number line.
• Subtracting a negative often turns into an addition, resulting in a movement to the right.

This is why for the operation (-23 - 38), you're actually adding -23 and -38, because moving left on the number line is the same as adding negative values.

Addition of Integers

Adding integers can involve both positive and negative numbers, and it's essential to comprehend how each affects the sum. Here are some critical points:

• If both integers are positive, the sum is positive.
• If one integer is positive and the other negative, the sum depends on which number has the larger absolute value.
• If both integers are negative, the result is negative, as you are essentially combining quantities that decrease value, much like debt.

When adding:

• The same signs lead to maintaining the sign, like in -23 + (-38), resulting in -61.
• Different signs require subtracting the smaller absolute value from the larger, and then using the sign of the number with the larger absolute value.

This means that the operation -23 - 38 is the same as adding negative integers (-23) + (-38). This consistent process allows you to navigate the rules of integer addition smoothly, resulting in the simplified answer, -61.