Problem 15 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. \(-21-17\)

Step-by-Step Solution

Answer

The result of \(-21-17\) is \(-38\).

Step 1: Convert Subtraction to Addition

Subtracting a number is the same as adding its negative. The expression \(-21 - 17\) can be rewritten as \(-21 + (-17)\).

Step 2: Add the Numbers

Now, we just need to add these two numbers together. When you add two negative numbers, you get a negative result. The magnitude (absolute value) of the result is the sum of the magnitudes of the numbers you are adding. So, if we add 21 and 17, we get 38. Therefore, \(-21 + (-17) = -38\).

Key Concepts

Understanding Negative Numbers

Negative numbers represent values less than zero. They are usually symbolized with a minus sign, such as -1, -2, or -21. In mathematics, negative numbers are essential for expressing values below zero. For instance, they often convey debt in financial contexts.
Negative numbers also differ significantly in behavior from positive numbers. Unlike positive values that denote quantities moving upwards or forwards, negatives suggest descending or moving backwards. This directional aspect is vital in arithmetic operations and helps in visualizing problems.
When you involve negative numbers in operations like subtraction or addition, it's important to think of them as movements on a number line. If we start at zero and move to the left for negatives, every step signifies a unit decrease. Understanding this concept underlies integer arithmetic, making calculations more intuitive.

The Role of Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of direction.
It is always a non-negative number. For example, the absolute value of both 7 and -7 is 7. Mathematically, the absolute value of a number \( n \) is denoted as \( |n| \).
Understanding absolute value is crucial when working with negative numbers and subtraction. It essentially allows for the comparison and addition of numbers without considering their signs. This is particularly useful in integer subtraction because to subtract two numbers, you can instead add the opposite (or negative) of the number being subtracted.
Therefore, the absolute value helps us in calculations by focusing on magnitude first, then determining the sign based on the operation's rules.

Arithmetical Operations Involving Integers

Arithmetical operations with integers often confuse students, primarily due to the presence of negative numbers. However, if you break it down into steps, it becomes simple. The core operations are addition, subtraction, multiplication, and division. Here, we focus on subtraction.
Subtraction and addition are close relatives. A subtraction like \(-21 - 17\) can be rephrased as \(-21 + (-17)\). This transformation highlights the relationship between subtraction and addition and makes solving the problem easier.
When adding two negative numbers, the absolute values combine to give the magnitude of the answer, while the sign remains negative. That's why in our given exercise, adding 21 (the absolute value of -21) and 17 (the absolute value of -17) gives us 38, and hence the answer -38.
For effective problem-solving:

• Always convert subtraction into addition of a negative number
• Combine absolute values for the magnitude
• Apply the correct sign based on the operation

Implementing these steps will enhance understanding and efficiency when dealing with integer arithmetic.