Problem 41 - Beginning and Intermediate Algebra

Question

In your own words, explain why \(5-8\) simplifies to a negative number.

Step-by-Step Solution

Answer

\(5-8\) is negative because 8 is larger than 5, so we end up below zero.

Step 1: Understanding the Operation

The expression \(5-8\) is a subtraction operation. This means we want to find out what number you get if you start with 5 and then take away 8 from it.

Step 2: Identifying the Larger Number

In the expression \(5-8\), note that 8 is larger than 5. When the number you are subtracting is larger than the number you start with, the result will be negative.

Step 3: Performing the Subtraction

Start with the number 5 and attempt to take away 8. Since 8 is more than 5, you can't do this directly without going below zero.

Step 4: Calculating the Difference

Think of it as counting backwards starting from 5. You take away all 5, which takes you to 0, and you still need to take away 3 more (because 8 - 5 = 3) to complete the subtraction. This takes you to -3.

Step 5: Conclusion

The answer \(5-8 = -3\) because you had to move 3 steps below zero after subtracting all of the 5. Therefore, the result is a negative number.

Key Concepts

Subtraction

Subtraction is one of the four fundamental arithmetic operations. It involves taking away one number from another, which we call the minuend and the subtrahend, respectively. When performing subtraction, we start with the minuend and remove the value of the subtrahend from it. In our example, 5 is the minuend and 8 is the subtrahend.

• If the subtrahend is larger than the minuend, as in our case of 5 - 8, the resulting number will be negative. This happens because you "subtract" more than what is available.
• If the subtrahend is smaller, the process is straightforward, resulting in a positive outcome.
• If both numbers are equal, the difference will be zero, balancing out the equation.

Understanding subtraction with negative numbers can defy intuition initially, especially when the numbers involved "cross over" zero, resulting in a negative difference.

Number Line

A number line is a visual representation of numbers in a straight horizontal line. It helps us to comprehend the ordering and relative sizes of numbers, including negative numbers.
On a number line:

• Zero is usually positioned in the middle.
• Numbers to the right of zero are greater (positive) while those to the left are smaller (negative).

When you solve 5 - 8 using a number line, you start at 5. You then "step" to the left 8 spaces to find the result. Since there are only 5 steps to get to zero, you need to "cross" zero and move 3 more steps. This takes you to -3.
Using a number line to perform subtraction helps us visualize and understand especially when the operation leads to negative numbers.

Integers

Integers are a set of numbers that include all whole numbers and their negative counterparts, as well as zero. They form a fundamental part of number systems and arithmetic operations.
Key points about integers include:

• They do not include fractions or decimals, only "whole" values.
• They are written without any fractional or decimal component. For example, -3 and 5.

When dealing with subtraction and integers, like in 5 - 8, understanding that you can have negative results is crucial. Negative integers indicate a value less than zero, providing a complete picture when numbers are subtracted and go "below" zero.

Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is always expressed as a non-negative number.
Consider the following about absolute value:

• The absolute value of a positive or negative number is always positive. For instance, \(|-3|\) and \(|3|\) are both 3.
• Absolute value helps when comparing the size or magnitude of numbers without considering their signs.

In subtraction operations, absolute values can be used to understand the amount or distance between numbers. In 5 - 8, the absolute value of the difference is 3, representing the number of steps moved "beyond" zero on the number line to reach the result of -3.