Problem 36 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ -4-(-1) $$

Step-by-Step Solution

Answer

The result is -3.

Step 1: Identify the Terms

Identify the terms involved in the expression: The expression given is \(-4 - (-1)\). This consists of subtracting a negative number (-1) from another negative number (-4).

Step 2: Replace Subtraction of Negative with Addition

Subtracting a negative number is the same as adding its positive equivalent. So, change \(-4 - (-1)\) to \(-4 + 1\).

Step 3: Perform the Addition

Perform the addition for the expression: Start with \(-4 + 1\). When you add 1 to -4, you move one unit towards the positive direction on the number line, giving you \(-3\).

Step 4: Verify with a Calculator

Enter the expression \(-4 - (-1)\) into a calculator. It should reach the same result of \(-3\), confirming our manual calculation is correct.

Key Concepts

Subtracting Negative Numbers

Subtracting negative numbers can often be confusing, but there's an easy trick to simplify this process. Whenever you encounter subtraction of a negative number, you can replace it with addition. Here's how it works:

• Recognize that subtracting a negative number is equivalent to adding its positive counterpart.
• For instance, if you have \( -4 - (-1) \), you can change it to \( -4 + 1 \).
• This transformation is based on the mathematical principle that two negative signs cancel each other out.

By keeping this in mind, subtracting negative numbers becomes much less daunting. Just remember, if you see a \(-(-x)\), you're adding \(x\) instead.

Addition and Subtraction

Understanding the relationship between addition and subtraction is crucial in mastering integer operations. Both processes involve moving numbers along the number line, acting in opposite directions:

• Addition involves moving to the right on the number line, making the number larger.
• Subtraction moves you to the left, decreasing the number's value.

When you connect addition and subtraction, you use the concept of inverse operations. This means what one operation does, the other can undo.

This is evident in scenarios like \(-4 + 1\). Here, you're adding 1 to \(-4\), which shifts your position one step to the right on the number line. Thus, knowing how to switch between these operations lets you handle integer problems more effectively.

Number Line

A number line is an excellent tool for visualizing integer operations. Imagine a horizontal line that stretches infinitely in both directions, with zero in the center:

• Numbers to the right of zero are positive.
• Numbers on the left are negative.

When given a problem like \(-4 + 1\), you start at \(-4\) on the number line. By adding 1, you move one unit to the right, landing at \(-3\).

The number line helps with understanding not only addition but subtraction as well. It provides a more visual way of seeing how numbers relate in space. Using this tool enhances comprehension by illustrating how operations like subtracting a negative number are effectively "moving" back towards the positive side. This way, you gain a better grasp of how mathematical operations interact, beyond just numerical calculations.