Problem 47 - Fundamentals of Mathematics

Question

Find the value of \(-6-(-8)\).

Step-by-Step Solution

Answer

The value is 2.

Step 1: Identify the Expression

We are given the expression \(-6-(-8)\) which involves subtraction of a negative number.

Step 2: Understand Subtraction of Negative Numbers

Subtracting a negative number is equivalent to adding its positive. Therefore, \(-(-8)\) can be rewritten as \(+8\).

Step 3: Rewrite the Expression

Based on the understanding from the previous step, rewrite the expression \(-6-(-8)\) into \(-6 + 8\).

Step 4: Perform the Addition

To solve \(-6 + 8\), we start from \(-6\) on the number line and move 8 units in the positive direction. This takes us to \(2\).

Step 5: Solution Interpretation

The solution means that the value of \(-6-(-8)\) is \(2\).

Key Concepts

Subtraction of Negative Numbers

Subtracting negative numbers might initially seem confusing because we're so used to the idea of taking things away. However, when you subtract a negative number, you are essentially adding its positive equivalent.
For example, consider the expression \(-6 - (-8)\). Here, \(-(-8)\) effectively becomes \(+8\). This is because negating a negative number results in a positive number. So, it's like saying "I'm owed 8 more chocolates," instead of "I lost 8 chocolates." In arithmetic terms, subtraction of a negative number is the same as addition.
This concept is crucial when solving math problems that include multiple negative and positive integers. Knowing this will help ensure correct calculations and an easier understanding of complex expressions.

Number Line

A number line is a visual representation that allows us to better understand the relationships between numbers, whether positive or negative.
When using a number line, you can easily see how numbers are spaced and observe the effect of addition or subtraction.

• Positive numbers are to the right of zero.
• Negative numbers are to the left of zero.
• Zero is the central point between positive and negative numbers.

Using the number line can make the concept of subtracting negative numbers clearer. For instance, starting from \(-6\), moving 8 spaces to the right (which makes the number more positive) lands you at \(+2\).
The number line assists by providing a straightforward method to "see" the mathematical operations taking place, making it easier to visualize complex problems.

Addition of Integers

Adding integers involves understanding how positive and negative numbers interact.
When adding a positive integer, you move right on the number line, increasing the total value. Conversely, adding a negative integer is akin to moving left on the number line, decreasing the total value.
In our example, after rewriting the expression \(-6 - (-8)\) as \(-6 + 8\), you add 8 to \(-6\).

• Starting point is \(-6\).
• Add 8 by moving right on the number line.
• This results in the number positive 2.

The results show that when you're adding a positive integer to a negative one, you're finding the balance between the two, leading to a result that depends on which magnitude is greater.