Problem 13 - Fundamentals of Mathematics

Question

Find each value. \(-8+6\)

Step-by-Step Solution

Answer

The result is \(-2\).

Step 1: Identify the Operation

The expression provided is \(-8 + 6\). This is an addition problem where we need to add \(-8\) and \(6\).

Step 2: Apply Addition Rules for Signed Numbers

When adding a negative number and a positive number together, you subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. Here, the numbers \(-8\) and \(6\) have absolute values of \(8\) and \(6\), respectively. Since \(8\) is larger, you perform \(8 - 6 = 2\) and keep the sign of \(-8\), which is negative.

Step 3: Compute the Result

Compute the result by subtracting the absolute values and applying the sign from the larger absolute value: \(-8 + 6 = -2\).

Key Concepts

Signed Numbers

Signed numbers are numbers that have a sign to indicate whether they are positive or negative. A positive number often has a '+' sign, whereas a negative number is marked with a '-' sign. For example, in the expression

• +3 means the number is positive 3.
• -5 indicates the number is negative 5.

When working with signed numbers, remember that the sign changes how you add or subtract them. Positive numbers are straightforward because they are the ones you normally count with. Negative numbers, however, represent values less than zero and can make understanding operations like addition slightly more complex.
Understanding and recognizing these signs is crucial because it directly affects how you calculate the result when combining these numbers.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of its direction. This means absolute value is always non-negative. When we talk about absolute value, we use vertical bars around the number, like this:

• \(|-5| = 5\)
• \(|3| = 3\)

The absolute value helps simplify operations by letting us focus purely on the size of the number without considering its sign.
For example, when you see the expression

• -8 + 6

You first convert these to their absolute values which are 8 and 6. Then, use these values to determine the result of any operations after you've applied any relevant rules. Understanding absolute value is crucial in making mathematical operations involving signed numbers easier to handle.

Addition Rules

Addition rules for signed numbers tell us how to handle sums involving both positive and negative numbers. These rules are simple, yet they form the backbone of correctly solving problems like

• -8 + 6.

Here are the steps you generally need to follow:

• If both numbers are positive, just add them together as usual.
• If both numbers are negative, add them together and keep the negative sign.
• If one number is positive and the other is negative, subtract the smaller absolute value from the larger absolute value.

Once you find the difference in absolute values, you keep the sign of the number with the larger absolute value. In our example, since the absolute value of -8 is larger than that of 6, the sign is negative.
This makes

• -8 + 6 = -2

Mastering these rules enables you to tackle any integer addition problem, no matter which signed numbers come your way.