Problem 30 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ \frac{-16}{-8} $$

Step-by-Step Solution

Answer

The value is 2.

Step 1: Identify the Operation

The given expression is a division problem: \( \frac{-16}{-8} \). We need to divide \(-16\) by \(-8\).

Step 2: Divide the Absolute Values

Ignore the signs for a moment and simply divide the absolute values: \( \frac{16}{8} = 2\).

Step 3: Determine the Sign of the Result

Since both the numerator and the denominator are negative, the division of two negative numbers results in a positive number. Therefore, the result is \(+2\).

Step 4: Verify with a Calculator

Use a calculator to confirm the division: Enter \(-16 \div -8\) into the calculator and observe that the result is indeed \(2\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are typically represented with a minus sign (-) before the number. In the number line, they are found to the left of zero.

When dealing with operations involving negative numbers, specially division, it is crucial to pay attention to their signs. Understanding how negative numbers work will help you make sense of many arithmetic operations that include them.

• A negative number divided by a negative number, as in our example, results in a positive number.
• It's important to remember that a negative number divided by a positive number gives a negative result.
• Conversely, a positive number divided by a negative number also gives a negative result.
• Finally, any number divided by itself is 1, and this is true for both positive and negative numbers (excluding zero).

By keeping these rules in mind, you'll have an easier time managing expressions that involve negative integers.

Absolute Value

Absolute value refers to the distance a number is from zero on the number line, without considering their direction. It is always a non-negative number.

In mathematical notation, it is expressed as two vertical bars surrounding the number, like this: \(|x|\). The absolute value of both positive and negative numbers is always positive. For example:

• The absolute value of -16 is written as \(|-16|\) and is 16.
• Similarly, the absolute value of -8 is \(|-8|\) and is 8.

When you are asked to find the result of dividing negative numbers, calculating absolute values first can simplify the process by removing signs temporarily from the equation. This makes it easier to handle before determining the final sign of the result.

Sign of Result

The sign of a result in division problems determines whether the answer will be positive or negative. This depends on the signs of the numbers involved.

In our example, we divided \(-16\) by \(-8\). Here is a simple guideline to help you determine the sign:

• When both numbers have the same sign, the result is positive. Negative divided by negative or positive divided by positive gives a positive outcome.
• When the numbers have different signs, the result is negative. A negative divided by a positive or a positive divided by a negative gives a negative outcome.

This approach is logical and helps keep your calculations consistent and correct. In our case, because both \(-16\) and \(-8\) are negative, the final result is \(+2\), which demonstrates the positive result from two negative numbers.