Problem 69 - Fundamentals of Mathematics

Question

Perform each operation. $$ \frac{-28}{7} $$

Step-by-Step Solution

Answer

The result is -4.

Step 1: Identify the Operation

The problem requires dividing two numbers: -28 by 7. Division is the operation being performed.

Step 2: Divide the Absolute Values

Divide the absolute value of the numerator, 28, by the absolute value of the denominator, 7. This gives 28 ÷ 7 = 4.

Step 3: Determine the Sign of the Result

One of the numbers being divided, -28, is negative and the other, 7, is positive. The rule for division is that if the signs are different, the result is negative. Thus, the result is -4.

Step 4: Write the Final Answer

Combine the results from Steps 2 and 3 to write the final answer, which is -4.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are found to the left of zero on the number line.
The key property of a negative number is its sign, which is indicated by a minus (-) symbol in front of it. For example,

• -3
• -45
• -128

Each of these numbers is negative. Negative numbers are used in many real-world contexts, such as temperatures below freezing or financial debts. When we divide a negative number by a positive one, like in our exercise with -28 and 7, the result will always be negative.
This is due to the rules of division for negative numbers, which we will explore further.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction.
This is represented as the same number without any negative sign. For example,

• The absolute value of -5 is 5.
• The absolute value of 7 is 7.

To denote absolute value, we use vertical bars around the number, like this: -28 is written as \(|-28| = 28\).
When we calculate division involving negative numbers, it's often helpful to first consider the absolute values. This makes the arithmetic simpler, as seen in our example where -28 divided by 7 was calculated as the absolute values 28 divided by 7.

Division Rules

Division is one of the four fundamental operations in arithmetic, the others being addition, subtraction, and multiplication. When dividing numbers, one must remember the basic division rules concerning signs:

• When both numbers have the same sign (both positive or both negative), the result is positive.
• When they have different signs (one positive, one negative), the result is negative.

In our exercise, -28 (negative) divided by 7 (positive) results in -4, following the rule that if the signs differ, the result is negative.
These rules ensure consistency in mathematical calculations and help maintain the logic of math operations.

Numerator and Denominator

In a division operation represented as \( \frac{numerator}{denominator} \), the top number is the numerator and the bottom number is the denominator. The numerator indicates the number of parts being considered, while the denominator shows the number of parts into which the whole is divided.
In the exercise \(\frac{-28}{7}\), -28 is the numerator, meaning we are distributing this among the 7 equal parts indicated by the denominator.
Understanding the roles of the numerator and denominator is crucial in division.
They help us interpret and perform operations correctly, ensuring that we're dividing the correct quantity into the designated number of parts. This basic structure is fundamental to fractions and division.