Problem 14 - Algebra for College Students

Question

Perform the following operations with real numbers. $$(-81) \div(-3)$$

Step-by-Step Solution

Answer

The answer is 27.

Step 1: Identify the Operation

The exercise is to perform division with the two numbers presented: \(-81\) and \(-3\). Our goal is to find out how many times \(-3\) can be subtracted from \(-81\).

Step 2: Simplify the Signs

When dividing two numbers with the same sign (either both positive or both negative), the result will always be positive. Since both \(-81\) and \(-3\) are negative, our result will be positive.

Step 3: Divide the Absolute Values

Ignore the negative signs and divide the absolute values: \(81 \div 3\). This results in \(27\), since 81 can be divided by 3 exactly 27 times.

Step 4: Assign the Correct Sign

Since the original numbers were both negative, the quotient should be positive (as concluded in Step 2). Therefore, the answer to \(-81 \div -3\) is \(27\).

Key Concepts

Division of Integers

Division of integers is a fundamental arithmetic operation where one integer is divided by another. Integers include all whole numbers, both positive and negative, as well as zero. The operation involves finding how many times one integer can be subtracted from another. For example, when dividing \(-81\) by \(-3\), you are determining how many times \(-3\) fits into \(-81\).

• Negative divided by negative results in a positive.
• Negative divided by positive results in a negative.
• Positive divided by negative results in a negative.
• Positive divided by positive results in a positive.

It's important to remember the rules of signs. In the example provided, both integers were negative, which means the result is positive, as negatives cancel each other out. Once you ignore the signs (or use absolute values), the division becomes straightforward, like dividing two positive numbers.

Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. This means that the absolute value is always positive or zero. It is denoted by vertical bars around the number, like this: \(|-81| = 81\).The concept of absolute value is crucial when dealing with integers, especially negative numbers, as it helps simplify calculations by eliminating the effect of the negative sign.

• Absolute value makes negative numbers easier to work with.
• For example, \(|-81| = 81\), and \(|-3| = 3\).
• In division or multiplication, you often calculate using absolute values first.

Understanding and using absolute values helps transform a problem into a more straightforward calculation, as shown in the example where you would take \(|-81|\) and \(|-3|\) and divide them as natural numbers.

Positive and Negative Numbers

Positive and negative numbers are a basic part of mathematics, indicating quantities with direction. Positive numbers are greater than zero and are found to the right of zero on the number line. Negative numbers, on the other hand, are less than zero and are situated to the left of zero.When performing operations like division, knowing how positive and negative numbers interact is crucial. Dividing negative numbers with the same sign (both positive or both negative) results in a positive number. This is because the negative signs "cancel" each other out.

• Negative numbers represent opposite directions or values compared to positive ones.
• They are essential in various real-world applications, like temperature or financial losses.
• The sign of the number greatly influences the outcome of mathematical operations.

When dividing \(-81\) by \(-3\), both are negative, leading to a positive result of 27, allowing you to view them just as you would positive integers for ease of calculation.