Problem 52 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$\frac{-66}{-6}$$

Step-by-Step Solution

Answer

The result of the division \(-66/-6\) is 11.

Step 1: Identify the Divisor and Dividend

In the exercise, we are required to perform division. The dividend is -66, and the divisor is -6.

Step 2: Check if the Divisor is Zero

Before proceeding with division, it's essential to ensure that we are not dividing by zero as that would make the expression undefined. However, in the exercise, the divisor is -6, and not zero, so we can proceed.

Step 3: Perform the Division Operation

When we divide -66 by -6, we get 11. In mathematics, a negative number divided by another negative number yields a positive number.

Key Concepts

Dividend and Divisor

Understanding the roles of the dividend and divisor in a division operation is fundamental in mathematics. In the division equation, the dividend is the number that is being divided, while the divisor is the number by which you are dividing the dividend. For instance, in the division expression \( \frac{-66}{-6} \), -66 is the dividend, and -6 is the divisor. It's important to note that the divisor must never be zero because division by zero doesn't have a defined value in mathematics and would make the entire expression undefined.

When you're performing division, you can think of it as dividing a 'whole' into equal 'parts'. The divisor represents how many parts you are creating, and the dividend represents the size of the whole. If you were to share -66 apples among -6 baskets, it's like asking, 'How many apples would each basket get if they each got an equal amount?' Of course, in reality, we would not have negative baskets, but in the mathematical sense, the concept still holds true.

Undefined Mathematical Expressions

In mathematics, undefined expressions are those that don't have a meaning or value under the current rules of arithmetic. The classic example is division by zero. If you have an expression where the divisor is zero, like in \( \frac{5}{0} \) or \( \frac{-66}{0} \), there is no defined value for these calculations. No matter what number you choose as the dividend, division by zero is undefined because you cannot have zero parts of a whole.

It's akin to asking 'How can you divide something into nothing?' It doesn't make logical sense, and hence, math upholds this by providing no answer. In the case of your textbook exercise, it's clarified in step 2 that the divisor is -6, not zero. Therefore, the expression \( \frac{-66}{-6} \) is not undefined and we can proceed with the calculation. Always check whether a divisor is zero to avoid undefined expressions in your mathematical operations.

Negative Number Division

Dividing negative numbers might initially seem confusing, but the concept follows a simple rule: dividing a negative by a positive or a positive by a negative results in a negative number, and dividing a negative by a negative results in a positive number. This is due to the properties of multiplication that division is based upon, since division is the inverse of multiplication.

For the division expression \( \frac{-66}{-6} \), both numbers involved are negative. Applying the rule mentioned above, a negative divided by a negative gives us a positive result. This means \( \frac{-66}{-6} = 11 \). It’s crucial to remember this rule whenever dealing with division involving negative numbers. Forming a mental model of this can be like thinking of a double negative in language; two 'nots' cancel each other out and create an affirmative.