Problem 68 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ 0 \div(-47) $$

Step-by-Step Solution

Answer

The quotient is 0.

Step 1: Understand Division by Negative Numbers

Recall that division by any non-zero number, including negative numbers, is always defined. The operation 0 divided by any non-zero number is always 0.

Step 2: Perform the Division

Divide 0 by -47 following the division rule: $$ 0 \div (-47) = 0.$$

Step 3: Verify the Result

Check the result by multiplying the quotient by the divisor: $$ 0 \times (-47) = 0.$$ Since the original dividend is 0, this confirms the correctness of the division.

Key Concepts

division by negative numbers

Division by negative numbers has some unique properties. First, it is important to remember that division by any non-zero number, positive or negative, is always defined. For instance, when you divide zero by a negative number, the quotient is always zero. This is because anything divided by zero results in zero. In mathematical terms: \( \frac{0}{-47} = 0 \). This rule holds true no matter what the negative number is.

quotient verification

Verifying the quotient is a key step to ensure your division is correct. After performing a division, you can check your answer by multiplying the quotient and the divisor. In this problem, we divided zero by -47 and got a quotient of zero. We verify this by multiplying: \( 0 \times (-47) = 0 \). If the product matches the original dividend, your solution is correct. This step confirms that our division was performed correctly. Always use multiplication to verify your division results.

division rules

Understanding division rules is crucial for solving problems efficiently. Here are some key points to remember about division:

• Dividing zero by any non-zero number always results in zero.
• Dividing a number by a negative number flips the sign of the quotient. For example, \( \frac{10}{-2} = -5 \).
• A number divided by 1 remains the same, and any number divided by itself is 1.
• Division by zero is undefined and cannot be performed.

These basic rules will help you navigate through most division problems with confidence.