Problem 53 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ -26 \div(-13) $$

Step-by-Step Solution

Answer

The quotient of \(-26 \div (-13)\) is 2.

Step 1 - Understand the Division

In this problem, -26 is being divided by -13. The format is \(\frac{-26}{-13}\).

Step 2 - Divide the Numbers

To divide -26 by -13, divide the absolute values of the numbers: \(\frac{26}{13} = 2\).

Step 3 - Determine the Sign of the Quotient

Since both -26 and -13 are negative, dividing two negative numbers gives a positive quotient. Therefore, the quotient is 2.

Step 4 - Check the Division

Multiply the quotient by the divisor to check: \(2 \times -13\). This should equal the dividend, -26. So \(2 \times -13 = -26\), which confirms that the division is correct.

Key Concepts

Division of Negative Numbers

When dealing with negative numbers in division, the signs are very important. Remember, division is the inverse operation of multiplication. Here are a few basic rules to keep in mind:

- If you divide two negative numbers, the result is a positive number.
- If you divide a positive number by a negative number, or a negative number by a positive number, the result is a negative number.

For the given exercise, \(-26 \div (-13)\), both numbers are negative. By dividing them, you get a positive result.
This happens because the negatives cancel each other out. In mathematical notation, \(-26 \div (-13) = 2\).

Absolute Value

Absolute value is a concept that helps us ignore the direction of a number (whether it's positive or negative) and just focus on its magnitude.

When dividing integers, find the absolute value first.
- The absolute value of a negative number is its opposite (e.g., the absolute value of -26 is 26).
- For positive numbers, the absolute value is the number itself (e.g., the absolute value of 13 is 13).

In our exercise, instead of dividing -26 straight by -13, we divide their absolute values: \(\frac{26}{13} = 2\).
This makes the division easier and straightforward.

Check Division Result

It's crucial to verify the result of any division operation to ensure accuracy. You can do this by multiplying the quotient with the divisor and checking if it equals the dividend.

Here's how to check:
- For the quotient 2 from \(-26 \div (-13)\), multiply it back with the divisor -13.
- So, \2 \times -13 = -26\.

Since \(-26\) is our original dividend, our division was done correctly. This step is very effective in confirming your work and avoiding mistakes in your calculations.