Problem 77 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{17}{-17} $$

Step-by-Step Solution

Answer

\( \frac{17}{-17} = -1 \).

Step 1: Understanding Division

The expression \( \frac{17}{-17} \) asks us to divide 17 by -17. Division involves finding how many times the denominator fits into the numerator, considering any sign changes.

Step 2: Identifying the Sign

When dividing numbers with different signs, the result will be negative. Here, 17 is positive and -17 is negative, so the result will be negative.

Step 3: Perform the Division

Divide the absolute values: \( 17 \div 17 = 1 \). Since we confirmed that the result will be negative, this means \( \frac{17}{-17} = -1 \).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are represented with a minus sign (\(-\)). These numbers are located to the left of zero on the number line. For example, -3, -7, and -17 are negative numbers. Negative numbers are opposites of positive numbers.

• A number and its opposite sum to zero. For example, 3 and -3 sum to zero.
• Negative numbers are used in real life to indicate a loss, temperature below zero, or debt.

When working with negative numbers in division, remember that:

• A positive number divided by a negative number results in a negative number.
• A negative number divided by a positive number also results in a negative number.
• Two negative numbers divided result in a positive number.

Understanding this helps when calculating the expression \( \frac{17}{-17} \), resulting in a negative number, specifically -1.

Absolute Values

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number. The absolute value of a number \( a \) is written as \( |a| \).
For example,

• \(|3| = 3\)
• \(|-8| = 8\)

Absolute values are particularly useful when performing operations like division or subtraction, where sign differences matter.
In the expression \( \frac{17}{-17} \), the absolute values of both numerator and denominator are 17. This helps simplify the division to \( 17 \div 17 \), which equals 1. However, considering the negative sign in the denominator, the actual result becomes -1. Absolute values simplify calculations by removing sign complexities, clarifying the actual division step.

Division of Integers

Division of integers involves determining how many times a divisor can fit into a dividend. It's crucial to consider the signs of the numbers being divided, as this affects the result's sign.
Key points to remember during the division of integers:

• If both integers have the same sign, the result is positive. For example, \( \frac{6}{3} = 2 \) or \( \frac{-6}{-3} = 2 \).
• If the integers have different signs, the result is negative. For example, \( \frac{-6}{3} = -2 \) or \( \frac{6}{-3} = -2 \).

In solving \( \frac{17}{-17} \), the steps followed involve dividing the absolute values, \( 17 \div 17 = 1 \), and then applying the negative sign due to the differing signs. Therefore, the result is -1. Understanding how division affects sign changes helps ascertain the correct result quickly and accurately.