Problem 65 - Fundamentals of Mathematics

Question

Perform each operation. $$ -16 \div-16 $$

Step-by-Step Solution

Answer

The result of the division is 1.

Step 1: Understand the Problem

The problem requires dividing -16 by -16. Division involves finding how many times the divisor fits into the dividend.

Step 2: Apply Division Rule for Negative Numbers

When dividing two negative numbers, the result is a positive number. Hence, dividing \(-16\) by \(-16\) will yield a positive result.

Step 3: Perform the Division

Now that we know the result is positive, we calculate the division: \(-16 \div -16 = 1\), since 16 divides into 16 exactly one time when considering the absolute values.

Key Concepts

Understanding Negative Numbers

Negative numbers are an extension of the number line that help to express values below zero. They are used in various mathematical operations such as addition, subtraction, multiplication, and division. Negative numbers have specific characteristics:

• They are represented with a minus (-) sign, for example, -3 or -16.
• In terms of quantity, they are the opposite of their positive counterparts.
• The more negative a number, the less its value in comparison to others (e.g., -16 is smaller than -1).
• Commonly used in real-world contexts like temperatures below zero or bank overdrafts.

Understanding how to work with negative numbers is crucial, especially when performing operations that involve both positive and negative values.

Division Rules for Negative Numbers

When dealing with division involving negative numbers, certain rules simplify the process and help determine the sign of the result:

• The quotient of two negative numbers is positive. For example, dividing \(-16 \div -16\) gives a positive 1.
• If one number is negative and the other is positive, the quotient will be negative, such as \(-16 \div 8 = -2\).
• Remember, these rules apply to multiplication too, as they are closely related operations.
• Always confirm the absolute values first, then apply sign rules as needed.

By applying these division rules, you ensure that you compute the sign of your answer correctly, avoiding any common pitfalls when working with negative numbers.

Absolute Value in Division

The absolute value of a number is essentially its distance from zero on the number line, regardless of direction. It simplifies calculations involving negative numbers:

• The absolute value is always non-negative. For example, the absolute value of \(-16\) is 16, written as \(|-16| = 16\).
• In division, consider the absolute values to calculate the result before worrying about the sign.
• For \(-16 \div -16\), compute using absolute values: \(16 \div 16 = 1\).
• Apply sign rules afterward to determine that the final answer is positive in this case.

Using absolute values helps break down problems, making complex operations more manageable by separating magnitude and sign.