Problem 14 - Prealgebra

Question

Find each of the following quotients. (Divide.) [Examples 1–5] $$\frac{0}{-5}$$

Step-by-Step Solution

Answer

The quotient is 0.

Step 1: Interpreting the Problem

The problem asks us to divide 0 by -5. We can think of this as asking how many times -5 can fit into 0, or what is the value of zero divided by a negative number.

Step 2: Applying Division Rules

According to basic division rules, any number divided by zero results in zero. This is because zero pieces cannot fit even once into any number, positive or negative.

Step 3: Final Calculation

Based on the division rules, we see that \( \frac{0}{-5} = 0 \), meaning zero divided by any nonzero number is zero.

Key Concepts

Basic Division Rules

Division is one of the fundamental operations in mathematics. It involves determining how many times one number, called the divisor, fits into another number, known as the dividend. Here are a few key points about division:

• The basic formula for division is: Dividend ÷ Divisor = Quotient.
• Division is essentially the opposite of multiplication. If you multiply the quotient by the divisor, you'll get back the original dividend.
• Division by zero is undefined because no number of "zeros" can fit into any number.

It's crucial to understand these simple rules as they form the foundation for more advanced mathematical concepts. When dealing with zero as either the dividend or the divisor, special rules apply which we will explore further.

Zero Property of Division

The zero property of division specifically addresses scenarios where zero is involved in a division problem, either as the dividend or divisor:

• If 0 is the dividend (the number being divided), the result is always 0, provided the divisor is not zero. For example, \( \frac{0}{5} = 0 \).
• Zero cannot be the divisor. If you try to divide any number by zero, the division is undefined. This is because you cannot determine how many times zero goes into another number since zero has no value or size.

Understanding the zero property helps avoid common mistakes, especially in algebra and calculus. Knowing that \( \frac{0}{a} = 0 \) where \( a eq 0 \) simplifies many mathematical problems.

Dividing by Negative Numbers

When dividing by negative numbers, it is important to understand how the sign affects the quotient. Here are the rules:

• Dividing a positive number by a negative number results in a negative quotient. For instance, \( \frac{5}{-1} = -5 \).
• Similarly, dividing a negative number by a positive number also yields a negative quotient. Example: \( \frac{-10}{2} = -5 \).
• When both dividend and divisor are negative, the quotient is positive. For example, \( \frac{-6}{-2} = 3 \).

This understanding is key in algebra where equations may include unknowns with negative values. The rules governing negative division help maintain consistency and accuracy in calculations.