Problem 7 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. \(\frac{-9}{3}=-3\) because _____ \(\cdot\) _____ = _____ b. \(\frac{0}{8}=0\) because _____ \(\cdot\) _____ = _____

Step-by-Step Solution

Answer

a. -3, 3, -9. b. 0, 8, 0.

Step 1: Analyze the expression in part a

We have the fraction \( \frac{-9}{3} = -3 \). This means when we divide \(-9\) by \(3\), the result is \(-3\).

Step 2: Determine the multiplication equality for part a

Since \( \frac{-9}{3} = -3 \), this multiplication equality needs to be \( -3 \cdot 3 = -9 \). This is because dividing by 3 should undo multiplication by 3, resulting in the original numerator \(-9\).

Step 3: Fill in the blanks for part a

Based on the multiplication equality, the blanks can be filled as: \(-3\), \(3\), and \(-9\). It reads: \(-3 \cdot 3 = -9\).

Step 4: Analyze the expression in part b

We have the fraction \( \frac{0}{8} = 0 \). This means when we divide 0 by \(8\), the result is \(0\).

Step 5: Determine the multiplication equality for part b

For division \( \frac{0}{8} = 0 \), the multiplication equality is \( 0 \cdot 8 = 0 \). This is because any number multiplied by zero results in zero.

Step 6: Fill in the blanks for part b

Based on the multiplication equality, the blanks are: \(0\), \(8\), and \(0\). It reads: \(0 \cdot 8 = 0\).

Key Concepts

What is Multiplication Equality?

To understand multiplication equality, we need to grasp the relationship between multiplication and division. When we say that \( \frac{-9}{3} = -3 \), we mean that dividing \(-9\) by \(3\) gives us \(-3\). But how does this connect to multiplication equality? Here, multiplication equality refers to rewriting a division statement as a multiplication one. This means that \(-3 \cdot 3 = -9\).

Therefore, after you perform division, if you take the result and multiply it by what you divided by, you should get the original number back. In essence:

• If \( \frac{a}{b} = c \), then \( c \cdot b = a \).
• This checks consistency between multiplication and division operations.
• The two are inverse operations, so they can be used to verify each other.

Understanding this concept is critical in mathematical operations as it helps verify answers and strengthen foundational arithmetic skills.

Numerator and Denominator: Understanding their Roles

In any given fraction, two parts play crucial roles—these are the numerator and the denominator. The numerator is the top number in a fraction and represents the number of parts we have. Meanwhile, the denominator is the bottom number, indicating the total equal parts something is divided into.

For example, in the fraction \(\frac{-9}{3}\), \(-9\) is the numerator, and \(3\) is the denominator. Here's what they mean:

• Numerator: Indicates the total quantity or parts of a whole (in this case, \(-9\)).
• Denominator: The total number into which the whole is divided (here, it's \(3\)).

Understanding these terms helps you comprehend the operations involving fractions. Knowing which is which simplifies solving problems—from performing basic operations to understanding their properties.

Dealing with Division by Zero

Division by zero is a fascinating and crucial concept in mathematics; it requires careful attention. To understand why it's impossible to divide by zero, imagine this: division is a way of breaking something into equal parts. If you try to divide a number by zero, you’re essentially asking, "How many groups of zero equal parts can I make from this number?"

The answer to that question is: it cannot be done. That's why mathematicians say division by zero is undefined.

• It's like having zero baskets but asking to distribute apples into those non-existent baskets; it's simply not feasible.
• In symbolic terms, given a fraction \( \frac{a}{0} \), there’s no multiplication equation \( c \cdot 0 = a \) because anything multiplied by zero yields zero.

This is a fundamental rule in mathematics: any attempt to divide by zero is ruled as undefined. It helps avoid paradoxes and maintains the integrity of arithmetic operations.