Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$0 \div(-10)$$

Step-by-Step Solution

Answer

The result of the division \(0\div(-10)\) is 0.

Step 1: Analyze the given expression

The given expression to evaluate is \(0 \div(-10)\).

Step 2: Apply the rules of division

In mathematics, the division of zero by any non-zero number always results in zero. The rationale behind this is simple: if we have zero items and distribute them into any number of groups, no group will have an item, i.e., each group would have zero items.

Step 3: Result

Therefore, \(0\div(-10)\) results in 0.

Key Concepts

Dividing Zero

In algebra, dividing zero by any non-zero number is a concept that often baffles students. Let's approach it with a real-world analogy to demystify it. Imagine you have 0 cookies. If you attempt to divide these non-existent cookies among any number of friends, say, -10 friends, each friend would end up with, unsurprisingly, 0 cookies. Mathematically, this is just as straightforward. When you divide 0 by any non-zero number, the result is always 0, no matter the sign of the divisor.

In our textbook exercise, we're presented with the task of dividing 0 by -10. Following the rationale of our example, the outcome is 0. This is consistent whether you're dividing by a positive or a negative number. By understanding this concept clearly, you can effortlessly tackle similar problems involving zero.

Rules of Division

Division is one of the basic operations in algebra that necessitates understanding certain rules to solve equations correctly. Here are the essential rules you should remember:

• If you divide any number by 1, the result is the number itself.
• Division by zero is undefined, which we will explore further in our next section.
• Dividing two positive or two negative numbers yields a positive result.
• Dividing a positive number by a negative number, or vice versa, yields a negative result.

When applying these rules, it's crucial to understand that division is essentially the inverse of multiplication. Think of it as asking the question, 'What number, when multiplied by the divisor, gives me the dividend?' This is a helpful way to wrap your head around division, especially when you encounter variables instead of outright numbers.

Undefined Expressions in Algebra

In algebra, not all expressions yield a nice, tidy number. Some are considered 'undefined.' One classic example of an undefined expression is division by zero. You might wonder why this is the case. To understand, consider this: if 3 divided by 1 gives us 3, because 1 fits into 3 exactly 3 times, how many times does 0 fit into 3? Well, it can't. Zero can't 'contain' any number, as it represents nothingness.

Think of it as trying to split a pizza among zero friends. It's an impossible situation. Mathematically, if we tried to divide a number by zero, we are essentially asking, 'What number multiplied by 0 gives us our original number?' Since any number multiplied by 0 is always 0, there's no number that satisfies this condition. Therefore, any expression that attempts to divide by zero is labeled undefined. Recognizing undefined expressions helps prevent errors in calculations and ensures a deeper understanding of algebraic principles.