Problem 64 - Introductory Algebra for College Students

Question

Perform the indicated division or state that the expression is undefined. $$0 \div(-10)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Numerator and the Denominator

In the given expression \(0 \div(-10)\), 0 is the numerator and -10 is the denominator.

Step 2: Apply the Arithmetic Rule

The rule of division by zero states that a number divided by zero is undefined, but zero divided by any number is always zero.

Step 3: Perform the Division

So, \(0 \div(-10) = 0\)

Key Concepts

Understanding the Numerator

In any division problem, the numerator is the number that's being divided. Think of it as the 'top' part of a fraction or division statement. It represents the amount you'll be splitting into smaller, equal parts or distributing across a specific number. In our exercise, the numerator is 0. This may seem a bit odd at first because dividing zero by any number will always lead to zero. The reason behind this is that if you have nothing (zero) to start with, no matter how you try to divide it, you still end up with nothing.
This concept is foundational as it helps you understand how division behaves differently compared to other arithmetic operations when zero is involved.

Delving into the Denominator

The denominator indicates how many parts the numerator is divided into. Consider it as the 'bottom' part of a fraction or division issue. In this scenario, the denominator is -10. While the specific value of the denominator doesn't affect the result when the numerator is zero, understanding the denominator is vital in other cases.

• A positive denominator simply shows the number of equal parts available.
• A negative denominator, like -10, suggests a direction or movement on the number line, often related to calculations in algebra and calculus.

But remember, if the denominator were zero, we would say the operation is undefined. That's because you can't divide something into zero parts — it's mathematically impossible.

Key Arithmetic Rules in Division

Arithmetic rules in division guide us in how to correctly handle numbers, especially zero and negative numbers. The exercise solution involves a crucial rule: zero divided by any non-zero number is zero. This is different from other operations like multiplications or additions where zero has different impacts.

• If you divide any number by zero, it is always undefined. This means there is no number you can multiply by zero to get back the original number.
• When negative numbers are involved, the sign of the quotient is determined by the rules of multiplying signs: positive divided by negative (or vice-versa) results in a negative quotient.

Understanding these arithmetic rules allows you to handle more complex algebraic equations with confidence, ensuring you know why certain operations are defined or undefined.