Problem 43 - Basic College Mathematics

Question

Simplify. $$ \frac{0}{8} $$

Step-by-Step Solution

Answer

0

Step 1: Understand the Concept of Division by Zero

When a number is divided by a non-zero number, the quotient is 0 if the numerator is 0. In this case, we have 0 divided by 8.

Step 2: Apply the Division Rule

According to the division rule, \(\frac{0}{8}\) simplifies to 0 because the numerator is 0 and any number divided by a non-zero number is 0.

Key Concepts

division by zero

One key concept to understand in division is what happens when you divide by zero. Division by zero is undefined. This means you cannot divide any number by zero in standard arithmetic.
Unlike division by other numbers, if the denominator (the number you divide by) is zero, the answer is not zero either; it simply does not exist in the realm of real numbers.
For example, \(\frac{5}{0}\) or \(\frac{0}{0}\) are both undefined. Remember: **Never divide by zero!**.

numerator and denominator

In a fraction, the top part is called the numerator, and the bottom part is called the denominator.
Understanding these terms is crucial, especially when simplifying or solving fractions.

• The **numerator** shows how many parts we have.
• The **denominator** indicates into how many parts the whole is divided.

For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.
When 0 is in the numerator, no matter what the denominator is (as long as it's not zero), the fraction's value is 0. Thus, \(\frac{0}{8}\) simplifies to 0.

division rules

Learning the rules of division helps make operations simpler. Here are some essential division rules:

• If you divide any number by 1, the result is the original number (e.g., \(\frac{5}{1} = 5\)).
• If you divide any number by itself (except zero), the result is 1 (e.g., \(\frac{8}{8} = 1\)).
• If 0 is the numerator (the number to be divided), and the denominator is any non-zero number, the result is always 0. That's why \(\frac{0}{8}\) equals 0.
• If you divide by zero, the operation is undefined, as mentioned in the division by zero section.

With these rules in mind, solving problems involving division becomes easier and more intuitive.