Problem 12 - Intermediate Algebra

Question

Complete each property of division. a. \(\frac{a}{1}=\square\) b. \(\frac{a}{a}=\square\) c. \(\frac{0}{a}=\square\) d. \(\frac{a}{0}\) is \(\square \)

Step-by-Step Solution

Answer

a. \(a\), b. 1, c. 0, d. undefined

Step 1: Division by One

The property of division states that any number divided by one is the number itself. Therefore, if you have \(\frac{a}{1}\), it equals \(a\). This is because dividing anything by one does not change its value.

Step 2: Division by Itself

When a number is divided by itself, the result is always one, provided that the number is not zero. For \(\frac{a}{a}\), this equals to 1 because any non-zero number divided by itself is 1.

Step 3: Zero Divided by Any Non-zero Number

Zero divided by any non-zero number is always zero. So, \(\frac{0}{a}\) equals 0, assuming \(a\) is not zero. Any number of zero objects shared among non-zero objects results in zero objects each.

Step 4: Division by Zero

Division by zero is undefined. The expression \(\frac{a}{0}\) cannot be evaluated, as dividing by zero does not result in a finite or meaningful value in mathematics.

Key Concepts

Division by One

A key property of division is that any number divided by one remains unchanged. This property is intuitive; consider slicing a cake into one piece—it is still the whole cake. When you write \( \frac{a}{1} \), you are essentially confirming that \( a \) divided by one equals \( a \). This holds true for all numbers, whether they are positive, negative, or even fractions. So remember, dividing by one is like looking at the number in its full, unchanged form.

Division by Itself

When a number is divided by itself, the outcome is always one, as long as the number isn't zero. This concept stands on the principle that any amount shared among itself results in a single unit portion. For instance, if you divide 5 cookies between precisely 5 people, each gets exactly 1 cookie. Thus, \( \frac{a}{a} \) results in 1 (when \( a eq 0 \)). Keep in mind that zero divided by itself is not included in this rule, as that leads to an undefined situation; division by zero is another concept altogether.

Zero Divided by a Non-zero Number

Zero can be a bit of an enigma in arithmetic, but dividing it by any non-zero number is straightforward. Zero divided by a non-zero number equals zero because there’s nothing to share. Visualize this as having zero candies to distribute among a group of friends; each friend ends up with none. Therefore, \( \frac{0}{a} = 0 \) (as long as \( a eq 0 \)). It illustrates the rule that the absence of items remains the absence, no matter how you try to share.

Division by Zero

Division by zero is a topic that often perplexes students. The expression \( \frac{a}{0} \) is undefined, meaning it doesn't have a value that fits into our number system or everyday experience. This stems from the fact that you cannot divide something into zero parts. Imagine trying to distribute 5 apples into zero groups—it simply doesn’t make sense. That is why mathematicians consider division by zero as undefined territory, cautioning against attempting it during arithmetic operations. Always remember: you can’t divide by zero, no exceptions.