A Parent's Guide to Algebra's Basic Concepts - Why Division By Zero Is Undefined

We are almost ready to start solving equations! I know it seems that I've been saying that forever; but we have actually accomplished a great deal.
We have defined Algebra and looked at it's needed terminology.
We have reviewed all the sets of numbers we will be using.
We have gone into depth with all of the Algebraic properties and tools we will need to solve equations. So what's left? We just need to discuss some final issues that often cause students problems.
If we discuss these now, you will have an easier time when we actually encounter them.
Division by zero is undefined. You have probably heard your child quote this sentence.
Students understand what this means when dealing just with numbers (sort of) and they feel smart saying it.
But if you ask WHY division is undefined, they struggle for an explanation. And when working with the more complicated equations where it is extremely important to keep this thought in mind at all times, students don't understand how it applies. Hopefully, we can avoid their troubles.

Are 5/0 and 0/5 the same thing? They don't look the same.

Do they have the same "answer?" Most Algebra students think that both of these expressions have the value 0.
This is problematic.

Remember that fractions have two meanings.
The symbol 3/4 can mean the fraction three-fourths or it can indicate the division problem three divided by four.
First, let's look at some division examples with numbers: 10/2 = 5 or 21/7 = 3 or 48/8 = 6. It is important to remember that each of these problems can be checked by using the inverse operation of multiplication.

Multiply the bottom number (divisor) by the answer (quotient) and the product should be the top number (dividend). Let's check our examples: 10/2 = 5 because (2)(5) = 10 and 21/7 = 3 because (7)(3) = 21 and 48/8 = 6 because (8)(6) = 48. Now, let's revisit 0/5 and 5/0. For 0/5 we have to think (5) (?) = 0, so 0/5 = 0 because it checks, and it makes sense when you consider its meaning.

Zero is being divided into 5 parts.
It makes sense that there would be zero in each of the 5 parts.
For 5/0 we have to think about (0)(?) =5. This is problematic to say the least.

There is simply no answer to this.
Since zero times everything is zero, it is impossible to get 5.

And logically, it is impossible to divide 5 into NO parts.
Thus, we say that division by zero is undefined. This is the first case of "you can't do that" that isn't given a new symbol as its solution. The problem 0/0 is a special case of division by zero because there are several logical answers.

0/0 = 0 seems right because it checks.
But 0/0 = 1 also seems right if you remember that any number divided by itself is one and it checks also.
But then, so does 0/0 = 2 and 0/0=3 and 0/0=4, etc.
It would seem logical to say that 0/0 is infinity. Since all of these are logical but cannot all be true, we say that not only is 0/0 undefined, it is indeterminate.
By now you may be thinking that this is logical and easy, so why the fuss? At this point it is easy.

But as soon as we encounter expressions with variables in the denominator (on the bottom) of a fraction, the situation becomes more complicated.

Consider the fraction 5/y or the fraction 7/(a - 5).

We have said that variables can represent any number, but now that isn't quite true if the variable is in the denominator of a fraction. Now we must remember that division by zero is undefined and we must be sure to consider if there are any numbers that would cause the denominator to become zero.

For example, in our expression 5/y, y can have any value except zero.

Likewise, in the expression 7/(a - 5), a can have any value except.

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5 since 5 - 5 = 0. As equations get more complicated, we must first consider if there are any values that cause the "undefined" problem.
Solving equations with variables in the denominator involve three steps: (1) determine if there are any values that will cause an "undefined" situation, (2) solve the equation, and (3) compare the number we think is the solution to the "cannot allow" numbers.
If our number is in that group of "cannot allow" numbers, then our equation has no solution.
Yes, equations often have no solution.
We will deal with this after we get into solving equations and graphing.
Then we will be able to understand why some equations don't have solutions.
Algebra students have difficulty remembering to do that first step so they end up with incorrect solutions.