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### Course: Algebra 1>Unit 1

Lesson 6: Division by zero

# Undefined & indeterminate expressions

Revisiting the problems of dividing any number by zero and dividing zero by zero. Using general mathematical considerations, we see why those are undefined and indeterminate problems. Created by Sal Khan.

## Want to join the conversation?

• If you have 0/0, how come it isn't 1? If you cancel it out you will get 1, right? Also, lets say you have 0 apples, and 0 people. So for each (nonexistent) person you have a (nonexistent) apple. I'm so confused...
• If you have 0 apples and 0 people, then it is vacuously true (http://en.wikipedia.org/wiki/Vacuous_truth) to say that all the people have 1 apple, but it is equally true to say that all the people have 100 apples since there are no people.

It's like saying all the cars I own are Ferraris, which is true because I don't own any cars. I have no cars and no Ferraris. The point is we can say all the people have any number of apples and it would be (vacuously) true, hence the number is undefined.
• what is the difference between undefined and indeterminate?
• Broadly speaking, undefined means there is no possible value (or there are infinite possible values), while indeterminate means there is no value given the current information. For example in the equation ax + by = c, the relation between x and y is indeterminate because you can't determine it without knowing the values of a, b and c.

However, there is some overlap in terminology and in calculus 0/0 is called an indeterminate form: http://en.wikipedia.org/wiki/Indeterminate_form
• So what would ∞*0 be?
• ∞*0 is considered to be an indeterminate form. Generally, an expression that yields ∞*0 can be rewritten to come out as 0/0 or ∞/∞ (this comes up in the context of L'Hôpital's Rule in calculus)
• Couldn't you just simplify 0/0=1, because they would just cancel each other out?
• One of the biggest issues with this reasoning is that you'd have to introduce exceptions to one of the most cherished properties of multiplication: It's associative.

What this means is that for any values a, b and c, it holds true that (a*b)c = a(b*c)
That is, the order of multiplication does not affect the result.

If you introduce 0/0 = 1, you'll get situations such as this:

4 * (0 * 1/0) = 4 * 0/0 = 4*1 = 4
(4 * 0) * 1/0 = 0 * 1/0 = 0/0 = 1
So, you either give up associativity, or accept that 4 = 1. Neither of these options is very tempting.

It's also worth remembering that the very idea behind division is that it undoes multiplication; if you multiply X by 4, you can then divide by 4 and end up where you started.
Division by zero does not accomplish this: Because X*0 = 0 for all values of X, 0/0 would have to take on different values each time in order to be an anti-multiplier, and that's not allowed.

Lastly, one must ask oneself: What would the use of adding 0/0 = 1 be?
If it is useful to your purposes, then, sure, use it. However, usually when you have a need to do anything similar to dividing zero by zero (such as when calculating limits in calculus), you usually require much more complex methods.

We've also established that this would constitute an exception to the associative nature of multiplication, and to the rule that division undoes multiplication. Since both of these properties are used in a lot of mathematical proofs, these proofs would be invalid whenever they involved a factor 0/0; another reason why defining 0/0 just isn't very useful a lot of the time.
• How does this relate to algebra?
• Great Question!
As I am sure you know by now, algebra has lots of rules, just like any game. To play the game, you follow the rules. The rules of the game of math are so constructed that if you follow them, you will ALWAYS arrive a TRUE result.
One of the rules you learned was that division by zero is not allowed. We say that division by zero is undefined. WHY? It is because if you do divide something by zero, you can arrive at an UNTRUE result. Most of the time this happens when it is not obvious that we are dividing by zero. How so? Follow this simple example.
Suppose we have two variables called `a` and `b`.
Let `a = 1` and let `b = 1`
Obviously then, `a = b` is true since `a=1` and `b = 1` thus `a = b` means `1 = 1`, which is true.

Now multiply both sides of the equation `a = b` by `a` and we get:
`a·a = a·b`, and we can rewrite that as `a² = a·b`

Now let us subtract `b²` from both sides of the equation so `a²=a·b` becomes:
`a² - b² = a·b - b²`

Now we can factor the expressions on each side of the equals sign.
I hope you recognize that the expression `a² - b²` is a difference of squares and has a well defined factor form : `a² - b² = (a + b)(a - b)`
We can factor the expression `a·b - b²` like so: `a·b - b² = b(a - b)`

To summarize so far:
1) `a = b` // given
2) `a² = a·b` // we multiplied both sides of `1)` by `a`
3) `a² - b² = a·b - b²` // we subtracted `b²` from both sides of `2)`
4) `(a + b)(a - b) = b(a - b)` // we factored the expressions in `3)`

Now, since there is an `(a - b)` term on both sides of the equation, we can get rid of it by dividing both sides of the equation by `(a - b)` like so:
5) `(a + b)(a - b)/(a - b) = b(a - b)/(a - b)`, giving
6) `(a + b) = b`

Now, remember we said at the beginning that `a = 1` and `b = 1`
Well let's substitute those values into `6)`
`(a + b) = b` means `(1 + 1) = 1` which means `2 = 1` But that is not true

So what happened?
Well, since we said `a = 1` and `b = 1`, that meant that when we divided both sides of the equation in step `5)` by `(a - b)` we were actually dividing by zero! since if `a = 1` and `b = 1`, then `a - b = 1 - 1 = 0`. Therefore, Step 5 was an error, that is, it was an illegal step since it violated the do not divide by zero rule.

To move forward in algebra we must be aware at all times if the forms we are working with are undefined or indeterminate. What we did in step `5)` turned the absolutely fine equation we had in step `4)` into an indeterminate form that gave us the wacky `2 = 1` result, which, I hope you agree, is not true.

I hope this helped a bit - I know it was a bit long winded.
This topic becomes more and more important as you get further into algebra.
• What is the Difference Between Undefined and Indeterminate ? It seems to me they are the same things ?
• The difference is subtle. When I say that a/b = c, that is another way of saying that c is the unique number that satisfies the equation bc = a. For instance 6/2 = 3, because 2 * 3 = 6.

Given that, you can see the difference between 1/0 and 0/0. We say that 1/0 is undefined because there is no number c that satisfies 0c = 1. On the other hand, any number c satisfies 0c = 0 and there's no reason to choose one over any of the others, so we say that 0/0 is indeterminate.
• To quote Siri from Apple "image you have 0 cookies and zero friend, and you want to divide those cookies evenly. See? It doesn't make sense. Cookie Monster is sad that there are no cookies, and you are sad because because you have no friends"
• Hehehe
• Is it possible that zero is not a number?
Mathematics is a system of the relationships of numbers, real or imaginary.
Zero is not a number. It is the absence of a number.