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Why Dividing by Zero is Undefined

In this video we’re going to explore why dividing by zero is undefined. But first, what we need to do is familiarize ourselves with the definition of division. The definition of division states that if "a" divided by "b" equals "c" and "c" is unique, then "b" times "c" equals "a." So let’s do something - divide two numbers that we know. So let's say that 6 divided by 2 equals 3. We can all agree with that. Notice, we can say that "c" is unique. 3 is unique because we know that 3 is the only number that equals 6 divided by 2. We can also figure out what the second part means. If we multiply our "b" times "c" then we should get "a". So our "b" is 2 times "c" which is 3 equals "a" which is our 6. Both of these are satisfied. So that means that 6 divided by 2 does equal 3. And we can also say that this is "defined" because it satisfies the whole definition of division. Likewise, if it only satisfies one part of the definition, it would mean that it is "undefined." Let's look at examples with zero in them and see what happens to them. So let me clear this, and let's start with zero divided by 1. I am going to say that this equals zero because 1 times zero equals zero. It satisfies this second part of the definition. And this first part, if you were to plug in, say, a 1, a 2, or any other number, then it wouldn't equal that so we can actually say that "c" is unique. So it satisfies that this is actually the only number that you can put there to actually equal zero. We can say that zero divided by 1 equals zero and we can also say that this is "defined" as well. Our next example is going to be 1 divided by zero. And a lot of people like to guess that it would be zero. So, let's try that out. We take our "b" which is zero and multiply it by our "c" which is zero. We don't get what "a" is because of course, zero times zero does not equal 1. So it doesn’t satisfy this part of the equation. Since it doesn't satisfy at least one part of that definition, then that means that it is considered "undefined." So this does not work and that means that it's going to be "undefined." Now, for our next example, sometimes we come across this idea where we actually have zero divided by zero. Well, I think all of us can agree that we can obviously put in a zero there and the second part will be defined. Because we’ve got zero which is our "b" times zero which is our "c," that does equal our "a" which is zero. So, this part works. Well, we can also put in a 5 if we wanted to because zero times 5 equals zero, so it still works for that second part. We can actually plug in anything into there. We can say, zero over zero equals x. We still have zero times x equals zero. But what I'm getting at is that it is the first part that is not being satisfied. Because what happens is that if we can say that zero, 5, or basically any number, then that means that that "c" is not unique. So, in this scenario the first part doesn't work. So, that means that this is going to be undefined. So zero divided by zero is undefined. So, let's label it as that. Make sure that when you are faced with something of this nature, where you are dividing by zero make sure you don't put an actual number down, or a variable down. Just say that it equals "undefined." In summary with all of this, we can say that zero over 1 equals zero. We can say that zero over zero equals "undefined." And of course, last but not least, that we’re a lot of times faced with, is 1 divided by zero, which is still undefined.

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