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Why dividing by zero is undefined

CCSS.Math: ,

Video transcript

comedian Steven Wright and I guess we should also credit him maybe with being a bit of a philosopher once commented that black holes are where God divided by zero and I won't get into the physics of it and you know obviously there the metaphor breaks down in certain ways but it is strangely appropriate because black holes are where our current understanding of physics seemed to break down and dividing by zero as simple of an ideas that seems to be is where our mathematics also breaks down this is undefined and sometimes when you see undefined when you're in math class that seems like a very strange thing it seems like a very bizarre idea but it really means exactly what the word means mathematicians have never said what defined what it must mean to divide by zero what is that value and the reason they haven't done it is because they couldn't come up with a good answer there's no good answer here no good definition and because of that any nonzero number divided by zero is just left undefined you know seven divided by zero eight divided by zero negative one divided by zero we say all of these things are just undefined now you might say hey you know well if we can just define it let's at least try to come up with a definition of what it means to take a nonzero number divided by zero so let's do that right now and we could just take the simplest of all nonzero numbers we'll just do it with one but we could have done this with any nonzero number let's take the example of one and since we don't know what it means or we're trying to figure out what it means to divide by zero let's just try out really really really small positive numbers let's divide by really really small positive numbers and see what happens as we get close to zero so let's divide by 0.1 well this will get us to ten if we divide one by 0.01 that gets us to 100 if I let me go really really close to zero if I divide one by 0.000001 this gets us so this is not a tenth hundredth thousandth ten-thousandth hundred thousand this is a millionth one divided by a millionth that's go to give us 1 1 million so we see a pattern here as we divide one by smaller and smaller and smaller positive numbers we get a larger and larger and larger value so based on just this you might say hey well I've got something somewhat of a definition for 1 divided by 0 maybe we can say that 1 divided by 0 is positive infinity if we put smaller and smaller positive numbers here we get super super super large numbers right over here but then your your friend might say well that worked when we divided by positive numbers close to 0 but what happens when we divide by negative numbers close to 0 so let's try those out well 1 divided by 0.1 or actually say negative zero point 1 that's going to be negative 10 1 divided by negative 0.01 that's going to be negative 100 and if we go all the way to 1 divided by negative 0.000001 yep I drew the same number of zeros that gets us to negative 1 million and so when we see when we keep dividing 1 by negative numbers that are closer and closer and closer and closer to 0 we get a very different answer we actually start approaching negative infinity so over here we said maybe it would be positive infinity but you can make an equally strong argument that it could be a very different number but negative infinity is going the exact opposite direction so you can make an equally strong argument that it should be negative infinity and this is why mathematician says well there's just no good answer here that especially one that's consistent with the rest of mathematics they could have just said it's equal to you know 42 or something like that but that would have that would make no sense it's neither one of these values and it wouldn't be consistent with everything else we know so they just left the whole thing undefined