It is the entire number a/0 that is impossible, not just the 0 at the bottom. If the number is impossible, then who are we to take a part of it out (the zero), and assume the remaining part is valid? The problem with this is that a/0 is impossible, so when the zeros are 'cancelled,' what's really getting cancelled (on the left side) (along with the zero we added) is a part of an impossible number. So next would be why this classic example meant to show that we can't divide by zero is actually flawed:Įach side is multiplied by 0 in order to prepare to cancel out the zeros, like this: I think that dividing by zero, regardless of what you mean by 'divide,' is impossible. (So that is part of the reason why it seems to me that we are not being consistent in our meaning and usage of division, and that is affecting how we think about dividing by zero.) (So, now it seems that 0/2 is undefined!) (Though, using a different meaning of division, we know we can have zero halves and that that simply equals 0!) On a similar note, if you say you have zero halves (0/2),īut you could also read those as zero divided by two (0/2)ĭivide (per some online definition) also means calculate how many times one number goes into another. having a 'zeroth' as a denominator), I think, is impossible, then there is no such thing as a zeroth of a pie. I've never heard of a '0th' and I don't think they exist. So with 3/0, would you have 3 number of '0ths'.

So with two divided by zero, we could continue adding zero to itself forever, but still not get to two.ĭoesn't that make 2/0 impossible, not undefined? (Now, to look at what we mean by 'divide' or if we actually mean more than one thing.)ĭivide (per somewhere online, I read), means to calculate how many times we could add one number to itself to get another number. (So, my first hypothesis is that dividing by zero is not undefined, but is defined as impossible.) Wouldn't that be impossible, rather than undefined? So if we have something and divide it by 0, wouldn't that be like making it disappear? If we have something and we divide it by 2, then don't we separate it into two pieces?Īnd if we have something and divide it by 1, then don't we 'separate' it into one piece? The first two can not be proved false using this method, nor can the latter, since it is not exactly defined as division anyway. If 0/0 is undefined, then you can't multiply back. If 0/0 is 0, then 0 times 0 is 0, so it is also correct. If 0/0 is 1, then 1 times 0 is, so it is correct. For example, 6 divided by 2 is 3, and it can be undone by multiplying 2 by 3 to get 6. So 0/0 must be undefined.Īlso, if you think about it more closely, (Sal also says this in the next video.) division must be able to be undone by multiplication. These thoughts can not merge, as 0 is not 1 or undefined, and vice versa. These statements are impossible and don't work, since a mathematical expression consisting of only constants (numbers like 1, not variables like x and y) can only have one value. All of these points of view are logical and reasonable, yet they contradict each other. Others say that 0/0 is obviously one because anything divided by itself, just like 20/20 is 1. However, some say anything divided by zero is undefined, since 4/0 and 5/0 are and so on. They say zero divided by anything is zero. I'm just stating what Sal said in the video, but some people say that 0/0 is obviously 0, since 0/4, for example, is zero.

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