Lesson Ten: There Is No 1/0

Remember when I said that n(1/n) = 1? Well, that is only mostly true, the single exception is that we cannot let n = 0 and expect that to be true, because there is no number (1/0). You might be asking yourself, "what right does he have to tell me what number cannot exist?" Bear with me for a moment and you too will see why (1/0) simply cannot make sense.

The Reciprocal of Zero:

Keep in mind that the purpose of (1/n) is to undo multiplying by n. So if a*n = b, then b*(1/n) = a, because to get from a to be you multiplied by n, so multiplying by (1/n) undoes this and takes b back to a. For example 3*2 = 6, so 6*(1/2) = 3.

However, zero times anything is zero. So 3*0 = 0 and we expect 0 * (1/0) = 3. But 5*0 = 0 so 0*(1/0) = 5 also needs to be true. Since multiplying by zero takes everything to zero, there is no way to undo it, we lose track of where things come from and cannot "send them back."

To use a metaphor, suppose we live in a small town with an airport that has flights that come in from New York. If someone arrives at our airport, we know that they just came from New York, because there is only one place that they can come from. If we consider a larger airport which has incoming flights from multiple places, then we cannot say where an incoming passenger is coming from without knowing more, because there are multiple possibilities. Since multiplying by 0 sends everything to 0, (1/0) does not have enough information to undo multiplication by 0. Because (1/n) is defined to undo multiplication by n, (1/0) cannot exist. Since m/0 would need to be m*(1/0), m/0 cannot exist either. In short, the denominator of a fraction CANNOT be zero

There are two very important concepts in this post. The fact that (1/0), and consequentially (m/0), cannot exist is something that even advanced math students forget or gloss over. You would be surprised at the ways the value 0 can sneak up on you. However, of even deeper importance is the notion that some things cannot be undone, because too much information is lost.