Q: Can you fix the “1/0 problem” by defining 1/0 as a new number?

The original question was: I’ve been relearning some things about imaginary numbers and the concept behind the number got me thinking about something else. Is there a reason that the quantity could not be defined in a similar way to , so that functions could a real component, imaginary component, and an undefined component? If so, what implications would this have to mathematics? I can’t see it being very useful but I was curious to see what you think.

Physicist: If that worked it would be extremely useful. However, it just doesn’t jive with the axioms of arithmetic.
People in the math biz are always making up place holders for things that aren’t known or, in some cases, can’t exist. Euler (pronounced “Oiler”, as in “one who oils”) wanted to come up with a number system that included a solution to x²+1=0. He called that solution “ “, for “Imaginary number” or possibly “Incredibly awesome number” (I think it’s the latter).

As it happens, there are no problems involved with defining .  In fact, it really cleans up a lot of math.  1/0 on the other hand is kind of a train wreck.

Define (for “Quite a bit more awesome than “) as the solution to . That is, define as “1/0”, so it’s the “multiplicative inverse” of 0.  Right off the bat there’s a problem:

This is because zero is defined as “x-x” for any number or variable x.  So by assuming (and the laws of arithmetic) you reach an impossible conclusion!

You could try to patch this problem, for example by declaring that .  Even so:

Again, by defining to be true, you’re led to a contradiction.  Mo’ logic, mo’ problems.  You could fix this problem by declaring that associativity doesn’t apply to .  That is, .  Losing associativity is a big deal though.  Without it you can barely do anything.

You can keep going, finding more problems and declaring more “fixes”, but in short order you’ll find that by the time you’re done patching things you’ll have more problems, exceptions, and caveats than just “1/0 is undefined”.

Best to just leave 1/0 undefined.