Dividing by smaller and smaller numbers gives bigger and bigger answers, so calling 1/0 infinity looks like the natural extension rather than an error.
Calculators refuse it, and my notes say it is undefined rather than infinite. What actually goes wrong if you define it?
Accepted Answer
Division is defined by multiplication: a divided by b is the number that gives a when multiplied by b.
So 1/0 would have to be a number which, multiplied by 0, gives 1. Nothing does, because everything multiplied by 0 gives 0. There is no candidate, so there is nothing to define.
0/0 fails the opposite way. Every number works, so there is no single answer to pick.
Notice that neither of those is about infinity being too big to handle. The problem is that the defining equation has no solution in the first case and too many in the second.
The "it approaches infinity" intuition has a specific hole in it, and it is the reason your calculator will not play along.
Approach zero from the positive side and 1/x grows without bound. Approach from the negative side and it falls without bound. The two sides disagree, so there is no single value being approached.
Even in systems that do add a point at infinity, and they exist, the price is giving up other arithmetic you rely on. It is not that nobody thought of it. It is that it costs more than it is worth.
Quick test of why it breaks things: if 1/0 were a number k, then 1 = 0 times k = 0. Once 1 equals 0, every number equals every other number and arithmetic is over.
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