eli5: Why haven’t mathematicians invented a symbol for x/0 like they have pi and imaginary numbers?
In: 277
Because there is no one value that x/0 could possibly be equal to. We know this because of a thing called limits.
The limit is the mathematical idea of the value that an expression approaches as some terms approach some value. It’s useful for assigning numerical values to expressions that wouldn’t normally have one.
So, let’s try to find a value for x/0 using limits. Let’s assume x=1, because we should just be able to multiply 1/0 by x to get the value we want. 1/0 can be thought of as the limit of 1/z as z approaches 0. So let’s plug in smaller and smaller values for z and see where the solutions tend towards.
1/0.1=10, 1/0.01=100, 1/0.001=1000, and so on. You can see that the value just keeps getting larger and larger, it’s tending towards infinity. Now, in general usage, infinity isn’t treated as a numerical value. But that’s fine, we could just redefine infinity and give it the numerical value 1/0 if we thought it would be useful. But there is one problem. For a limit to exist, the expression must approach it from both sides, so to speak. So let’s start taking negative values for z closer and closer to 0.
1/-0.1=-10, 1/-0.01=-100, 1/-0.001=-1000, and so on. Now, as z goes to 0, 1/z is getting smaller and smaller, tending towards _negative_ infinity. The limit is different for different directions of approach, meaning the limit doesn’t exist.
Some people have attempted to reconcile this by saying “maybe negative infinity and positive infinity are the same thing”, but they still run into problems down the line. It just doesn’t work.
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