Definition: Infinity refers to something without any limit, and is a concept relevant in a number of fields, predominantly mathematics and physics. The English word infinity derives from Latin infinitas, which can be translated as " unboundedness ", itself derived from the Greek word apeiros, meaning " endless ".
I know that $\infty/\infty$ is not generally defined. However, if we have 2 equal infinities divided by each other, would it be 1? if we have an infinity divided by another half-as-big infinity, for
Similarly, the reals and the complex numbers each exclude infinity, so arithmetic isn't defined for it. You can extend those sets to include infinity - but then you have to extend the definition of the arithmetic operators, to cope with that extended set. And then, you need to start thinking about arithmetic differently.
The mathematical reason: because there's no good way to decide what the product of the complex unit i and the symbol for infinity should mean (and further, because if you want to consider the product to be the limit of I * x as x tends to infinity, there's no preferred direction in which to let x move).
L'Hospital's Rule of infinity over infinity Ask Question Asked 2 years, 8 months ago Modified 2 years, 8 months ago
This "$1^\infty$" (in regards to indeterminate forms) actually means: when there is an expression that approaches 1 and then it is raised to the power of an expression that approaches infinity we can't determine what happens in that form. Hence, indeterminate form.
The problem is that the laws of addition and multiplication you are using hold for real numbers, but infinity is not a natural number, so these laws do not apply. If they did, you could use a similar argument that multiplying anything by infinity, no matter how small, gives infinity, thus $\infty \times 0 = \infty$. More sophisticated arguments can also be made, like $\infty \times 0 = \lim ...