Infinity is not a number. It is another way of saying that something gets larger and larger without an end. So, you cannot add 1 to some thing which is not a number. However, in mathematics, there are infinities that could be added, multiplied, or subtracted. There are cardinal, ordinal, surreal and hyperreal numbers to all of which 1 can be meaningfully added. For infinite cardinal numbers, adding one does not change the number. For ordinal numbers, the number can be changed depending on whether you add 1 to the number, or the number to 1 - there is no commutativity. With hyperreal numbers, one can do pretty much anything one would expect of a real number. As to the surreal ones, the arithmetic operations produce magic results increasing the number of various infinities to fantastic proportions. In general, to add 1 to a number means to take the successor of that number. The successor of a set A is the set which consists of all the elements of A along with A itself. In other words the successor of A is A U {A}, and it has "one" more element than A. So, infinity + 1 = infinity U {infinity}.
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infinity
Negative infinity plus negative infinity equals negative infinity.
Infinity divided by any finite number is infinity. Here are the rules: 1. Infinity divided by a finite number is infinite (I / f = I); 2. Any finite number divided by infinity is a number infinitesimally larger than, but never equal to, zero (f / I = 1 / I); 3. Infinity divided by infinity is one (I / I = 1), or in fact any other positive number (I / I = and so on...); 4. Infinity multiplied by zero (no infinity) is zero (I * 0 = 0); 5. Infinity divided by a positive finite number is infinity (I / +f = I); 6. Infinity divided by a negative finite number is minus infinity (I / -f = -I); 7. Infinity divided by zero is not possible; 8. Infinity plus infinity is infinity (I + I = I); 9. Zero divided by infinity (nothing divided into infinity) equals zero (0 / I = 0); 10. Infinity plus a finite number is infinity (I + f = I); 11. Infinity minus a finite number is infinity (I - f = I); but 12. Infinity minus infinity, due to the nature of infinity, can be zero, infinity, or minus infinity (I - I = -I, 0, I).
If you mean the arcsin function then the range is the whole of the real numbers - from "minus infinity" to "plus infinity". If you mean the cosecant function, the answer is the whole of the real numbers excluding (-1, 1).
Infinity is a concept, not a number. Even if it were considered such, infinity plus one is also infinity, and so infinity minus one is still infinity.