What happens if you divide multiply add or subtract infinity from any number?

What happens if you divide multiply add or subtract infinity from any number?
Originally Posted by An inconvenient lie
, such that x > 0.
, such that x < 0.
, such that x = 0.
, such that x > 0.
, such that x < 0.
In the above case, . What that means is that x is an element of the set of real numbers (it is any real number, but only a real number [this inludes sets that are subsets of the real numbers, such as the integers or rational numbers]).
Infinity is not a real number and operations such as you suggest do not result in real numbers. They are not defined.Originally Posted by An inconvenient lie
There is a theory of cardinal numbers that can be used to address your question in part. In that theory there are infinite cardinal numbers, which arise from infinite sets. There are in fact different sizes of infinity. For instance the infinity that corresponds to the real numbers is strictly larger than the infinity that corresponds to the natural numbers. But the infinity of the natural numbers is the same as the infinity of the integers and of the rational numbers, called "countable infinity". In that theory the product of any nonzero number with an infinite cardinal is the larger of the two numbers. Zero times anything is zero. Similarly the sum of any number with an infinitge cardinal is the larger of the two. Subtraction and division are not defined.
http://www.sjsu.edu/faculty/watkins/cardinals.htm
http://euclid.colorado.edu/~monkd/se...y/m6730_06.pdf
There is also a theory of ordinal numbers and ordinal arithmetic. However, ordinal addition is not commutative, so we will not discuss it here.
Then there there is the "extended real numbers" which one commonly finds in treatments of the theory of measure and integration. In that theory one adds two elements to the real numbers, and . In that system one simply defines the operations as in this Wiki article.
http://en.wikipedia.org/wiki/Extended_real_number_line
This is done simply for convenience in developing the abstract theory of integration and order topologies that compactify the real numbers. It eliminates the need to consider a number of special cases in the developoment of the theory, reduciing tedium. The resulting "number system" is not a field, or even a group. Note that in this case is not defined. This reflects the ambiguity one finds in calculus with the limit of a product in which the limit of one term is 0 while the other is infinite.
So, basically the answer to your question is very much dependent on the context in which it is asked, and there are no hard and fast rules. This is to be expected, as the term "infinity" itself is not welldefined except when the context is specified. In particular "infinity" is not a number. To say that a set is infinite is not to say what the set is but rather what it is not  it is not finite.
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