Types of Infinity
Most students have run across infinity at some point in time prior to a calculus class.
However, when they have dealt with it, it was just a symbol used to represent a really,
really large positive or really, really large negative number and that was the extent of
it. Once they get into a calculus class students are asked to do some basic algebra
with infinity and this is where they get into trouble. Infinity is NOT a number and for
the most part doesn’t behave like a number. However, despite that we’ll think of
infinity in this section as a really, really, really large number that is so large there isn’t
another number larger than it. This is not correct of course, but may help with the
discussion in this section. Note as well that everything that we’ll be discussing in this
section applies only to real numbers. If you move into complex numbers for instance
things can and do change.
So, let’s start thinking about addition with infinity. When you add two non-zero
numbers you get a new number. For example,
. With
infinity this is not true. With infinity you have the following.
In other words, a really, really large positive number (
) plus any positive
number, regardless of the size, is still a really, really large positive number. Likewise,
you can add a negative number (
i.e.
) to a really, really large positive
number and stay really, really large and positive. So, addition involving infinity can
be dealt with in an intuitive way if you’re careful. Note as well that the
a
must NOT
be negative infinity. If it is, there are some serious issues that we need to deal with as
we’ll see in a bit.
Subtraction with negative infinity can also be dealt with in an intuitive way in most
cases as well. A really, really large negative number minus any positive number,
regardless of its size, is still a really, really large negative number. Subtracting a
negative number (
i.e.
) from a really, really large negative number will
still be a really, really large negative number. Or,