The union of the irrational and rational numbers forms the *real
number line*.^{10} Real numbers are of great
importance in physics. They are closed under the arithmetical
operations of addition, subtraction, multiplication and division, where
one must exclude only division by zero. Real exponential functions such
as
or
(where
are all presumed to be real) will
have real values, as will algebraic functions such as
where
is an integer.

However, as before we can discover arithmetical operations such as the square root operation that lead to problems with closure. For positive real arguments , is real, but probably irrational (irrational for most possible values of ). But what happens when we try to form the square root of negative real numbers? In fact, what happens when we try to form the square root of ?

This is a bit of a problem. All real numbers, squared, are positive.
There *is* no real number that can be squared to make
. All we
can do is *imagine* such a number, and then make our system of
numbers bigger to accomodate it. This process leads us to the *imaginary* unit
such that
, and thereby to numbers with
both real and imaginary parts: Complex numbers.