COMPLEX NUMBERS 43
assume the existence of a large number system that contains all the
solutions to all equations of the form f (x) = a
x + a
= 0, where the coefﬁcients a
, ..., a
In this case, you must remember that the same root can occur for
many different polynomials. A sophisticated method is required for
keeping track of these solutions, but it can be done.
One advantage of working with complex numbers is that each
solution comes with its own personality. For example, the solutions
of the equation of x
+ 1 = 0arei =
two complex numbers have all the same algebraic properties, but
they are not equal to each other—they are “twins.” (For more about
this mystery, you can refer to Imagining Numbers (Mazur, 2003).)
But if we have the complex numbers sitting before us, we can call
one of them i and the other −i. How, you might ask, do you tell i
and −i apart? Easy. Multiplication by i rotates the complex plane
counterclockwise as we look down upon it, while −i rotates it
the same amount clockwise.
Now, we deﬁne complex numbers, starting with the real
DEFINITION: A real number is any number that can be
expressed a s a decimal.
For example, 0 = 0.0, 1 = 1.0,
= 0.6, and −
2 =−1.4121356 ...
are real numbers. A real number can be expressed as a terminating
or repeating decimal if and only if it is the ratio of two integers. The
set of all real numbers is usually denoted by the symbol R.
Complex numbers were forced upon the world when mathemati-
cians began solving cubic
equations. You may think that complex
numbers would have shown up as soon as someone tried to solve
the quadratic equation x
+ 1 = 0, but in fact mathematicians were
quite happy to proclaim that this equation simply had no solutions.
After all, the equation 0x = 1 has no solutions, so why should it be
a problem if x
+ 1 = 0 has no solutions?
The complex plane is the (x, y)-plane on which we plot the complex number x + iy as the
point (x, y). See below for how to multiply complex numbers.
Degree 3: For example, x
+ x − 1 = 0.