44 CHAPTER 5
The situation with cubic equations was more complicated.
Mathematicians discovered a formula for solving cubic equations
similar to but more complicated than the quadratic formula
which solves the quadratic equation ax
+ bx + c = 0. In
some cases, however, some of the numbers in the cubic formula
were square roots of negative numbers, even though in the end all
of the solutions to the cubic equation were real numbers. We are not
going to go through the algebra, but here is a very explicit example.
The equation x
− 7x + 6 = 0 has the three solutions x = 1, x = 2,
and x =−3. If you try to solve this equation by using the cubic
formula (which is like the quadratic formula, only much more com-
plicated), along the way you unavoidably encounter the square root
of a negative number.
Whether or not we believe that complex numbers really exist, we
need to come up with rules for manipulating them. And the rules
are simple enough.
DEFINITION: A complex number is a number of the form
a + bi,wherea and b are real numbers.
So 3 + 4i and
2 + πi are complex numbers;
so is 0 + 0i (which
is another w ay of writing 0). Sometimes, we write the “i” before the
second number: 2 − i sin 2 is also a complex number.
In the complex number a + bi, we call a the real part and bi
the imaginary part. (Actually, mathematicians normally reserve
the term “imaginary part” for the real number b, but that is a bit
confusing, and we will not follow that usage.) We consider every
real number x to be a complex number too, but write x instead of
x + 0i.
The symbol π stands for the area of a circle of radius 1, which is approximately
3.14159265. It is a nonrepeating inﬁnite decimal.
If b is negative, we can write a − (−b)i rather than a + bi.