2.4 Complex Numbers
For centuries mathematics has been an ever-expanding field because of one particular “trick.”
Whenever a notable mathematician gets stuck on a problem that seems to have no solution, they make
up something new. This is how complex numbers were “invented.” A simple quadratic equation would
be
210x
. However, in trying to solve this it was found that
21x
and that was confusing. How
could a quantity multiplied by itself equal a negative number?
This is where the genius came in. A guy named Cardano developed complex numbers off the base of the
imaginary number
1i
, the solution to our “easy” equation
21x
. The system didn’t really get
rolling until Euler and Gauss started using it, but if you want to blame someone it should be Cardano.
My Definition – The imaginary unit
i
is a number such that
21i
. That is,
1i
.
Definition of a Complex Number – If a and b are real numbers, the number a + bi is a complex number,
and it is said to be written in standard form. If b = 0, the number a + bi = a is a real number. If b
≠
0, the
number a + bi is called an imaginary number. A number of the form bi, where b
≠
0, is called a pure
imaginary number.
Equality of Complex Numbers – Two complex numbers a + bi and c + di, written in standard form, are
equal to each other
a bi c di
if and only if a = c and b = d.
Addition and Subtraction of Complex Numbers – If a + bi and c + di are two complex numbers written in
standard form, their sum and difference are defined as follows.
Sum:
a bi c di a c b d i
Difference:
a bi c di a c b d i
To multiply complex numbers, use the distributive property keeping in mind that
21i
.
