A
complex number is a number that has two separate components, the
first is a real number and the second is also a real number with the
letter 'i' appended to it. (the letter 'j' is used in Physics and
particularly Electrical Engineering.)
Definition:
a complex number has the form (a + bi), where i =
Examples:
12 +
3i - 4.5 + 6.1i 3.1
5.3i
The
last two examples have one of their components missing; 3.1 is
missing thee second component, and 5.32 is missing the first
component. These numbers are still complex numbers (3.1 + 0i) and
(0 + 5.3i)
The
square root function has the domain: non-negative Reals. Since
negative numbers are not in this domain, by the limitation imposed by
this domain, we simply do not apply the square root to negative
numbers. It's common to hear that the square root of a negative
number does not exist, but oftentimes in mathematics we extend the
concept of a number by adding components, in this case a single
component, to get over these limitations, and the square root of
negative numbers in particular. (Friederich Gauss proved that all
the solutions of all polynomial equations are complex numbers.)
Geometric
interpretation:
The
complex number is a convenient way to name a unique point in the XY
coordinate system. Given the complex number (a + bi), (a) is the x
coordinate, and (bi) is the y coordinate of the point in the plane.
So (a + bi) is equivalent to (x,y). In the first case we have a
number, in the second a coordinate pair. The complex number allows
us to think of the point in terms of a single number rather than a
coordinate pair; this number is actually a vector.
Final
Note on Naming:
Often
you will here the word imaginary used for these numbers. We
use the Real numbers without a second thought. The Pythagoreans
could not accept the fact that a right triangle whose legs were both
1 gave a hypotenuse with a length (we know to be
)
that did not exist in their concept of a number, however they could
demonstrate that some number must exist to represent this length.
Unfortunately the term imaginary was used for complex numbers since
their definition, but they do model certain phenomena as do all the
other numbers. So in this sense they are anything but imaginary.
The word complex is somewhat a misnomer as well, since other number
definitions are more complex to this one.
Now
using the property
and
the definition for i
we have:
Example:
5i
· 6i = 30 ·
i ·i = 30 · i2 = -30
Operations
with complex numbers:
Addition/Subtraction:
Add/Subtract
the respective components.
example:
2 + 7i + 3 + 6i = (2 + 3) + (7i + 6i) = 5 + 13i
example:
6 + 14i – (4 + 5i) = 6 – 4 + (14i – 5i)
= 2 + 9i
example:
-4.2 – 1.2i – ( - 6.1 – 2i ) = -4.2 + 6.1
– 1.2i + 2i = 1.9 – 0.8i
In the
last two examples the – sign must be distributed across the
parenthesis.
Multiplication:
Treat
the complex numbers as binomials and multiply accordingly using the
fact the i2 = -1.
example:
(2 +
3i) ( 4 + 6i) = 2(4 + 6i) + 3i(4 + 6i)
= 2 · 4 + 2 ·
6i + 3 · 4i + 3i ·
6i
= 8 + 12i + 12i – 18i2
=
8 + 24i – 18( -1)
=
8 + 18 + 24i
=
26 + 24i
Division:
some
examples:
explanation:
Example:
Example:
Now
the case :
The
problem here is the 'di' in the denominator. If we multiple the
denominator by (c - di) we get
.
Using this we can eliminate the di from the denominator by
multiplying both the numerator and the denominator by (c – di)
and get:
Example:
Example:
