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
5.32i
The
last two examples have one of their components missing; 3.1 is
missing the second component, and 5.32 is missing the first
component. These numbers are still complex numbers and can be written as (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. (for example, Friederich Gauss proved that all
the solutions, zeroes, 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
= -10 + 24i
Division:
some
examples:
explanation:
Now
the case :
The
problem here is the 'di' in the denominator. It needs to be
eliminated.
If
we multiply (c + di) by (c - di) we get
c2
- (di)2 = c2 - d2i2 =
c2 - d2(-1) = c2 + d2
Notice
this eliminates 'i' from the the starting expression, c + di.
The
multiplier (c - di) is called the complex conjugate, or more
briefly,
the
conjugate of (c + di).
Given
a complex number, change the sign of the 'i' term and you have its
conjugate.
examples:
|
complex number
|
its conjugate
|
|
3 + 6i
|
3 - 61
|
|
-4 + 10i
|
-4 - 10i
|
|
13 – 12i
|
13 + 12i
|
|
+16i
|
–16i
|
|
2
|
2
|
Now,
back to
Recall
that we can multiply any non-zero expression by another (which
represents 1) and only change the way it appears, not its value.
32 and 128/4 look different but represent the same value, 32.
If
we multiple both the numerator and the denominator by the conjugate
of the denominator (c - di) we get:
If
you can remember this formula, then it does make this work easier.
The first few examples use this formula, the remainder do not. You
need to write the denominator and numerator in terms of the original
involving addition only first.
Visually
it is easy to remember. Add the products vertically, subtract the
cross products. The example will illustrate.
Example:
Example:
Powers
of i.
|
power
|
value
|
|
i0
|
1
|
|
i1
|
i
|
|
i2
|
-1 (i)(i)
= -1
|
|
i3
|
-i
(i)(i)(i) = -1(i)
|
|
i4
|
1
(i)(i)(i)(i) = -1(-1) = 1
|
|
i5
|
i
(i)(i)(i)(i)(i) = -1(-1) i = i
|
|
i6
|
-1 etc...
|
|
i7
|
-i
|
|
i8
|
1
|
|
in
|
depends on n, note
the previous examples
odd powers yield
(+/-) i, even powers yield (+/-) 1
First determine if
n is even or odd. Then ask yourself how many pairs of i do you
have as factors. Then you will have your answer.
examples:
i13
n = 13 is odd, so we have either i or -i as the answer,
i13 =
i • i12
now, 12/2 = 6, even factors of i2 (= -1)
(-1)(-1)(-1)(-1)(-1)(-1)
= 1
i13 =
i
i30
n = 30 is even so we will have either 1 or -1 as the
answer
30 / 2
= 15, odd number of i2 factors so
i30 =
-1
i91
n = 91 is odd, i91 = i •
i90 and 90/2 = 45 which is
odd, so we have an
odd number of i2 factors so
i91 =
-i
NOTICE that in all
odd cases I first factored a single i out and looked at the
exponent of the remaining factor to determine the sign of the
result.
|