A. An
imaginary number is represented by *i *which is _{}.
Usually we write imaginary numbers in the form: a + b*i*.

B. There are many ways of working with imaginary numbers. Below are some things you should know:

1. Powers
of *i:*

*i*^{0}
= 1; *i*^{1} = *i*; *i*^{2} =
-1; *i*^{3} = -*i*.

Note: These are
the only values for *i*^{n}. After this, the values repeat
themselves. For example, *i*^{10} = -1. In general, *i*^{n}
= *i*^{(n MOD 4)}

Ex [1] *i*^{244}
= _______

a. Since
244 MOD 4 = 0, *i*^{244} is the same as *i*^{0} = 1.

b. The answer is 1.

2. Conjugate:

a. The
conjugate of an imaginary number, a + b*i*, is defined as being a - b*i*.

Ex [1] The
conjugate of 4+5*i* is a+b*i*. b = ____.

a. The
conjugate of 4+5*i* is 4-5*i*. So the answer is -5.

3. Modulus:

a. If you
were to graph a+b*i* on a graph, the coordinates would be (a,b).
Modulus is the distance from the origin (0,0) to the point (a,b).

b. Therefore, the formula for the modulus is:

Ex [1] The
modulus of 5 + 12*i* is _______.

a. You
should know the Pythagorean Triple (5,12,13). The answer is 13. If
you don't you can see _{}.

c.
Therefore, knowing *Pythagorean Triples*
will be very advantageous.

4. Multiplying 2 imaginary numbers together:

a. This
method is going to use the same idea as the *FOIL
Method*.

b. Multiplying
2 imaginary numbers gives an answer in the form of a+b*i*, since *i*^{2}
= -1.

Ex [1]
(3-2*i*)(5+4*i*) = a + bi. b = _____.

a. Since the question is asking for the 'b' value and not the 'a' value we are only concerned with the "OI" in the FOIL method.

b. (-2*i*)(5)
+ (3)(4*i*) = 2*i*. The answer is 2.

c. If the
question had asked for the 'a' value, then we would only be concerned with the
"F" and "L" in the FOIL method. So we would want
(3)(5) + (-2*i*)(4*i*) = 15 - 8*i*^{2} = 15 + 8 =
23. So a is 23.

5. Powers
of a+b*i*:

a. For
any integral value of 'n', (x+y*i*)^{n}, can be written in the form
a+bi. Most of the time, on number sense tests, the power will be 2.

b. This
method will use the fact that (x+y)^{2} = x^{2 }+ 2xy + y^{2}.
Remember that *i*^{2} = -1.

Ex [1] (3
- 4*i*)^{2} = a + b*i*. The b = _____

a. If we are looking for the b, then the answer is 2(x)(y). So in this case, the answer is 2(3)(-4) = -24.

b. If we
are looking for the a, then the answer is x^{2} - y^{2}.
So in this case, the answer would be 3^{2} - 4^{2} = -7.

6.
Dividing by a+b*i*:

a. The
rules of imaginary numbers are similar to the rules of square roots since
technically an imaginary number is a square root. One of these rules is
you cannot have an *i* in the denominator. So when you are dividing
by 2 imaginary numbers, you must multiply the numerator and the denominator by
the conjugate of the denominator.

^{Ex
[1]} _{}^{=
a+bi. a = ____.}

a. To
solve this problem we have to multiply the numerator and the denominator by the
conjugate or by 2-3*i*.

b.
Anytime you multiply a+b*i* by its conjugate, you get a^{2} + b^{2}.
So the denominator becomes 2^{2} + 3^{2} or 13.

c. Now,
to find the numerator, we have to multiply (3-4*i*)(2-3*i*).
Since the question just wants the 'a' value we are only concerned with: 3(2)+12*i*^{2
}= 6 - 12 = -6. If the question wanted the 'b' value we would need to know
(-4)(2) + 3(-3) or -17.

d. The
answer is -^{6}/_{13}.