Transcript Complex Numbers

Complex Numbers
Or
I’ve got my “i ” on you.
COMPLEX NUMBERS
Real Numbers
Rational
Numbers
Irrational
Numbers
Imaginary
Numbers
Standard Form
of a Complex Number
a + bi
REAL PART
IMAGINARY
PART
i = -1
2
i 2 = -1 is the basis of everything
you will ever do with complex
numbers.
Simplest form of a complex
number never allows a power of i
greater than the 1st power to be
present, so ………
Simplifying Powers of i
i
i2
i3
i4
i5
i6
i7
i8
Simplification
None needed
By definition
i2 x i = -1 x i =
( i2)2 = ( -1)2 =
( i2)2 x i = ( -1)2 x i
( i2)3 = ( -1)3=
( i2)3 x i = ( -1)3 x i
( i2)4 = ( -1)4 =
Simplest Form
i
-1
-i
1
i
-1
-i
1
Simplification Examples
i42 =
Divide exponent by 2 (42 ÷2 = 21 R 0)
Quotient is exponent; Remainder is
extra power of i
21
2
Write as power of i (i )
Simplify = (-1)21 = - 1
Simplification Examples
i27 =
Divide exponent by 2 (27÷2 = 13 R 1)
Quotient is exponent; Remainder is
extra power of i
13
2
Write as power of i (i )  i
Simplify = (-1)13 i = - 1 i = - i
Adding/Subtracting
Complex Numbers
Adding and subtracting complex numbers
is just like any adding/subtracting you
have ever done with variables.
Simply combine like terms.
(6 + 8i) + (2 – 12i) = 8 – 4i
(7 + 4i) – (10 + 9i) = 7 + 4i – 10 – 9i =
-3 – 5i
Multiplying Complex Numbers
This will be FOIL method with a slight
twist at the end.
An i2 will ALWAYS show up. You will
have to adjust for this.
(4 + 9i)(2 + 3i) = 8 + 12i + 18i + 27i2 =
8 + 30i – 27 = -19 + 30i
(7 – 3i)(6 + 8i) = 42 + 56i – 18i – 24i2 =
42 + 38i + 24 = 66 + 38i
Binomial Squares
and Complex Numbers
You can still do the five-step shortcut, or
you can continue to do FOIL.
You will still have to adjust for the i2 that
will show up.
(7 + 3i)2 = 49 + 42i + 9i2 = 49 + 42i – 9 =
40 + 21i
(8 – 9i)2 = 64 – 144i + 81i2 =
64 – 144i – 81= -17 – 144i
D2S and Complex Numbers
Situations that in the real numbers would
have been differences of two squares
(D2S) demonstrate in the complex
numbers what are known as conjugates.
(3 + 4i)(3 – 4i) = (3)2 – (4i)2 = 9 – 16i2 =
9 + 16 = 25
When conjugates are used, there will be
no i in the answer.
Things Not Allowed
in a Denominator
1. Negative sign
2. Radical
3. Fractional Exponent
4. Complex Number
Each one of these must be
adjusted out of the problem.
Clearing Complex Numbers
from the Denominator
If there is a pure imaginary number in the
bottom, multiply by i with the opposite
sign.
Example: If denominator contains 3i,
multiply both sides by -i.
– Why? This also takes care of a negative in
the denominator. 3i  ( - i ) = -3i2 = 3
Example: If denominator contains -2i,
multiply both sides by i. -2i  i = -2i2 =
(-2)  (-1) = 2
Clearing the Denominator (continued)
If the denominator is of the form a + bi,
then multiply both sides of the fraction
by the conjugate.
If denominator contains (7 + 3i),
multiply both sides by (7 – 3i).
(7 + 3i)(7 – 3i) = 49 – 9i2 = 49 + 9 = 58
If denominator contains (8 – i), multiply
both sides by (8 + i).
(8 – i)(8 + i) = 64 – i2 = 64 + 1 = 65
Simplifying Square Roots of
Negative Numbers
√–9
does not exist in the reals
because there is no number that can be
squared to give a negative answer.
Therefore, you must use i2 to replace the
negative.
√ – 9 = √9i2 = 3i
√ – 20 = √20i2 = √45i2 = 2i√5
Multiplying Square Roots of
Negative Numbers
Any time multiplication of square roots
involves the square root of a negative
number, you MUST replace the negative
with i2 before doing any computation.
√6  √ – 3 = √6  √3i2 = √18i2 = √92i2 =
3i√2
√– 2  √ – 8 = √2i2  √8i2 = √16i4 = 4i2 =
–4
Solving Equations in the
Complex Numbers
x2 + 4 = 0
Remember this equation that we used to
show why a sum of two squares never
factors in the reals?
x2 = - 4  √x2 = √-4
x =  √-4 =  √4i2 =  2i
Complex solutions always come in
conjugate pairs.