Complex Numbers 22 11 Definitions Graphing 33 Absolute Values.
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Transcript of Complex Numbers 22 11 Definitions Graphing 33 Absolute Values.
Complex Numbers
2
1Definitions
Graphing
3Absolute Values
2
Imaginary Number (i)
Defined as:
Powers of i
1i
1i
12 i
ii 3
14 i
Complex Numbers
A complex number has a real part & an imaginary part.
Standard form is:
bia
Real part Imaginary part
Example: 5+4i
4
Definitions
Pure imaginary number Monomial containing i
Complex Number An imaginary number combined with a real
number Always separate real and imaginary parts
ii
5
3
5
2
5
32
The Complex plane
Imaginary Axis
Real Axis
Graphing in the complex plane
i34 .
i52 .i22 .
i34
.
Absolute Value of a Complex Number
The distance the complex number is from the origin on the complex plane.
If you have a complex number the absolute value can be found using:) ( bia
22 ba
Examples
1. i52
22 )5()2(
254 29
2. i622 )6()0(
360
366
9
Simplifying Monomials
Simplify a Power of i Steps
Separate i into a power of 2 or 4 taken to another power
Use power of i rules to simplify i into -1 or 1 Take -1 or 1 to the power indicated Recombine any leftover parts
10
Operations
Simplify a Power of iSimplify
11
Simplifying Monomials Example Square Roots of Negative NumbersSimplify
12
Addition & Subtraction
Add and Subtract Complex Numbers Treat i like a variableSimplify
ii 4523
ii 4523
i22
ii 3146
ii 3146
i7
Ex: )33()21( ii
ii 3231 i52
Ex: )73()32( ii )73()32( ii
i41
Ex: )32()3(2 iii iii 3223
i21
Addition & Subtraction Examples
)7332 ii
14
Multiplying Complex Numbers Multiply Pure Imaginary Numbers Steps
Multiply real parts Multiply imaginary parts Use rules of i to simplify imaginary parts
15
Monomial Multiplication Example
Multiply Pure Imaginary NumbersSimplify
16
Multiplication Example
Multiply Complex NumbersSimplify ji 5731
)57(3)57(1 iii 2152157 iii 2152157 iii
)1(152157 ii152157 ii
i1622
17
Solving ax2+b=0
Equation With Imaginary SolutionsSolve
Note: ± is placed in the answer because both 4 and -4 squared equal 16
Multiply the numerator and denominator by the complex conjugate of the complex number in the denominator.
7 + 2i3 – 5i The complex conjugate
of 3 – 5i is 3 + 5i.
Multiplying Complex Numbers
19
Dividing Complex Numbers
Divide Complex Numbers No imaginary numbers in the
denominator! i is a radical
Remember to use conjugates if the denominator is a binomial
Simplify
i
i
i
iEx
21
21*
21
113 :
)21)(21(
)21)(113(
ii
ii
2
2
4221
221163
iii
iii
)1(41
)1(2253
i
41
2253
i
5
525 i
5
5
5
25 i
i 5
21
Division Example
Simplify
7 + 2i3 – 5i
21 + 35i + 6i + 10i2
9 + 15i – 15i – 25i221 + 41i – 10
9 + 25
(3 + 5i)(3 + 5i)
11 + 41i 34
Try These.
1. (3 + 5i) – (11 – 9i)
2. (5 – 6i)(2 + 7i)
3. 2 – 3i 5 + 8i
4. (19 – i) + (4 + 15i)
Try These.
1. (3 + 5i) – (11 – 9i) -8 + 14i
2. (5 – 6i)(2 + 7i) 52 + 23i
3. 2 – 3i –14 – 31i 5 + 8i 89
4. (19 – i) + (4 + 15i) 23 + 14i