complex numbers any number in form a+bi, where a and b are real numbers and i is imaginary. what is...
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Complex Numbers
� Any number in form a+bi, where a and b are real numbers and i is imaginary.
What is an imaginary number?
Definition of imaginary numbers:
−b2 = b2 ⋅ −1=bi
i = −1
i2 =−1
Simplify complex numbers
Ex: −4i ⋅7i=−28⋅ i2 =−28⋅−1=Remember i
2 =−1
28
Ex#2: −8⋅ −5 =
i 8⋅ i 5 = Remember that −1=i
i2 ⋅ 40= −1⋅2 10=
−2 10
Ex#3: i19
i19 =i18⋅i i18⋅i = i2( )
9⋅ i
i2( )9⋅ i= −1( )
9⋅ i
Answer: -i
Try these problems:
1. −5i ⋅−3i =
2. 3 −6⋅ −3=
3. i13=
-15
9i 2
i
When adding or subtracting complex numbers, combine like terms.
Ex: 8−3i( ) + 2+5i( )8+2( ) + −3i+5i( )
10+2i
Try these on your own
1. (16−4i)−(12−3i)
2. (3i+7)+(34−6i)
3. (−4+7i)+(8−9i)
4. (−1−i)−(−3−4i)
ANSWERS:
1. 4−i
2. −3i+41
3. 4−2i
4. 2+3i
Multiplying complex numbers.
To multiply complex numbers, you use the same procedure as multiplying polynomials.
Examples to do together:
3⋅ −12=
−36= −1⋅ 36
6i
Lets do another example.
3−2i( ) 5+3i( )
15+9i−10i−6i2F O I L
15+9i−10i+6 i2 =−1Next
Answer:
Now try these:
21-i
1. −5⋅ 20
2. 4+5i( ) 4−5i( )
3. (3−2i)2
Next
Answers:
1. 10i
2. 41
3. 5−12i
Conjugates
In order to simplify a fractional complex number, use a conjugate.
What is a conjugate?
a b−c d and a b+c d
are said to be conjugates of each other.
Ex: 3 2i −5 and 3 2i +5
Lets do an example:
Ex: 8i
1+3i
8i1+3i
⋅1−3i1−3i
Rationalize using the conjugate
Next
8i−24i2
1+9=8i+24
10
4i +125
Reduce the fraction
Lets do another example
Ex: 4+i2i
4+i2i
⋅ii=
4i +i2
2i2
Next
4i +i2
2i2=
4i−1−2
Try these problems.
1. 3
2 −5i
2. 3- i2- i
1. 2 +5i9
2. 7+i
5