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Imaginary and Complex Numbers
15 October 2010
Question:
If I can take the , can I take the ?
Not quite….
25 25
Answer???
But I can get close!
1512512525
= Imaginary Number or
????1
1 i
12 i
Simplifying with Imaginary Numbers
Step 1: Factor out -1 from the radicand (the number or expression underneath the radical sign)
Step 2: Substitute for
Step 3: If possible, simplify the radicand
i 1
Example: 8
Your Turn:
1. 2.
3. 36
2 12
*Powers of
4
3
2
1
i
i
i
i
4
3
2
)1(
)1(
)1(
1
i
1
1
i
i
*Powers of , your turn:
Observations?
i
Simplifying Powers of i
To simplify a power of i, divide the exponent by 4, and the remainder will tell you the appropriate power of i.
Example: i54
54 ÷ 4 = 13 remainder 2 i54 = i2 = -1
Complex Numbers
Real Part Imaginary Part
a + bi
where a and b are both real numbers, including 0.
Complex Number System
Complex Numbers: -5, -√3, 0, √5, ⅝, ⅓, 9, i, -5 + -4i
Imaginary Numbers:
-4i
2i√2
√-1
i
Real Numbers: -5, -√3, 0, √5, ⅝, ⅓, 9
Irrational Numbers
-√3
√5
Rational Numbers: -5, 0, ⅓, 9
Integers: -5, 0, 9
Whole Numbers: 0, 9
Natural Numbers: 9
The complex numbers are an algebraically closed set!
Writing Complex Numbers
Step 1: Simplify the radical expression Step 2: Rewrite in the form a + bi.
Examples:
i
i
36
63
619
619
69
Examples, cont.
327
327
347
347
1127
1127
127
i
i
i
i
Your Turn:
1. 2.
3. 4.
481 361
726 11136
Operations on Complex Numbers
Operations (adding, subtracting, multiplying, and dividing) on complex numbers are the same as operations on radicals!!! Remember: the imaginary number is really just a
radical with a negative radicand.
Operations on Complex Numbers, cont.
Addition and Subtraction You can only add or subtract like terms.
Translation: You must add or subtract the real parts and the imaginary parts of a complex number separately.
Step 1: Distribute any negative/subtraction signs. Step 2: Group together like terms. Step 3: Add or subtract the like terms.
Addition Example
)4()42( ii
Subtraction Example
)88(6 i
Your Turn:
1.
2.
3.
4.
)5()97( ii )24()53( ii
)31()76( ii )(2)512( ii
Operations on Complex Numbers, cont.
Multiplication Doesn’t require like terms!
Translation: You can multiply real parts by imaginary parts and imaginary parts by real parts!
Multiply complex numbers like you would multiply expression with radicals. Monomials: Multiply everything together. Binomials: FOIL!
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