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[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§7.7 Complex§7.7 ComplexNumbersNumbers
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §7.6 → Radical Equations
Any QUESTIONS About HomeWork• §7.6 → HW-29
7.6 MTH 55
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt3
Bruce Mayer, PE Chabot College Mathematics
Imaginary & Complex NumbersImaginary & Complex Numbers Negative numbers do not have square
roots in the real-number system. A larger number system that contains the
real-number system is designed so that negative numbers do have square roots. That system is called the complex-number system.
The complex-number system makes use of i, a number that with the property (i)2 = −1
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt4
Bruce Mayer, PE Chabot College Mathematics
The “Number” The “Number” ii
i is the unique number for which i2 = −1 and so 1i
Thus for any positive number p we can now define the square root of a negative number using the product-rule as follows .
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt5
Bruce Mayer, PE Chabot College Mathematics
Imaginary NumbersImaginary Numbers
An imaginary number is a number that can be written in the form bi, where b is a real number that is not equal to zero
Some Examples
5 2973
37
ii
i
i is called the “imaginary unit”
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt6
Bruce Mayer, PE Chabot College Mathematics
Example Example Imaginary Imaginary NumbersNumbers Write each imaginary number as a
product of a real number and ia) b) c)16 21 32
SOLUTIONa) b) c)16
1 16
1 16 4i
21
1 21
1 21 21i
32
1 32
1 32 16 2i
4 2i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt7
Bruce Mayer, PE Chabot College Mathematics
ReWriting Imaginary NumbersReWriting Imaginary Numbers
To write an imaginary number in terms of the imaginary unit i:
n
1. Separate the radical into two factors 1 .n
2. Replace with i
3. Simplify
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt8
Bruce Mayer, PE Chabot College Mathematics
Example Example Imaginary Imaginary NumbersNumbers Express in terms of i:
a) b)
SOLUTIONa)
b)
1 9 3, or 3 .i i
1 16 3 4 3 4 3i i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt9
Bruce Mayer, PE Chabot College Mathematics
Complex NumbersComplex Numbers
The union of the set of all imaginary numbers and the set of all real numbers is the set of all complex numbers
A complex number is any number that can be written in the form a + bi, where a and b are real numbers. • Note that a and b both can be 0
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt10
Bruce Mayer, PE Chabot College Mathematics
Complex Number ExamplesComplex Number Examples
The following are examples of Complex numbers
7 2
12
3
11
i
i
i
Here a = 7, b =2.
Here 2, 1/3.a b
Here 0, 11.a b
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt11
Bruce Mayer, PE Chabot College Mathematics
The complex numbers:
a = bi
Complex numbers thatare real numbers:
a + bi, b = 0
Rational numbers:
Complex numbers thatare not real numbers:
a + bi, b ≠ 0
Irrational numbers:
Complex numbers (Imaginary)
2
3
, 0, 0 :
3 , , 17 ,...
a bi a b
i i i
32, , 7,...
2
, 7, 18, 8.7...3
Complex numbers
2 73 5
, 0, 0:
2 2 ,5 4 ,
a bi a b
i i i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt12
Bruce Mayer, PE Chabot College Mathematics
Add/Subtract Complex No.sAdd/Subtract Complex No.s
Complex numbers obey the commutative, associative, and distributive laws.
Thus we can add and subtract them as we do binomials; i.e.,• Add Reals-to-Reals
• Add Imaginaries-to-Imaginaries
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt13
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Add & Sub Complex Add & Sub
Add or subtract and simplify a+bi
(−3 + 4i) − (4 − 12i)
SOLUTION: We subtract complex numbers just like we subtract polynomials. That is, add/sub LIKE Terms → Add Reals & Imag’s Separately• (−3 + 4i) − (4 − 12i) = (−3 + 4i) + (−4 + 12i)
• = −7 + 16i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt14
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Add & Sub Complex Add & Sub
Add or subtract and simplify to a+bia) b)
SOLUTIONa)
b)
10 (2 8) 10 10i i
Combining real and imaginary parts
1 3i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt15
Bruce Mayer, PE Chabot College Mathematics
Complex MultiplicationComplex Multiplication
To multiply square roots of negative real numbers, we first express them in terms of i. For example,
6 5 1 6 1 5
6 5i i 2 30i
1 30 30.
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt16
Bruce Mayer, PE Chabot College Mathematics
Caveat Complex-MultiplicationCaveat Complex-Multiplication
CAUTIONCAUTION With complex numbers, simply
multiplying radicands is incorrect when both radicands are negative:
3 5 15. The Correct Multiplicative Operation
151515315311
51315131532
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt17
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Multiply Complex Multiply
Multiply & Simplify to a+bi forma) b) c)
SOLUTIONa)
2 10i i 2 20 1 2 5 2 5i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt18
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Multiply Complex Multiply
Multiply & Simplify to a+bi forma) b) c)
SOLUTION: Perform Distributionb)
210 6i i
10 6 6 10i i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt19
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Multiply Complex Multiply
Multiply & Simplify to a+bi forma) b) c)
SOLUTION : Use F.O.I.L.c)
8 2 3i
11 2i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt20
Bruce Mayer, PE Chabot College Mathematics
Complex Number CONJUGATEComplex Number CONJUGATE
The CONJUGATE of a complex number a + bi is a – bi, and the conjugate of a – bi is a + bi
Some Examples
231 Conjugate231
13 Conjugate13
ii
ii
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt21
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Conjugate Complex Conjugate
Find the conjugate of each number a) 4 + 3i b) −6 − 9i c) i
SOLUTION:a) The conjugate is 4 − 3i
b) The conjugate is −6 + 9i
c) The conjugate is −i (think: 0 + i)
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt22
Bruce Mayer, PE Chabot College Mathematics
Conjugates and DivisionConjugates and Division
Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators.
Note the Standard Form for Complex Numbers does NOT permit i to appear in the DENOMINATOR• To put a complex division into Std Form,
Multiply the Numerator and Denominator by the Conjugate of the DENOMINATOR
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt23
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Division Complex Division
Divide & Simplify to a+bi form
SOLUTION: Eliminate i from DeNom by multiplying the Numer & DeNom by the Conjugate of i
i
i
1
312232
2
i
i
ii
i32
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt24
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Division Complex Division
Divide & Simplify to a+bi form
SOLUTION: Eliminate i from DeNom by multiplying the Numer & DeNom by the Conjugate of 2−3i
NEXT SLIDE for Reduction
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt25
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Division Complex Division
SOLN2 3
2 3
i
i
2
2(4 )(2 3 ) 8 12 2 3
(2 3 )(2 3 ) 4 9
i i i i i
i i i
8 14 3 5 14
4 9 13
i i
5 14
13 13i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt26
Bruce Mayer, PE Chabot College Mathematics
Example Example Complex Division Complex Division
Divide & Simplify to a+bi form
SOLUTION: Rationalize DeNom by Conjugate of 5−i
3 5
5
i
i
3 5
5
i
i
53
5
5
5
i
i
i
i
2
2
15 3 25 5
25
i i i
i
15 3 25 5( 1)
25 ( 1)
i i
15 3 25 5
25 1
i i
10 28
26
i
10 28
26 26
i
5 14
13 13
i
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt27
Bruce Mayer, PE Chabot College Mathematics
Powers of Powers of ii → → iinn
Simplifying powers of i can be done by using the fact that i2 = −1 and expressing the given power of i in terms of i2.
The First 12 Powers of i
i 1
i2 1
i3 i2 • i 1 1
i4 i2 • i2 1• 11
i5 1
i6 1
i7 1 1
i8 1
i9 1
i10 1
i11 1 1
i12 1
• Note that (i4)n = +1 for any integer n
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt28
Bruce Mayer, PE Chabot College Mathematics
Example Example Powers of Powers of ii
Simplify using Powers of i a) b)
SOLUTION : Use (i4)n = 1a)
b)
= 1 Write i40 as (i4)10.
84 = i i
= 1 i = i
Write i32 as (i4)8.
Replace i4 with 1.
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt29
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §7.7 Exercise Set• 32, 50, 62, 78, 100, 116
Ohm’s Law of Electrical Resistance in the Frequency Domain uses Complex Numbers (See ENGR43) ZIV
Law sOhm' AC
Law sOhm' DC
r iv
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt30
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
ElectricalEngrs Use j instead
of i
jj
j
i
23or 17 :Examples
DefEngr 1
DefMath 1
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt31
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt32
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt33
Bruce Mayer, PE Chabot College Mathematics
Graph Graph yy = | = |xx||
Make T-tablex y = |x |
-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6
x
y
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt34
Bruce Mayer, PE Chabot College Mathematics
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls -5
-4
-3
-2
-1
0
1
2
3
4
5
-10 -8 -6 -4 -2 0 2 4 6 8 10
M55_§JBerland_Graphs_0806.xls
x
y