BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.
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BMayer@ChabotCollege.edu • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
§7.7 Complex§7.7 ComplexNumbersNumbers
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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 .
BMayer@ChabotCollege.edu • 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”
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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.
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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)
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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.
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt31
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
BMayer@ChabotCollege.edu • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt32
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • 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
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