bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

[email protected] • MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§7.7 Complex§7.7 ComplexNumbersNumbers



Review §Review §

Any QUESTIONS About• §7.6 → Radical Equations

Any QUESTIONS About HomeWork• §7.6 → HW-29

7.6 MTH 55



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



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 .



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”



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



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



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



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



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



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



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



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



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



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.



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



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





SOLUTION: Perform Distributionb)

210 6i i

10 6 6 10i i





SOLUTION : Use F.O.I.L.c)

8 2 3i

11 2i



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



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)



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



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





SOLUTION: Eliminate i from DeNom by multiplying the Numer & DeNom by the Conjugate of 2−3i

NEXT SLIDE for Reduction




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





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



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



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.



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



All Done for TodayAll Done for Today

ElectricalEngrs Use j instead

of i

jj

j

i

23or 17 :Examples

DefEngr 1

DefMath 1



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



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



-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

bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

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