COMPLEX NUMBER COMPLEX NUMBER SYSTEMSYSTEM

1

COMPLEX NUMBER NUMBER OF THE FORM C= a+Jb a = real part of C b = imaginary part.

2

Definition of a Complex Number Definition of a Complex Number If a and b are real numbers, the number a + bi is a

complex number, and it is said to be written in standard form.

If b = 0, the number a + bi = a is a real number.

If a = 0, the number a + bi is called an imaginary number.

Write the complex number in standard form

81 81 i 241 i 221 i

Real NumbersImaginary Numbers

Real numbers and imaginary numbers are subsets of the set of complex numbers.

Complex Numbers

Conversion between Rectangular and polar form

Convert Between Form C = a + jb (Rectangular Form) C = C<ø ( Polar Form) C is Magnitude a = C cos ø and b=C sin ø where C = √ a2 + b2

ø = tan-1 b/a

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Complex Conjugates and Complex Conjugates and Division Division

Complex conjugates-a pair of complex numbers of the form a + bi and a – bi where a and b are real numbers.

( a + bi )( a – bi )a 2 – abi + abi – b 2 i 2

a 2 – b 2( -1 )a 2 + b 2

The product of a complex conjugate pair is a positive real number.

Complex PlaneComplex PlaneA complex number can be plotted on a plane with

two perpendicular coordinate axes The horizontal x-axis, called the real axis The vertical y-axis, called the imaginary axis

P

z = x + iy

x

y

O

Represent z = x + jy geometrically as the point P(x,y) in the x-y plane, or as the vector from the origin to P(x,y). OP

::::::::::::::

The complex plane

x-y plane is also known as the complex plane.

Complex plane, polar form of a complex number

x

y1tan

Pz = x + iy

x

y

O

Im

Re

θ

Geometrically, |z| is the distance of the point z from the origin while θ is the directed angle from the positive x-axis to OP in the above figure.

From the figure,

θ is called the argument of z and is denoted by arg z. Thus,

For z = 0, θ is undefined.

A complex number z ≠ 0 has infinitely many possible arguments, each one differing from the rest by some multiple of 2π. In fact, arg z is actually

The value of θ that lies in the interval (-π, π] is called the principle argument of z (≠ 0) and is denoted by Arg z.

0tanarg 1

z

x

yz

,...2,1,0,2tan 1

nn

x

y

Consider the quadratic equation x2 + 1 = 0. Solving for x , gives x2 = – 1

12 x

1x

We make the following definition:

1i

Complex Numbers

1i

Complex Numbers : power of j

12 iNote that squaring both sides yields:therefore

and

so

and

iiiii *1* 13 2

1)1(*)1(* 224 iii

iiiii *1*45

1*1* 2246 iiii

And so on…

Addition and Subtraction of Addition and Subtraction of Complex Numbers Complex Numbers

If a + bi and c +di are two complex numbers written in standard form, their sum and difference are defined as follows.

i)db()ca()dic()bia(

i)db()ca()dic()bia(

Sum:

Difference:

Perform the subtraction and write the answer in standard form.

( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i

234188 i

234298 ii

234238 ii

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Multiplying Complex NumbersMultiplying Complex Numbers

Multiplying complex numbers is similar to multiplying polynomials and combining like terms.

Perform the operation and write the result in standard form. ( 6 – 2i )( 2 – 3i )

F O I L12 – 18i – 4i + 6i2

12 – 22i + 6 ( -1 )6 – 22i

Consider ( 3 + 2i )( 3 – 2i )9 – 6i + 6i – 4i2

9 – 4( -1 )9 + 4 13

This is a real number. The product of two complex numbers can be a real number.

This concept can be used to divide complex numbers.

To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator.

dic

bia

dic

dic

dic

bia

22

2

dc

bdibciadiac

22 dc

iadbcbdac

Perform the operation and write the

result in standard form.

i

i

21

76

i

i

i

i

21

21

21

76

22

2

21

147126

iii

41

5146

i

5

520 i

5

5

5

20 i i4

ii

i

4

31 i

i

ii

i

i

i

4

4

4

31

Perform the operation and write the result in standard form.

222

2

14

312

i

i

ii116

312

1

1

ii

ii17

3

17

121 ii

17

3

17

121

i17

317

17

1217

i

17

14

17

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Expressing Complex NumbersExpressing Complex Numbers in Polar Form in Polar Form

Now, any Complex Number can be expressed as:X + Y i That number can be plotted as on ordered pair

in rectangular form like so…

6

4

2

-2

-4

-6

-5 5

Expressing Complex NumbersExpressing Complex Numbers in Polar Form in Polar Form

Remember these relationships between polar and

rectangular form: x

ytan 222 ryx

cosrx sinry

So any complex number, X + Yi, can be written inpolar form: irrYiX sincos

)sin(cossincos irirr

rcisHere is the shorthand way of writing polar form:

Expressing Complex NumbersExpressing Complex Numbers in Polar Form in Polar Form

Rewrite the following complex number in polar form: 4 - 2i

Rewrite the following complex number inrectangular form: 0307cis

Expressing Complex NumbersExpressing Complex Numbers in Polar Form in Polar Form

Express the following complex number inrectangular form: )

3sin

3(cos2

i

Expressing Complex NumbersExpressing Complex Numbers in Polar Form in Polar Form

Express the following complex number inpolar form: 5i

Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form

)sin(cos 111 ir

The product of two complex numbers, and

Can be obtained by using the following formula:)sin(cos 222 ir

)sin(cos*)sin(cos 222111 irir

)]sin()[cos(* 212121 irr

Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form

)sin(cos 111 ir

The quotient of two complex numbers, and

Can be obtained by using the following formula:)sin(cos 222 ir

)sin(cos/)sin(cos 222111 irir

)]sin()[cos(/ 212121 irr

Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form

Find the product of 5cos30 and –2cos120

Next, write that product in rectangular form

Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form

Find the quotient of 36cos300 divided by 4cis120

Next, write that quotient in rectangular form

Products and Quotients of Products and Quotients of Complex Numbers in Polar FormComplex Numbers in Polar Form

Find the result ofLeave your answer in polar form.

Based on how you answered this problem, what generalization can we make aboutraising a complex number in polar form toa given power?

4))120sin120(cos5( i