1 Simplify Problems With Imaginary Roots Standards 5 & 6 Equation That Results In Imaginary Roots...

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Simplify Problems With Imaginary Roots

Standards 5 & 6

Equation That Results In Imaginary Roots

COMPLEX NUMBERS

Complex Numbers In Context

END SHOW

Imaginary Roots

Graph and Add Complex Numbers

Simplify Expressions With Complex Conjugates

A Problem Of Iterations

Simplify Expressions With Complex Numbers

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

http://www.mrperezonlinemathtutor.com/

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STANDARD 5:

Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.

STANDARD 6:

Students add, subtract, multiply, and divide complex numbers.

ALGEBRA II STANDARDS THIS LESSON AIMS:



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ESTÁNDAR 5:

Los estudiantes demuestran conocimiento de como los números reales y complejos están relacionados aritméticamente y gráficamente. En particular, ellos pueden graficar números como puntos en el plano.

ESTÁNDAR 6:

Los estudiantes suman, restan, multiplican, y dividen números complejos.



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Let’s remember the case where the INDEX is EVEN and the RADICAND NEGATIVE:

Standards 5 & 6

- 4 = (-1)(4)

= (-1) 4

= i 4= 2i



5

IN GENERAL:

Standards 5 & 6

- b = (-1)(b)

= (-1) b

= i b

(-1)i =

i = -1

2



6

Standards 5 & 6

a = bn a a a a = b.... n b = a

n

EVEN

ODD

b > 0, positive b < 0, negative

n b+

n b–

n b+

None negative

i n b

n b–

None positive

one

one

one

one

POWERS FACTORS ROOTS



7

Standards 5 & 6Simplify:

– 63 = (–1)(9)(7)

= –1 9 7

= i 3 7

= 3i 7

–200x y3 = (–1)(100) x 2 x y2

= –1 100 x 2xy2

= i 10 x 2xy

= 10ix 2xy

6i 9i= 54i2

= 54( )2–1

= 54(–1)

= –54



8

Standards 5 & 6Simplify:

–2 –8 –5 = (–1)(2) (–1)(4)(2) (–1)(5)

= –1 2 –1 4 2 –1 5i i i

= i i 2 4 5

2

2–1

= i(–1)(4) 5

= – 4i 5

4i77 = 4i i76 1

= 4(i ) i2 38

= 4(–1 ) i38

= 4(1) i

= 4i



9

Standards 5 & 6Solve: x +2x + 4 = 02

x +2x + 4 = 02 Using the quadratic equation:

a= 1b= 2c= 4

x=-( ) ( ) - 4( )( )

2( )

2+_

1

1 2 2 4

x= – 2 4 – ( 4 )( 4 )

2

+_

– 2 -12x=

2

+_

– 2 ( -1)(4)(3)x=

2

+_

– 2 -1 4 3 x=

2

+_

– 2 ( i )(2) 3 x=

2

+_

–1 + i 3x= –1 – i 3x=



10


– 1 + i 3

Checking the solution:

( ) +2( ) + 4 = 02

1 – 2i 3 + i 3 – 2 + 2i 3 + 4 = 0

22

– 1 + i 3

1 –2i 3 – 3 – 2 + 2i 3 + 4 = 0

0 = 0

– 1 – i 3( ) +2( ) + 4 = 02

1 + 2i 3 + i 3 – 2 – 2i 3 + 4 = 0

22

– 1 – i 3

1 +2i 3 – 3 – 2 – 2i 3 + 4 = 0

0 = 0

True

True




x +2x + 4 = 02 Using the quadratic equation:

a= 1b= 2c= 4

x=-( ) ( ) - 4( )( )

2( )

2+_

1

1 2 2 4

x= – 2 4 – ( 4 )( 4 )

2

+_

– 2 -12x=

2

+_

– 2 ( -1)(4)(3)x=

2

+_

– 2 -1 4 3 x=

2

+_

– 2 ( i )(2) 3 x=

2

+_

–1 + i 3x= –1 – i 3x=PRESENTATION CREATED BY SIMON PEREZ. All rights reserved


12

Standards 5 & 6

0 1 2 3 4 5 6 7 8 9 10 11 12-1-2-3-4-5-6-7-10-11-12 -8-9

WHOLE NUMBERS

NATURAL NUMBERS

POSITIVE INTEGERS

INTEGERS

NEGATIVE INTEGERS

THE NUMBER LINE

NATURAL NUMBERS: 1, 2, 3, 4, …

WHOLE NUMBERS: 0, 1, 2, 3, 4, …

POSITIVE INTEGERS: 1, 2, 3, 4, …

NEGATIVE INTEGERS: -1, -2, -3, -4, …

INTEGERS: …, -3, -2, -1, 0, 1, 2, 3, …

REVIEW:



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REAL NUMBERS

R= reals

I

I= irrationals

Q

Q= rationals

Z

Z= integers

W

W= Wholes

N

N= naturals

Standards 5 & 6



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Standards 5 & 6

COMPLEX NUMBERS

a + bi

then

REAL NUMBERS

then

IMAGINARY NUMBERS

then

PURE IMAGINARY

then

NON-PURE IMAGINARY

If b = 0 If b = 0

RATIONALES IRRATIONALS If a = 0 If a = 0

4i

- 7i3 + 2i

0.25

7

25

14

=

71

=

5=

-8 -8 1

=

51

=

2

3.1416

1.4142

5 - 7i



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Standards 5 & 6

(10 + 5i) + 6i

SIMPLIFY:

= 10 + 5i + 6i

= 10 + (5 + 6)i

= 10 + 11i

(1 + i) + (3i – 5)

= (1 + i) + (–5 + 3i)

= 1 + i –5 + 3 i

= 1 – 5 +1i + 3i

= – 4 + (1 + 3)i

= – 4 + 4i

Grouping like terms

Distributive property

Grouping like terms




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Standards 5 & 6SIMPLIFY:

(4 – 7i) – (9 + i)

Grouping like terms


= 4 – 7i –9 – i

= 4 – 9 –7i – 1i

= –5 + (–7 – 1)i

= –5 + (–8)i

= –5 –8i



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4i

a

b

Imaginary

Real

i

Standards 5 & 6

Graph:

(-5, -7)- 5 - 7i

3 + 2i(3, 2)4i

(0, 4)

- 5 - 7i

3 + 2i

Complex numbers comply with vector’s properties.



184i

a

b

Imaginary

Real

i

Standards 5 & 6

– 5 – 7i

Now let’s add two of them graphically:

( ) + ( ) = - 5 – 3i

- 5 – 3i

(-5,-3)

• Draw parallel vectors.

• Draw a vector from the origin over the diagonal of the resulting parallelogram.

(-5, -7)- 5 – 7i

4i(0, 4)



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Standards 5 & 6Find x and y so the equation is true:

4 + 4i = 6x + 3yi

4 + 4i = 6x + 3yi

For two complex numbers to be equal the real part is equal in both of them, and the imaginary part is equal in both of them as well.

4 = 6x 4i = 3yi6 6 3i 3i

x = 46

x = 23

y = 43 Check:

4 + 4i = 6( ) + 3( )i

23

43

4 + 4i = + i12 3

12 3

4 + 4i = 4 + 4i true



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(5 – 3i)(2 + 4i)

F O I L

(5)(2) + (5)(4i) + (-3i)(2) + (-3i)(4i)

10 + 20i – 6i – 12i2

10 + 14i –12(-1)

10 + 14i + 12

22 + 14i


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(5 – 3i)(5 + 3i) = 25 – (3i)2

= 25 – 9i2

Difference of squares

COMPLEX CONJUGATES

= 25 – 9(-1)

= 25 + 9

= 34

The product of COMPLEX CONJUGATES is always a REAL NUMBER.


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Standards 5 & 6

1 – 3i 2 1 + 3i 2

4i –3 5 4i + 3 5

2 – 3i 2 + 3i

5 2 – i 5 2 + i

6 + 5i 6 – 5i

Complete the complex conjugates for the following:


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Standards 5 & 64

6 – 5i

4

6 – 5i 6 + 5i6 + 5i

36 – 25i

4(6 + 5i) 2

36 – 25i

24 + 20i2

36 – 25(-1)

24 + 20i

36 + 25

24 + 20i

61

24 + 20i

SIMPLIFY:

Multiply both numerator and denominator by the COMPLEX CONJUGATE of the denominator.

Difference of squares.


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Standards 5 & 6

3i9i – 1 SIMPLIFY:

3i9i – 1 i

i

Multiply both numerator and denominator by i to obtain i at the denominator and eliminate the imaginary part there.

2

3i9i – i 2

2

3(-1)

9(-1) – i

-3

-9 – i

3 + i 13


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Standards 5 & 6Iterate three times the following function with the initial value of x = 2i:f(x) = x + 12

f(2i) = ( ) + 1

22i

= 4i + 12

= 4(-1) + 1

= - 4 + 1

= -3 f(-3) = ( ) +12-3

= 9 + 1

= 10 f(10) = ( ) +1210

= 100 + 1

= 101

First Iteration:

Second Iteration:

Third Iteration:


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