A2-T UNIT 5 PART II IMAGINARY NUMBERS (DAY 5)

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A2-T UNIT 5 PART II

IMAGINARY NUMBERS (DAY 5)

Write the following in simplest radical form.

1) 8 2) 20 3) 75

Solve the following and express you answers in simplest radical form.

4) x2 – 32 = 0 5) x2 – 98 = 0

6) x2 – 54 = 0 7) x2 + 28 = 0

The reason for the name “imaginary” number is that when these numbers were first

proposed several hundred years ago, people could not “imagine” such a number. It is

said that the term “imaginary” was coined by Rene Descartes in the seventeenth century

and was meant to be a derogatory reference since, obviously, such numbers did not exist.

Today, we find the imaginary unit being used in mathematics and science. Electrical

engineers use the imaginary unit (which they represent as j) in the study of electricity.

Imaginary numbers occur when a quadratic equation has no roots in the set of real

numbers.

Solve the following for x: x2 + 1 = 0

To simplify notation, mathematicians use the lower-case letter i to represent the square

root of negative one, which is called the imaginary unit.

i = 1−

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Rewrite the following in terms of i and in simplest form.

8) 25− 9) 36− 10) 11−

11) 2 49− 12) 3 8− 13) -4 48−

Rewrite the following as a radical.

14) 6i 15) 7i 16) 10i

Solve for x in simplest radical form.

17) x2 + 64 = 0 18) x2 + 125 = 0 19) 3x2 + 96 = 0

CHANGE SETTINGS IN CALC

Go to MODE � change from REAL to a + bi (2nd row from bottom)

• This will NOT effect any other answers but allows you to get Imaginary answers.

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POWERS OF i Simplify the following.

1) 2)3( 2) 2)4( 3) 2)1( −

Higher powers of i

5) i3 = (i2) i = 6) i4 = 7) i5 = 8) i6 =

Fill in the following table.

i0 =

i1 =

i2 =

i3 =

i4 =

i5 =

i6 =

i7 =

i8 =

i9 =

i10 =

i11 =

i12 =

i13 =

i14 =

i15 =

RULE FOR FINDING HIGHER POWERS

Divide power by 4 (repeats in cycle of 4)

and use the remainder to determine the

new i power.

Once new i power is determined use calc

to find the value.

Examples (÷by 4)

Decimal

value

Equivalent

Fraction

(w/den of 4

always)

New poweri it

becomes,

put in calc 28i

33i

42i

51i

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Determine the value of the following:

9) i80 = 10) i125 = 11) i72 =

12) i45 = 13) i90 = 15) =215i

14) If f(x) = x3 – 2x2, then f(i) is equivalent to:

(1) -2 + i (2) -2 – i (3) 2 + i (4) 2 – i

15) Expressed in simplest form, i 16 + i 6 – 2i 5 + i 13, is equivalent to

(1) 1 (2) -1 (3) i (4) - i

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OPERATIONS WITH IMAGINARY NUMBERS (DAY 6)

If there is a negative under the radical, you MUST:

1) 8− + 18− 2) 4− - 25−

3) 8− • 2− 4) 16− ÷÷÷÷ 9−

5) 2 9− + 144− 6) 2 6− • 8−

7) 33

279

−

−− 8) 30075227 −−−+−

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COMPLEX NUMBERS (DAY 7)

Standard Form: a + bi

Real Component Imaginary Component

Two complex numbers are equal if and only if their real parts are equal and their

imaginary parts are equal.

If x + yi = 2i – 3, then x = -3 and y = 2

Adding and Subtracting Complex Numbers

Addition: (a + bi) + (c + di) =

Subtraction: (a + bi) – (c + di) =

Add or subtract the following complex numbers and simplify.

1) (-8 + 5i) + (5 – 7i) 2) (8 + 9− ) – (10 + 4− )

3) Subtract 6 + 14i from -9 – 3i 4) (5 – 50− ) – (-2 + 162− )

5) (3 + 4i) + ( 7 + 8i) 6) (2 + i) – (7 – 2i)

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GRAPHING COMPLEX NUMBERS

Every complex number may be represented as a point or as a vector on a complex

number plane. A complex number plane is similar to the real number plane (x, y) except

you graph the numbers (a, b).

For example: 3 + 2i

Point Vector

Graph the following complex numbers below:

1) -2 – i

2) –5 + 4i 1) 2)

3) –2i +1

4) 4

5) 2i

3) 4) 5)

a

bi

a

bi

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Graph the sum or difference of each following complex numbers.

6) (2 + 3i) + (3 - i) 7) (1 + 3i) – (3 + 2i)

8) (-2 + i) + (5 – 4i) 9) (2 – i) – (-1 + 3i)

10) On a stamp honoring the German mathematician Carl Gauss, several complex

numbers appear. The accompanying graph shows two of these numbers. Express

the sum of these numbers in a + bi form.

11) In which quadrant does the

difference (-5 + 11i) - (-2 + 7i) lie?

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MULTIPLICATION OF COMPLEX NUMBERS (DAY 8)

First remember that I = 1− and that i2 = -1.

Multiply and write your answer in simplest a + bi form.

1) (3 – 4i) (3 – 4i) 2) (3 – 4i) (6 + i)

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

5) )252)(163( −+−− 6) )12)(48( −+−+

7) (4 – 4i)2 8) (7 + 5i)2

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9) (6 - )26)(2 −+− 10) )73)(72( −+−−

11) (4 - 24− ) (2 + 54− ) 12) (1 + 18− )( 5 - 72− )

13) What is the product of 5 + 36− and 1 - 49− , expressed in simplest a + bi form?

(1) -37 + 41i (2) 5 – 71i (3) 47 + 41i (4) 47 – 29i

14) The expression (-1 + i)3 is equivalent to

(1) -3i (2) -2 – 2i (3) -1 – I (4) 2 + 2i

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RATIONALIZING DENOMINATORS AND DIVIDING COMPLEX NUMBERS (DAY 9)

Reason for Rationalizing the denominator:

To rationalize the denominator we will multiply the numerator and denominator by the

_______________ of the denominator.

Determine the conjugate of the following.

1) 4 + 6i 2) 1 – 7i 3) 5 – 11i

Rationalize the following and express your answers in simplest a + bi form.

4) i2

i1

+

+ 5)

i3

i312 −

6) i3

i2

+

− 7)

i54

i1

−

−

8) i4

i3 − 9)

i32

7

−

10) 33

32

−+

−+

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COMPLEX ROOTS OF A QUADRATIC EQUATION (DAY 10)

Quadratic Formula:

You have worked with the quadratic formula to find the roots of quadratic

equations earlier in this unit. Only this time the discriminant, b2 – 4ac, is going to be a

negative number. This will cause the value of the roots to become imaginary,

since i = 1− .

Simplify and write your answer in simplest a + bi form.

1) 3

369 −± 2)

2

644 −±

3) 14

987 −±− 4)

4

488

−

−±

Solve the following quadratic equations and express in simplest a + bi form.

5) x2 = 8x - 17

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6) 4x(x – 2) + 5 = 0

7) 2x

3x2 −=+

8) For what values of c does the equation 2x2 + 3x + c = 0 have imaginary roots.

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RELATIONSHIPS BETWEEN ROOTS AND COEFFICIENTS

OF QUADRATIC EQUATIONS (DAY 11)

Find the roots of the equation: x2 – 4x – 12 = 0

Write a quadratic equation whose roots are -5 and 4.

What conclusions can be made from the coefficients and the roots of the equation?

So, if we know the roots, could there be a shorter way to obtain the quadratic equation?

1) Write a quadratic equation whose roots are 1 ± 2i.

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2) What is the product of the roots of the equation –2x2 + 3x + 8 = 0?

3) What is the sum of the roots of the equation –x2 = 3x + 4?

4) For which equation does the product of the roots equal the sum of the roots?

(1) x2 – 2x + 3 = 0

(2) 4x2 – 4x + 1 = 0

(3) x2 – 13x + 13 = 0

(4) 5x2 – 5x = 0

5) If the sum of the roots of x2 + kx – 3 = 0 is equal to the product of the roots, what is

the value of k?

6) What is the value of c if x2 – 4x + c = 0 and the roots of the equation are 2 ± i?

7) Find the result if the sum of the roots of 2x2 – 3x + 7 = 0 is added to the product

of its roots.

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8) If the product of the roots of x2 + 4x – 6 = 0 is subtracted from the sum of its roots, find

the result.

9) If the roots of a quadratic equation are 5 ± 2i, what is the equation?

10) Write a quadratic equation with roots of 1 + 6 and 1 - 6 .

11) If one root of the equation x2 – 3x + c = 0 is 5, what is the other root?