Complex Numbers Basic Concepts of Complex Numbers …

[Pages:26]Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Identify the number as real, complex, or pure imaginary. 2i

The complex numbers are an extension of the real numbers. They include numbers of the form a + bi where a and b are real numbers. Determine if 2i is a complex number. 2i is a complex number because it can be expressed as 0 + 2i, where 0 and 2 are real numbers. The complex numbers include pure imaginary numbers of the form a + bi where a = 0 and b 0, as well as real numbers of the form a + bi where b = 0. Choose the correct description(s) of 2i: complex and pure imaginary.

Express in terms of i.

-196

First, write ? 196 as ? 1 times 196.

-196 =-1 (196 ) Next, write -1(196) as the product of two radicals.

-1 (196 )= 196-1

Finally, simplify each radical.

196 -1=14 i

Write the number as a product of a real number and i. Simplify all radical expressions.

-19

First, write ? 19 as ? 1 times 19.

-19=(-1)? 19 Next, write (-1)? 19 as the product of two radicals.

(-1)? 19 = -1 ? 19 Simplify -1 .

-1 19=i 19

Express in terms of i.

-150

(Work out same as above.)

5i 6

Express in terms of i.

--50

--50 = -5 i 2

Solve the equation. x2=-25

Take the square root of both sides.

x=? -25

Rewrite the square root of the negative number.

x=? i 25

Simplify: x=? 5 i The solutions are x = 5i, ? 5i.

Solve the quadratic equation, and express all complex solutions in terms of i.

x2=4 x-20

First, write the equation in standard form.

x2 ? 4 x+20=0

Use the quadratic formula to solve.

-b ? b2 ? 4ac

2a

a = 1, b = ? 4, c = 20

Substitute these values into the quadratic formula to get

-(-4)? (-4)2 ? 4(1)(20)

2(1)

Simplify the radical expression.

x=

4

?8 2

i

2 + 4i, 2 ? 4i.

Multiply.

-8 ? -8

Recall that -8=i 8 .

First, rewrite ? 8 as ? 1 ? 8.

-8 ? -8 = -1(8)-1(8)

Split into several radicals.

-1(8)-1(8) = -1 8 -1 8

Simply each radical, if possible.

-1 8 -1 8 = (i)(8)(i)(8)

Multiply.

(i)(8)(i)(8) = i2( 88)=-8

Divide.

-192 -64

-192 -64

=

3

Divide.

-175 7

Notice that the expression in the numerator is imaginary.

-175=i 175

Then simplify the radical.

i 175=5i 7

5i 7 7

Reduce the fraction to lowest terms by dividing out the common

factor, 7 .

5i 7 7

= 5i

Add and simplify. (7+5i)+(2 ? 4 i)

(7+5i)+(2 ? 4 i) = 9 + i

Multiply. (7+ 8 i )(5+i )

(7+8i)(5+i) = 47 i+27

Multiply. (-6+9 i )2

(-6+9 i)2 = -108 i-45

Multiply.

( 10+i )( 10-i )

(10+i)(10-i) = 11

Simplify. i17

Since 17 is odd, rewrite 17 as 16 + 1 and simplify. i^17 = i^{16+1} = i^16 ? i

i17=i16+1=i16 ? i Write 16 as 2(8).

i16 ? i =i2(8) ? i Write i2(8) as power of i2 .

i2(8)? i=(i2)8? i Remember that i2=-1 and simplify.

(i2)8? i=(-1)8? i Remember that a negative number raised to an even power is positive.

(-1)8? i =1? i = i

Simplify. i34

i34=-1

Simplify. i15

i15=-i

Find the power of i.

i-17

Use the rule for negative exponents.

a-m=

1 am

i-17=

1 a 17

Because i2 is defined to be ? 1, higher powers of i can be found. Larger powers of i can be simplified by using the fact that i4=1 .

1 i 17

=

i4

?

i4

1 ?i4

?

i

4

?

i

Since

i4=1 ,

1 i4? i4?i4?i4 ?i

=

1 i

To simplify this quotient, multiply both the numerator and denominator by -i , the conjugate of i.

1 i

=

1 (-i ) i (-i)

-i -i2

-i -(-1)

=

-i

Divide.

8+4 i 8-4 i

Reduce.

2+i 2-i

Multiply by a form of 1 determined by the conjugate of the denominator.

2+i 2-i

=

2+i 2-i

22

+i +i

=

4

+4 i+i2 4-i 2

Remember that i2=-1 and simplify.

4

+4 i+i2 4-i2

=

4+4 i+(-1) 4-(-1)

4+44+i1-1=

3+4 5

i

Write in the form a + bi.

3+ 4 i 5

=

3 5

+

4 5

i

Trigonometric (Polar) Form of Complex Numbers The Complex Plane and Vector Representation Trigonometric (Polar) Form Converting Between Trigonometric and Polar Forms An Application of Complex Numbers to Fractals

Graph the complex number as a vector in the complex plane. -4- 7 i

In the complex plane, the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis. The real part of the complex number is ? 4. The imaginary part of the complex number is ? 7. Graph the ordered pair (-4,-7) . Draw an arrow from the origin to the point plotted.

Write the complex number in rectangular form. 18(cos 180? + i sin 180?)

Rewrite the equation in the form a + bi. 18(cos 180? + i sin 180?) = 18 cos 180? + (18 sin 180?)i a = 18 cos 180? b = 18 sin 180? Simplify. a = 18 cos 180? a = (18)( ? 1) a = ? 18 Simplify. b = 18 sin 180? b = 18 ? 0

b = 0 18(cos 180? + i sin 180?) = ? 18 + 0i = ? 18

Write the complex number in rectangular form. 8(cos(30?) + i sin (30?))

4 3+4 i

Write the complex number in rectangular form. 14 cis 315?

Rewrite the equation.

14 cis 315? = 14(cos 315? + i sin 315?)

14(cos 315? + i sin 315?) = 14 cos 315? + (14 sin 315?)i

a = 14 cos 315? b = 14 sin 315?

Simplify.

a = 14 cos 315?

a=14

?

2

2

a=7 2

Simplify.

b = 14 sin 315?

b =

14

?-

2

2

b = -72

14 cis 315?=7 2 ? 7 2 i

Find trigonometric notation. 5 ? 5i

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