Complex Numbers

Once upon a time…

1 no real solution

Complex Number System

Reals

Rationals(Can be written as fractions)

Integers(…, -1, -2, 0, 1, 2, …)

Whole(0, 1, 2, …)

Natural(1, 2, …)

Irrationals(Cannot be written as a

fraction),

Pure Imaginary

, , , etc.

Since all number belong to the Complex number field, C, all number can be classified as complex. The Real number field, R, and the imaginary numbers, i, are subsets of this field as illustrated below.

Real Numbersa + 0i

Pure Imaginary Numbers0 + bi

Complex Numbersa + bi

A Little History Math is used to explain our universe. When a recurring phenomenon is seen and can’t be explained by our present mathematics, new systems of mathematics are derived.In the real number system, we can’t take the square root of negatives, therefore the complex number system was created.Complex numbers revolutionized computer graphics

-In the set of real numbers, negative numbers do not have square roots.

-Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions.

-These numbers were devised using an imaginary unit named .

1i

-The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1.

-The first four powers of i establish an important pattern and should be memorized.

Powers of i1 2 3 41 1i i i i i i

Divide the exponent by 4No remainder: answer is 1.remainder of 1: answer is i.

remainder of 2: answer is –1.remainder of 3:answer is –i.

Divide the exponent by 4No remainder: answer is 1.remainder of 1: answer is i.

remainder of 2: answer is –1.remainder of 3:answer is –i.

i4 1

i

i2 1

i i3

Powers of i

1.) Find

2.) Find

3.) Find

4.) Find

i1i1

1.) 5 1 5 5i

1 7 7i

1 99

3 11i

Simplify.3.)

2.) 7 4.)

3.) 995.)

-Express these numbers in terms of i.

¿ 𝑖√9√11

You try…

6.

7.

7

36

1608.

i 7

6i

4 10i

94i

22 5i 2 52 21i

( 1) 21 21

Multiplying47 2i

2 5i

3 7

2 1 5i 2 5i i

i i3 7

9.

10.

11.

Complex Numbers Day 2

Recap:

1)1()(

)1(

1)1(

2224

23

22

ii

iiiii

i

a + bi

Complex Numbers

real imaginary

The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

7.) 7 9i i 16i

8.) ( 5 6 ) (2 11 )i i 3 5i

9.) (2 3 ) (4 2 )i i 2 3 4 2i i 2 i

Add or Subtract

12.

13.

14.

Multiplying & Dividing Complex Numbers

REMEMBER: i² = -1

12 2i 12( 1) 12

2 27 i 49( 1) 49

Multiply3 4i i

27i

1)

2)

You try…3)

4)

7 12i i

211i

284i )1(84 84

2211 i )1(121 121

28 8i 21i 26i228 29 6i i

28 29 6( 1)i 28 29 6i 22 29i

ii 2734 5)

Multiply

You try… ii 1032 6)

2103206 iii

i1716

210176 ii 110176 i

10176 i

25 35i 35i 249i

25 49( 1)

25 49

74

You try…

ii 7575 7)

Conjugate-The conjugate of a + bi is a – bi

-The conjugate of a – bi is a + bi

Find the conjugate of each number…

3 4 i 3 4 i

4 7i 4 7i

5i 5i

6 6

8)

9)

10)

11)

iiBecause 06 as same theis 06

1

1

i

i

2

14 4

1

i

i

14 4

2

i 7 2i

Divide…

5 9

1

i

i

12)

5 5 9 9

1

2

2

i i i

i i i

3 5

3 5

i

i

2

9 19

9 25

i

i

9 19

34

i

2 3

3 5

i

i

13)You try…

6 10 9 15

9 15 15 25

2

2

i i i

i i i