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# Imaginary and Complex Numbers

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by

Tweet## Ainsley Su-Williams

on 7 May 2013#### Transcript of Imaginary and Complex Numbers

By: Ainsley Su-Williams Imaginary and Complex Numbers Goals - Understand what an imaginary

number and a complex number is What is an Imaginary Number? Complex Numbers Complex numbers are a combination of a

real number and an imaginary number How to solve with a Large Exponent Questions Simplify: √-361, √-54 - Be able to solve simple problems/equations An Imaginary number is a number that can be written as a real number multiplied by the imaginary unit i. Pattern

i = -1

i = -i

i = 1

i = i

i = -1

i = -i

i = 1 i = i 0 1 i = 1 2 3 4 5 6 7 Starting from i, imaginary numbers follow a pattern:

i, -1, -i, 1 Looking at the pattern on the left, can you predict what i equals? 9 i = i 9 e.g. i 802 Steps:

1) Divide exponent by 4 (4 is the number of digits in the pattern,

i, -1, -i, 1)

2) Using remainder, determine the answer 802/4=200 R2 The remainder is 2, therefore i will be the same as i . i equals -1 802 Let's try One Simplify i 919 919/4=229 R3 The remainder is 3, therefore i will be the same as i . i equals -i 919 2 802 3 919 Simplifying Imaginary Numbers √-64 = √-1√64 = 8i Perfect Squares Simplifying numbers that aren't a perfect square These take longer to simplify. For example, √-20. √20 isn't fully simplified = √2*2*5 = 2√5 √-20= √-1√20 = i√20 √-20 equals 2i√5 How to Add, Subtract, Multiply and Divide Complex Numbers Examples:

1 + i

39 + 3i - Be able to simplify imaginary numbers 8 Adding (5 + 2i ) + (7 + 12i) Step 1) Group the real part of the complex number and the imaginary part of the complex number. (5 + 7) + (2i + 12i) Step 2) Combine the like terms and simplify The final answer is: 12+14i Subtraction

Step 1) Distribute the negative

Step 2) Group the real part of the complex number

and the imaginary part of the complex number.

Step 3) Combine the like terms and simplify (8 + 6i) - (5 + 2i) (8 + 6i) + (−5 − 2i) (8 + −5) + (6i + −2i) The final answer is 3+4i Multiplication (5 + 2i ) (7 + 12i) Step 1) Use F.O.I.L. (First, outer, inner, last) (a+bi)(c+di) = ac + adi + bci + bdi 2 5*7 + 5*12i + 2i*7 + 2i*12i 5*7=35

5*12i=60i

2i*7=14i

2i*12i=24i =-24 2 Step 2) Simplify by adding the terms (35+ -24) + (60i + 14i) The final answer is 11 + 74i Division (5 + 2i)

(7 + 4i) Step 1) Find the conjugate of the denominator The conjugate of (7 + 4i) is (7 - 4i) (FLIP THE SIGN) Step 2) Multiply the numerator and the denominator by the conjugate (5 + 2i)

(7 + 4i) (7 - 4i)

(7 - 4i) 35-20i+14i-8i

49-28i+28i-16i 2 2 35-6i+8

49+16 = = The final answer is 43-6i

65 Graphing Simplify: i 103 , i 81 HINT: THE PATTERN IS i, -1, -i, 1 (12 + 14i) + (3 -2i) 1) (−2 - 15i) - (-12i + 13i) 2) (9 + 7i ) (6 + 8i) 3) 3 + 5i

2 + 6i 4) Answers Answers i = 103/4= 25 R3 i =-i 103 103 1) (12+14i)+(3-2i) = (12+3)+(14i-2i)

= 15+12i √-54 = √-1√54

= i√54

√54 = √2*3*3*3

√54 = 3√6

√-54 = 3i√6 i =81/4=20 R1 i =i 81 81 √-361 = 19i 2) (-2-15i)-(-12+13i) = (-2-15i)+(12-13i)

= (-2+12)+(-15i-13i)

= 10-28i 3) (9 + 7i ) (6 + 8i) = 9*6 + 9*8i + 7i*6 + 7i*8i

= 54 + 72i + 42i - 56

= 54 + 114i - 56

= -2 + 114i 3 + 5i

2 + 6i 4) 2 - 6i

2 - 6i x 6-18i+10i+30

4-12i+12i+36 HINT: i is -1 2 = = 36-8i

40 = 9-8i

10 i =√-1 An imaginary number has a negative or zero square

e.g. 2i=-4, 3i=-9, 4i=-16, 5i=-25.....

Full transcriptnumber and a complex number is What is an Imaginary Number? Complex Numbers Complex numbers are a combination of a

real number and an imaginary number How to solve with a Large Exponent Questions Simplify: √-361, √-54 - Be able to solve simple problems/equations An Imaginary number is a number that can be written as a real number multiplied by the imaginary unit i. Pattern

i = -1

i = -i

i = 1

i = i

i = -1

i = -i

i = 1 i = i 0 1 i = 1 2 3 4 5 6 7 Starting from i, imaginary numbers follow a pattern:

i, -1, -i, 1 Looking at the pattern on the left, can you predict what i equals? 9 i = i 9 e.g. i 802 Steps:

1) Divide exponent by 4 (4 is the number of digits in the pattern,

i, -1, -i, 1)

2) Using remainder, determine the answer 802/4=200 R2 The remainder is 2, therefore i will be the same as i . i equals -1 802 Let's try One Simplify i 919 919/4=229 R3 The remainder is 3, therefore i will be the same as i . i equals -i 919 2 802 3 919 Simplifying Imaginary Numbers √-64 = √-1√64 = 8i Perfect Squares Simplifying numbers that aren't a perfect square These take longer to simplify. For example, √-20. √20 isn't fully simplified = √2*2*5 = 2√5 √-20= √-1√20 = i√20 √-20 equals 2i√5 How to Add, Subtract, Multiply and Divide Complex Numbers Examples:

1 + i

39 + 3i - Be able to simplify imaginary numbers 8 Adding (5 + 2i ) + (7 + 12i) Step 1) Group the real part of the complex number and the imaginary part of the complex number. (5 + 7) + (2i + 12i) Step 2) Combine the like terms and simplify The final answer is: 12+14i Subtraction

Step 1) Distribute the negative

Step 2) Group the real part of the complex number

and the imaginary part of the complex number.

Step 3) Combine the like terms and simplify (8 + 6i) - (5 + 2i) (8 + 6i) + (−5 − 2i) (8 + −5) + (6i + −2i) The final answer is 3+4i Multiplication (5 + 2i ) (7 + 12i) Step 1) Use F.O.I.L. (First, outer, inner, last) (a+bi)(c+di) = ac + adi + bci + bdi 2 5*7 + 5*12i + 2i*7 + 2i*12i 5*7=35

5*12i=60i

2i*7=14i

2i*12i=24i =-24 2 Step 2) Simplify by adding the terms (35+ -24) + (60i + 14i) The final answer is 11 + 74i Division (5 + 2i)

(7 + 4i) Step 1) Find the conjugate of the denominator The conjugate of (7 + 4i) is (7 - 4i) (FLIP THE SIGN) Step 2) Multiply the numerator and the denominator by the conjugate (5 + 2i)

(7 + 4i) (7 - 4i)

(7 - 4i) 35-20i+14i-8i

49-28i+28i-16i 2 2 35-6i+8

49+16 = = The final answer is 43-6i

65 Graphing Simplify: i 103 , i 81 HINT: THE PATTERN IS i, -1, -i, 1 (12 + 14i) + (3 -2i) 1) (−2 - 15i) - (-12i + 13i) 2) (9 + 7i ) (6 + 8i) 3) 3 + 5i

2 + 6i 4) Answers Answers i = 103/4= 25 R3 i =-i 103 103 1) (12+14i)+(3-2i) = (12+3)+(14i-2i)

= 15+12i √-54 = √-1√54

= i√54

√54 = √2*3*3*3

√54 = 3√6

√-54 = 3i√6 i =81/4=20 R1 i =i 81 81 √-361 = 19i 2) (-2-15i)-(-12+13i) = (-2-15i)+(12-13i)

= (-2+12)+(-15i-13i)

= 10-28i 3) (9 + 7i ) (6 + 8i) = 9*6 + 9*8i + 7i*6 + 7i*8i

= 54 + 72i + 42i - 56

= 54 + 114i - 56

= -2 + 114i 3 + 5i

2 + 6i 4) 2 - 6i

2 - 6i x 6-18i+10i+30

4-12i+12i+36 HINT: i is -1 2 = = 36-8i

40 = 9-8i

10 i =√-1 An imaginary number has a negative or zero square

e.g. 2i=-4, 3i=-9, 4i=-16, 5i=-25.....