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Introduction

Complex Numbers

The Argand Diagram

Modulus

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• Mathematicians have a concept called completeness. It is the need to be able to answer every single question.

• Historically it is this need for completeness which led the Hindus to discover negative numbers.

• Later the Greeks developed the idea of irrational numbers.

IntroductionIntroduction

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• During the sixteenth century, an Italian called Rafaello Bombelli came up with the question:

“If the square root of +1 is both +1 and –1, then what is the square root of –1?”

• He answered the question himself and declared that

• In your own life you develop through all these stages of number.

Rafaello BombelliRafaello Bombelli

– 1 = i, an imaginary number.

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• N: Natural numbers – these are whole positive numbers. These are the first numbers people understand, e.g. you are three years old and fighting with your little sister because she has three sweets and you have only two!

• Z: Integers – positive and negative whole numbers. Later in life, perhaps when you are five years old you understand the idea of minus numbers. You may have five sweets but you owe your friend two sweets, so you realise that in fact, you really only have three sweets, 5 – 2 = 3.

Rafaello BombelliRafaello Bombelli

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• R: Real Numbers – all numbers on the number line. • Later again in life you realise that there are fractions

and decimals. You may divide a bar of chocolate with eight squares and give your brother three squares and keep five for yourself.

• Other real numbers include:

Rafaello BombelliRafaello Bombelli

38__ 5

8__• He gets of the bar and you get of it.

THIS IS PROBABLY WHERE YOU ARE NOW!THIS IS PROBABLY WHERE YOU ARE NOW!

4 , 2·114, – 3·49, 3 , 9 101514 –

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Complex NumbersComplex Numbers

• These are numbers with a real and an imaginary part. • They are written as a + ib where a and b are real

numbers.

–1 = i

i2 = –1

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Simply add the real parts, then add the imaginary parts.

3 + 5i

6 + 2i+

––––––9 + 7i

AdditionAddition

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–

‘Change the sign on the lower line and add’

8 – 3i

3 2i–

––––––11 – 5i

SubtractionSubtraction

+ –++

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Each part of the first complex number must be multiplied by each part of the second complex number.

MultiplicationMultiplication

i2 = –1

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4(3 + 2i) = 12

i2 = –1

= 12i + 8

= 12i – 8

= – 8 + 12i

+ 8i

= 12i + 8i2

(–1)

4i(3 + 2i)

MultiplicationMultiplication

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(3 + 2i)(4 + 5i) = 12 i2 = –1

= 12 + 15i + 8i + 10

= 12 + 15i + 8i – 10

= 2

+ 15i + 8i + 10i2

(–1)

+ 23i

Collect real and imaginary parts

MultiplicationMultiplication

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To divide complex numbers we need the concept of the complex conjugate.

The conjugate of a complex number is the same number with the sign of the imaginary part changed.

The conjugate of 5 + 3i is 5 – 3i.

DivisionDivision

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To divide we multiply the top and bottom by the conjugate of the bottom.

DivisionDivision

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15

DivisionDivision

9

– 8i 2

+ 12i – 12i – 16i 2

= i 2 = –1

5 + 2i 3 + 4i

3 – 4i 3 – 4i

×– 20i + 6i (–1)

(–1)––––––––––––––––––

15 – 20i + 6i + 89 + 16

= ––––––––––––––

23 – 14i25

= –––––––

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A farmer has 100 m of fence to surround a small vegetable plot. The farmer wants to enclose a rectangular area of 400 m2. How long and wide should it be?

ApplicationApplication

y

x

50 – x

2 lengths + 2 widths = 100

2x + 2y = 100

x + y = 50

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The area = Length width

400 = x(50 – x)

400 = 50x – x2

x2 – 50x + 400 = 0

x

50 – x

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b 4ac2

2a x =

b

= 40 or 10

x2 – 50x + 400 = 0

a = 1 b = – 50 c = 400

––––––––––––––– (–50)

2 – 4(1)(400)

2(1)x =

–(–50) ––––––––––––––––––––––––

––––––––––––––––– 2500 – 1600

2 =

50

–––––––––––

2 =

50 30–––––––

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A farmer has 100 m of fence to surround a small vegetable plot. The farmer wants to enclose a rectangular area of 650 m2. How long and wide should it be?

ApplicationApplication

y

x

50 – x

2 lengths + 2 widths = 100

2x + 2y = 100

x + y = 50

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The area = Length width

650 = x(50 – x)

650 = 50x – x2

x2 – 50x + 650 = 0

x

50 – x

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b 4ac2

2a x =

b

x2 – 50x + 650 = 0

a = 1 b = – 50 c = 650

––––––––––––––– (–50)

2 – 4(1)(650)

2(1)x =

–(–50) ––––––––––––––––––––––––

––––––––––––––––– 2500 – 2600

2 =

50

––––––––––– ––––––

–––––––––––– – 100

2 =

50

–––

=

––

–––––––––––––– 100

2

50 –1 ––––––––

2

50

10i =

i2 = –1

= 25 5i

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• The German mathematician Carl Fredrich Gauss

(1777 – 1855) proposed the Argand diagram.• This has the real numbers on the x-axis and the

imaginary ones on the y-axis. • All complex numbers can be plotted and are usually

called z1, z2 etc.

The Argand DiagramThe Argand Diagram

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-3 -2 -1 1 2 3 Re

-3

-2

-1

1

2

3

Im

4

-4

The Argand DiagramThe Argand Diagramz1

z2

z4 z3

z5z1 = (2 + 4i)

z2 = (–2 – 4i)

z3 = (4 + 0i)

z4 = (–3 + 0i)

z5 = (–3 + 2i)

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| 2 + 4i | = 22 + 42

= 4 + 16

= 20

z = x2 + y2

z1 = 2 + 4i

| 4 + 0i | = 42 + 02

= 16

z3 = 4 + 0i

= 4

ModulusModulus

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