What are the complex Numbers Parti

IN 1,2 3 the natural numbers

2 I O l z the integers

Q Eg where a b integers bio the rationals

decimalsIR the real numbers

Key Geometric Picture The numberline

R i i i

All come with two fundamental operations and

Aim Extend and From line to plane

1122 a b where a b in IR xy plane

Let's identify R with the x axis ie a is identified withCa o

I ca b Cartesian coordinatesb

vector associated to Ca bt

Addition in R 2 3

What is 2 3 o siii

Tz 3vector associated to 2

What is 2 C 3zte 3

Ll 0

lil

ta

C Given a b in

atb Position on 112 given by laying a and b vectors end toend

Definition of Addition in 1122

Given a b and c d in RZ

a b C c d Position in 115 given by laying a b

and c d Vectors end to end

Example 1,2 t 3,111,2 t 3 l 4,3

1,2 Tq T it

Factsaib t c d a c b id

4 Identifying R with the x axis gives usual addition

Multiplication in 112 Uniformly stretchingnumber line Fixing 0

what is 2 3 to bring 1 to 3Then 2 moves to 2 3

O Zi

1 4i l 2 3I

o G1 i

3 3

What is 2 431 Rotate by about O

and uniformly stretch2 3fixing 0 to bring 1 to 3

g Fm z Then Z moves to 2 1 31i

I3 3

Conclusion Given a b in IR

ax b Position a ends up after rotating 117IEdfY2to bring l to b

Remark This way the product of two negtives is positiveRotating and stretching

uniformlyb a 0 a b

Definition of Multiplication in 1122

Given a b and c d in RZ

a b x C e d Position a b ends up after rotatinguniformly stretch radially to bring 1,0

Tto Cc d identified with1 in R

Example f i i x 1,1

stretchedbyT2

length TzC n i lt g

1,0 C i 1 11,14ft

2,0

Remain This is easier to understand in polarcoordinates

r O pr What is r O p rz.dz p

vid.jp Critz 01 04p O O r Ur 2 pCrr Odpyo

ft1,0

Rotateanticlockwiseby Oz andstretchbyr 2

Conclusion r Op Rz Or p r rz O 102 p

Identifying R with the x axis gives usual multiplication

Important Fact All the usual Laws of algebra hold for

these and on 1122

Fundamental Definition 1122 together with new

I and X

1122 t complex numbers

1

Terminology same as CR tixiso E is an extension of realnumbersL

Y X axis real axis Not really imaginary

4 y axis imaginary axis i

3 i oil 1 Iz p

Properties of i

a x i a rotated anticlockwise by Izi

i x i C 1,0 Iixi

Hence iz y negativenumbers havesquare roots in E

Can we interpret multiplication m 1C only using Cartesian

coordinates

et b be in 112 identified with bro in E

b o b o

SL

bi o b b

bi o b

Conclusion Given b in IR bi o b

Given a b mi IR thought of as begui E

a bi Ca o t o b a b

at bib

a b c d

t 1Laws A Algebra La bi cedi

act adi bei bdi2Not exactly

ac boil ad be ia memorableFormula T

ac lsd ad be

FiuThe complex numbers are the plane equipped with a

carefully chosen and x so the laws of algebra still

hold