Complex Numbers
Tryingto solve polynomials equations
Standardnotation
IR setof all reals
R is a fieldunderusual additionand multiplication
Thereis an orderin IR
a b as b
Completeness axiom
IN 1 2,3 3
I O I 1 2 2 3 Note In thefield z 0,13
Q Elm ne E n 03 where 0 0 0
1 0 1
Then IN E Z E Q EIR 1 1 0
1 0 0
IRI Q set of allirrationals 1 1 1
TI n IR x 1 0 has no solution
2
Proof XE IR X220 7 1
Of thereis a solution to x't1 0
Construction Existence of ComplexNumbers
startwith IR
a
Lookat IR IRXITY ab la beIR Visualization
prodigyorderedpair
Cartesian Y aib
ay
s
Notethat R is avectorspace over IR x axis IR
ab le d late bed wanttomake IR intoa field
TM
X a b Xa xb where lab c d ac bd
XEIR Then
1,0 0,1 10,079
40 Édditionidentity
Thiswill NI work
,Trythefollowing multiplication so in IR
la b o c d fac bd ad be a b c da c bed
withrespect totheprevious t andthe 10,0 istheadditiveidentity
abovemultiplication IR becomes a field ab 0,07 a b
If a b 0,07 then a b
at d d Taib la b multiplicative
identity inIR
1,0
IR isthex y plane cabs
Geometrically
aim b
y axisn
i 0,1 a aib
Def objects of IR addition multiplication
s are called complex numbers
cao a
x axis
Any complexnumbercan be represented
Let it 10,1 in IR as at bi where i is it 1
Here a and b are realnumbers
Calculate i i 0,1 o1 L1,07 1
Notation
Also Ca b E IR La b la 07 10 b A Z at bi la belt
la O b 0,1
at b i
Addition
at bi letdi ate bed i
Multiplication
atbilletdi ac bdl lad be i
Elements of are calledcomplexnumbers I is a field Fieldofcomplexnumbers
Another
way
Lookatall 2 2 matrices of theform
ba where a b EIR
, fi
addition
s a c
Multiplication ae bd adtbe
Lay fad betad ac bd
Now associate
at bi
b
Iba ath at c Ebi atte
, Z at bi la be IR
ComplexNumbersystem
Here 5 1
know Cl is a field
fatatbibi ctcedi
di e a c bed
ate b di
1 atbi cedi ac bdl ad_b
IR is an orderedfield
so ta eIR a 0 or a o or 0
a
a so b o at b 0 and ab o
However I cannotbe ordered
i Theaxioms of IR cannot gotowards a
Figo or o
ii o
so
y axis
Conjugate of a complexnumber Imaginary a is Z la b at bi
ZEC Z at bi where a b E IR
S
Here a Real part of z Re z
x axis
b Imaginarypartof z Im z
Real axis
Def For Z at bi E C
E a bi
Conjugate of z
Tryingto solve polynomials equations
Standardnotation
IR setof all reals
R is a fieldunderusual additionand multiplication
Thereis an orderin IR
a b as b
Completeness axiom
IN 1 2,3 3
I O I 1 2 2 3 Note In thefield z 0,13
Q Elm ne E n 03 where 0 0 0
1 0 1
Then IN E Z E Q EIR 1 1 0
1 0 0
IRI Q set of allirrationals 1 1 1
TI n IR x 1 0 has no solution
2
Proof XE IR X220 7 1
Of thereis a solution to x't1 0
Construction Existence of ComplexNumbers
startwith IR
a
Lookat IR IRXITY ab la beIR Visualization
prodigyorderedpair
Cartesian Y aib
ay
s
Notethat R is avectorspace over IR x axis IR
ab le d late bed wanttomake IR intoa field
TM
X a b Xa xb where lab c d ac bd
XEIR Then
1,0 0,1 10,079
40 Édditionidentity
Thiswill NI work
,Trythefollowing multiplication so in IR
la b o c d fac bd ad be a b c da c bed
withrespect totheprevious t andthe 10,0 istheadditiveidentity
abovemultiplication IR becomes a field ab 0,07 a b
If a b 0,07 then a b
at d d Taib la b multiplicative
identity inIR
1,0
IR isthex y plane cabs
Geometrically
aim b
y axisn
i 0,1 a aib
Def objects of IR addition multiplication
s are called complex numbers
cao a
x axis
Any complexnumbercan be represented
Let it 10,1 in IR as at bi where i is it 1
Here a and b are realnumbers
Calculate i i 0,1 o1 L1,07 1
Notation
Also Ca b E IR La b la 07 10 b A Z at bi la belt
la O b 0,1
at b i
Addition
at bi letdi ate bed i
Multiplication
atbilletdi ac bdl lad be i
Elements of are calledcomplexnumbers I is a field Fieldofcomplexnumbers
Another
way
Lookatall 2 2 matrices of theform
ba where a b EIR
, fi
addition
s a c
Multiplication ae bd adtbe
Lay fad betad ac bd
Now associate
at bi
b
Iba ath at c Ebi atte
, Z at bi la be IR
ComplexNumbersystem
Here 5 1
know Cl is a field
fatatbibi ctcedi
di e a c bed
ate b di
1 atbi cedi ac bdl ad_b
IR is an orderedfield
so ta eIR a 0 or a o or 0
a
a so b o at b 0 and ab o
However I cannotbe ordered
i Theaxioms of IR cannot gotowards a
Figo or o
ii o
so
y axis
Conjugate of a complexnumber Imaginary a is Z la b at bi
ZEC Z at bi where a b E IR
S
Here a Real part of z Re z
x axis
b Imaginarypartof z Im z
Real axis
Def For Z at bi E C
E a bi
Conjugate of z
