Properties of Complex Numbers
OVERVIEW
A complex number satisfy 11 properties. 5 properties are concerning addition, 5 are concerning multiplication and 1 are mixed of both.
{x+iy is written as (x,y)}
1) PROPERTIES USING ADDITION
I)Binary Composition:- If Z1 & Z2 are complex number, Z1 + Z2 be a complex numbers.
Example:- If (2,3) and (4,5) are complex numbers.
(2+4,3+5)=(6,8) is a complex numbers.(proved)
II) Commutative Law:- If Z1 & Z2 are complex number,
Z1 + Z2 = Z2 + Z1
Example:- If (2,3) and (4,5) are complex numbers.
(2+4,3+5)=(6,8)---LHS
(4+2,5+3)=(6,8)---RHS
LHS=RHS (proved)
III) Associative Law:- If Z1 , Z2 & Z3 are complex number,
( Z1 + Z2)+ Z3= Z1 +( Z2+ Z3)
Example:- If (2,3),(4,5) and (6,7) are complex numbers.
{(2,3)+(4+5)}+(6,7)=(6,8)+(6,7)=(12,15)---LHS
(2,3)+{(4+5)+(6,7)}=(2,3)+(10,12)=(12,15)---RHS
LHS=RHS (proved)
IV) Additive Identity :- If Z is any complex number, there is a unique complex number (0,0) such that
Z+0=0+Z=Z
Example:- If (2,3) and (4,5) are complex numbers.
(1)(2,3)+(0,0)=(0,0)+(2,3)=(2,3)
(2)(4,5)+(0,0)=(0,0)+(4,5)=(4,5)
V)Additive Inverse :- If Z is any complex number, there is a unique complex number (-Z) such that
Z+(-Z)=(-Z)+Z=0
Example:- If (2,3) and (4,5) are complex numbers.
(1)(2,3)+(-2,-3)=(-2,-3)+(2,3)=(0,0)
(2)(4,5)+(-4,-5)=(-4,-5)+(4,5)=(0,0)
2) PROPERTIES USING MULTIPLICATION
I)Binary Composition:- If Z1 & Z2 are complex number, Z1*Z2 be a complex numbers .
Example:-If (9,10) & (1,2) is a complex number.
(9,10)*(1,2)={(9*1-10*2),(9*2+10*1)}=(-11,28) is a complex number
II) Commutative Law :- If Z1 & Z2 are complex number,
Z1*Z2 = Z2*Z1
Example:-If (9,10) & (1,2) is a complex number.
(9,10)*(1,2)={(9*1-10*2),(9*2+10*1)}=(-11,28)---LHS
(1,2)*(9,10)=(1*9-2*10),{(2*9+1*10)}=(-11,28)---RHS
LHS=RHS [Proved]
III) Associative Law:- If Z1 , Z2 & Z3 are complex number,
( Z1*Z2)*Z3= Z1*( Z2*Z3)
Example:-If (9,10),(1,2) & (4,5) is a complex number.
{(9,10)*(1,2)}*(4,5)=(-11,28)*(4,5)=(-184,57)---LHS
(9,10)*{(1,2)*(4,5)}=(9,10)*(-6,13)=(-184,57)---RHS
LHS=RHS [Proved]
IV) Multiplicative Identity :- If Z is any complex number, there is a unique complex number 1 such that
Z*1=1*Z=Z
Example:-If (1,2) is a complex number.
(1,2)*(0,1)=(1,2)----LHS
(0,1)*(1,2)=(1,2)---RHS
LHS=RHS[Proved]
V)Multiplicative Inverse :- If Z(≠0) is any complex number, there is a unique complex number Z-1 such that
Z*Z-1=Z-1*Z=1
• Determination Of Multiplicative Inverse:-If (a,b) is any complex number and (a1,b1) is multiplicative inverse of this complex number.
(a,b)*(a1,b1)=(1,0)
ax - by=1
ay + bx=0
The value of (a1,b1) is [{a/(a2+b2)},{-b/(a2+b2)}]
[∵(a,b)≠0 ,∴ (a2+b2)≠0 ]
Example:-If (1,2) is a complex number. Multiplicative Inverse is (1/5,2/5)
(1,2)*(1/5,2/5)=1---LHS
(1/5,2/5)*(1,2)=1---RHS
LHS=RHS[Proved]
3) MIX PROPERTIES
I) Distributive Law:- If Z1 , Z2 & Z3 are complex number,
( Z1+Z2)*Z3= (Z1*Z2)+(Z2*Z3)
Example:- If (2,3),(4,5) and (6,7) are complex numbers.
{(2,3)+(4,5)}*(6,7)=(6,8)*(6,7)=(-20,90)---LHS
{(2,3)*(6,7}+{(4,5)*(6,7)}=(-9,32)*(-11,58)=(-20,90)---RHS
LHS=RHS [Proved]
THANK YOU FOR READING
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