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Math Formulas ► Complex numbers ► Sets of Numbers ► Set Identities

Math Formulas ► Complex numbers

  • Equality of complex numbers
  • Addition of complex numbers
  • Subtraction of complex numbers
  • Multiplication of complex numbers
  • Division of complex numbers
  • Polar form of complex numbers
  • Multiplication and division of complex numbers in polar form
  • De Moivre's theorem
  • Roots of complex numbers    

Math Formulas  ► Sets of Numbers 

  • Natural numbers (counting numbers )
  • Whole numbers ( counting numbers with zero )
  • Integers ( whole numbers and their opposites and zero )
  • Irrational numbers: Non repeating and nonterminating integers
  • Real numbers: Union of rational and irrational numbers  

Math Formulas ► Set Identities

  • Union of sets
  • Intersection of sets
  • Complement
  • Difference of sets
  • Cartesian product

Math Formulas ► Set Identities

► Set Identities involving union
  • Commutativity
  • Associativity
  • Idempotency

► Set Identities involving intersection
  • Commutativity
  • Associativity
  • Idempotency 
► Set Identities involving union and intersection
  • Distributivity
  • Domination
  • Identity 
► Set Identities involving union, intersection and complement
  • Complement of intersection and union
  • De Morgan's laws  
► Set identities involving difference

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Math Formulas | Complex numbers

Math Formulas ► Complex numbers ► Sets of Numbers ► Set Identities

Math Formulas ► Complex numbers

Definitions:

A complex number is written as a+bi where a and b are real numbers an i, called the imaginary unit, has the property that i2=−1.
The complex numbers z=a+bi and z−=a−bi are called complex conjugate of each other.

Formulas:

Equality of complex numbers

a+bi=c+di⟺a=c  and  b=d
Addition of complex numbers

(a+bi)+(c+di)=(a+c)+(b+d)i

Subtraction of complex numbers

(a+bi)−(c+di)=(a−c)+(b−d)i

Multiplication of complex numbers

(a+bi)⋅(c+di)=(ac−bd)+(ad+bc)i

Division of complex numbers

a+bic+di=a+bic+di⋅c−dic−di=ac+bdc2+d2+bc−adc2+d2i

Polar form of complex numbers

a+bi=r⋅(cosθ+isinθ)

Multiplication and division of complex numbers in polar form

[r1(cosθ1+i⋅sinθ1)]⋅[r2(cosθ2+i⋅sinθ2)]=r1⋅r2[cos(θ1+θ2)+i⋅sin(θ1+θ2)]


r1(cosθ1+isinθ1)r2(cosθ2+isinθ2)=r1r2[cos(θ1−θ2)+i⋅sin(θ1−θ2)]

De Moivre's theorem

[r(cosθ+isinθ)]n=rn(cos(nθ)+isin(nθ))

Roots of complex numbers

[r(cosθ+isinθ)]1/n=r1/n(cosθ+2kÏ€n+isinθ+2kÏ€n)  k=0,1,…,n−1

Under:
Math Formulas | Sets of Numbers

Math Formulas ► Complex numbers ► Sets of Numbers ► Set Identities


Math Formulas ► Sets of Numbers

Definitions:

N : Natural numbers
N0 : Whole numbers
Z : Integers
Z+ : Positive integers
Z− : Negative integers
Q : Rational numbers
C : Complex numbers

Formulas:

Natural numbers (counting numbers )

N={1,2,3,…}

Whole numbers ( counting numbers with zero )

N0={0,1,2,3,…}

Integers ( whole numbers and their opposites and zero )

Z={…,−2,−1,0,1,2,…}


Z+=N={1,2,…}


Z−={…,−3,−2,−1}


Z=Z−∪0∪Z

Irrational numbers: Non repeating and nonterminating integers

Real numbers: Union of rational and irrational numbers
Complex numbers:

C={x+iy | x∈R  and  y∈R}


N⊂N0⊂Z⊂Q⊂R⊂C

Under:
Math Formulas | Set Identities

Math Formulas ► Complex numbers ► Sets of Numbers ► Set Identities

Math Formulas ► Set Identities

Definitions:
Universal set : I
Empty set: ∅
Union of sets

A∪B={x:x∈A  or  x∈B}
Intersection of sets
A∩B={x:x∈A  and  x∈B}
Complement
A′={x∈I:x/∈A}
Difference of sets
A∖B={x:x∈A  and  x/∈B}
Cartesian product
A×B={(x,y):x∈A  and  y∈B}

Set identities involving union

Commutativity
A∪B=B∪A
Associativity
A∪(B∪C)=(A∪B)∪C
Idempotency
A∪A=A

Set identities involving intersection

Commutativity
A∩B=B∩A
Associativity
A∩(B∩C)=(A∩B)∩C
Idempotency
A∩A=A

Set identities involving union and intersection

Distributivity
A∪(B∩C)=(A∪B)∩(A∪C)


A∩(B∪C)=(A∩B)∪(A∩C)
Domination
A∩∅=∅


A∪I=I
Identity
A∪∅=∅


A∩I=A

Set identities involving union, intersection and complement

Complement of intersection and union

A∪A′=I


A∩A′=∅
De Morgan's laws

(A∪B)′=A′∩B ′


(A∩B)′=A′∪B ′

Set identities involving difference


B∖A=B∖(A∪B)


B∖A=B∩A′


A∖A=∅


(A∖B)∩C=(A∩C)∖(B∩C)


A′=I∖A

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