Math Formulas

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

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+2kπn+isinθ+2kπn)  k=0,1,…,n−1

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

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