Algebra Formulas-The Infinity Mathematics

Algebra Formulas

 

1.       Set identities

Definitions:

I: Universal set A’: Complement Empty set: Æ

Union of sets

A È B = {x | x ΠA or x ΠB}

Intersection of sets

Identity

A È Æ = A A Ç I = A

Set identities involving union, intersection and complement

complement of intersection and union

A È A¢ = I A Ç A¢ = Æ

De Morgan’s laws

A Ç B = {x | x ΠA and

Complement

A¢ = {x ΠI | x ΠA}

Difference of sets

x ΠB}

( A È B)¢ = A¢ Ç B¢

( A Ç B)¢ = A¢ È B¢

Set identities involving difference

B \ A = B ( A È B)

B \ A = {x | x ΠB and x Ï A}

Cartesian product

B \ A = B Ç A¢

A \ A = Æ

( A \ B ) Ç C = ( A Ç C ) \ ( B Ç C )

A ´ B = {( x, y )| x Î A and  y Î B}

Set identities involving union

Commutativity

A È B = B È A

Associativity

A È ( B È C ) = ( A È B ) È C

Idempotency

A È A = A

Set identities involving intersection

commutativity

A Ç B = B Ç A

Associativity

A Ç ( B Ç C ) = ( A Ç B ) Ç C

Idempotency

A Ç A = A

Set identities involving union and intersection

Distributivity

A È ( B Ç C ) = ( A È B ) Ç ( A È C ) A Ç ( B È C ) = ( A Ç B ) È ( A Ç C ) Domination

A Ç Æ = Æ

A È I = I

A¢ = I \ A

 

2.       Sets of Numbers

Definitions:

N: Natural numbers N_o: Whole numbers Z: Integers

Z⁺: Positive integers Z^-: Negative integers Q: Rational numbers C: Complex numbers

Natural numbers (counting numbers )

N = {1, 2, 3,... }

Whole numbers ( counting numbers + zero )

N_o = {0, 1, 2, 3,... }

Integers

Z ⁺ = N = {1,  2,  3,... }

Z ^- = {..., - 3, - 2, -1}

Z = Z^- È{0} È Z = .{.., - 3, - 2, -1, 0, 1, 2, 3,... }

Irrational numbers:

Nonerepeating and nonterminating integers

Real numbers:

Union of rational and irrational numbers

Complex numbers:

C = {x + iy | x ΠR and y ΠR}

N Ì Z Ì Q Ì R Ì C

3.       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 a+bi and a-bi are called complex conjugate of each other.

Equality of complex numbers

a + bi = c + di if and only if 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 + bi = a + bi × c - di = ac + bd + æ bc - ad ö i

                                                    c + di     c + di   c - di     c² + d ^{2        (}c2 + d 2 ⁾

Polar form of complex numbers

x +iy = r (cosq +isinq) r -modulus, q -amplitude

Multiplication and division in polar form

éër₁ (cosq₁ + i sinq₁ )ùû × éër₂ (cosq₂ + i sinq₂ )ùû =

= r₁r₂ éëcos (q₁ + q₂ ) + i sin (q₁ + q₂ )ùû

De Moivre’s theorem

éër (cosq + sinq )ùûn   = rⁿ (cos nq + sin nq )

4. Algebric equations

Quadric Eqation: ax² + bx + c = 0

Solution (roots):

if D=b²-4ac is the discriminant, then the roots are

(i)  real and unique if D > 0

    (ii)  real and equal if D = 0

    (iii)   complex conjugate if D < 0

5.       Factoring and product

Factoring Formulas

a² - b² = (a - b)(a + b)

a³ - b³ = (a - b)(a² + ab + b² )

a³ + b³ = (a + b)(a² - ab + b² )

a4 - b4 = (a - b)(a + b)(a2 + b2 )

a5  - b5  = (a - b)(a4  + a3b + a2b2  + ab3 + b4 )

Product Formulas

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3ab² - b³

(a + b)4 = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

(a - b)4 = a⁴ - 4a³b + 6a²b² - 4ab³ + b⁴

(a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc

(a + b + c +...)² = a² + b² + c² +...2(ab + ac + bc +...)