Algebra formulas
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The document provides definitions and formulas related to algebra concepts including sets, numbers, complex numbers, factoring, products, and algebraic equations. It defines types of sets and operations on sets like union, intersection, complement and difference. It also defines types of numbers like natural numbers, integers, rational numbers, irrational numbers, real numbers and complex numbers. Formulas are given for addition, subtraction, multiplication and division of complex numbers. Other formulas presented include factoring formulas, product formulas, formulas for solving quadratic, cubic and quartic equations.
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- 1. www.mathportal.org Algebra Formulas 1. Set identities Definitions: I: Universal set A’: Complement Empty set: ∅ Union of sets { }|A B x x x BrA o∪ = ∈ ∈ Intersection of sets { }|A B x x x BdA an∩ = ∈ ∈ Complement { }|A x I x A′ = ∈ ∈ Difference of sets { } |B A x x B x Aand= ∈ ∉ 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∪∅ = 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′ = 2. Sets of Numbers Definitions: N: Natural numbers No: Whole numbers Z: Integers Z + : Positive integers Z - : Negative integers Q: Rational numbers C: Complex numbers Natural numbers (counting numbers ) { }1, 2, 3,...N = Whole numbers ( counting numbers + zero ) { }0, 1, 2, 3,...oN = Integers { }1, 2, 3,...Z N+ = = { }..., 3, 2, 1Z− = − − − { } { }0 . .., 3, 2, 1, 0, 1, 2, 3,...Z Z Z− = ∪ ∪ = − − −
- 2. www.mathportal.org Irrational numbers: Nonerepeating and nonterminating integers Real numbers: Union of rational and irrational numbers Complex numbers: { }|C x iy x R and y R= + ∈ ∈ N Z Q R C⊂ ⊂ ⊂ ⊂ 3. Complex numbers Definitions: A complex nuber is written as a + bi where a and b are real numbers an i, called the imaginary unit, has the property that i 2 =-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 2 2 2 2 a bi a bi c di ac bd bc ad i c di c di c di c d c d + + − + − = ⋅ = + + + − + + Polar form of complex numbers ( )cos sin modulus, amplitudex iy r i rθ θ θ+ = + − − Multiplication and division in polar form ( ) ( ) ( ) ( ) 1 1 1 2 2 2 1 2 1 2 1 2 cos sin cos sin cos sin r i r i rr i θ θ θ θ θ θ θ θ + ⋅ + = = + + + ( ) ( ) ( ) ( )1 1 1 1 1 2 1 2 2 2 2 2 cos sin cos sin cos sin r r r r θ θ θ θ θ θ θ θ + ⋅ = − + − + De Moivre’s theorem ( ) ( )cos sin cos sin n n r r n nθ θ θ θ + = + Roots of complex numbers ( ) 11 2 2 cos sin cos sinnn k k r r n n θ π θ π θ θ + + + = + From this the n nth roots can be obtained by putting k = 0, 1, 2, . . ., n - 1 4. Factoring and product Factoring Formulas ( )( )2 2 a b a b a b− = − + ( )( )3 3 2 2 a b a b a ab b− = − + + ( )( )3 3 2 2 a b a b a ab b+ = + − + 4 4 2 2 ( )( )( )a b a b a b a b− = − + + ( )( )5 5 4 3 2 2 3 4 a b a b a a b a b ab b− = − + + + + Product Formulas 2 2 2 ( ) 2a b a ab b+ = + + 2 2 2 ( ) 2a b a ab b− = − + 3 3 2 2 3 ( ) 3 3a b a a b ab b+ = + + + 3 3 2 2 3 ( ) 3 3a b a a b ab b− = − + − ( ) 4 4 3 2 2 3 4 4 6 4a b a a b a b ab b+ = + + + + ( )4 4 3 2 2 3 4 4 6 4a b a a b a b ab b− = − + − + 2 2 2 2 ( ) 2 2 2a b c a b c ab ac bc+ + = + + + + + 2 2 2 2 ( ...) ...2( ...)a b c a b c ab ac bc+ + + = + + + + + + 5. Algebric equations Quadric Eqation: ax2 + bx + c = 0 Solutions (roots): 2 1,2 4 2 b b ac x a − ± − = if D=b 2 -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
- 3. www.mathportal.org Cubic Eqation: 3 2 1 2 3 0x a x a x a+ + + = Let 32 1 2 3 12 1 3 2 3 23 3 9 27 23 , 9 54 , a a a aa a Q R S R Q R T R Q R − −− = = = + + = − + then solutions are: ( ) ( ) ( ) ( ) 1 1 2 1 3 1 1 3 1 1 1 3 2 3 2 1 1 1 3 2 3 2 x S T a x S T a i S T x S T a i S T = + − = − + − + − = − + − − − if D = Q 3 + R 3 is the discriminant, then: (i) one root is real and two complex conjugate if D > 0 (ii) all roots are real and at last two are equal if D = 0 (iii) all roots are real and unequal if D < 0 Cuadric Eqation: 2 4 4 3 1 2 3 0x ax a x a x a ++ + + = Let y1 be a real root of the cubic equation ( ) ( )3 2 2 2 2 1 3 4 2 4 3 1 44 4 0y a y a a a y a a a a a− + − + − − = Solution are the 4 roots of ( ) ( )2 2 2 1 1 2 1 1 1 4 1 1 4 4 4 0 2 2 z a a a y z y y a+ ± − + + ± − =