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Algebra Formulas
1. Set identities
Definitions:
I: Universal set
A’: Complement
Empty set: ∅
Union of sets
A ∪ B = { x | x ∈ A or x ∈ B}
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
Intersection of sets
A ∩ B = { x | x ∈ A and x ∈ B}
Complement
A′ = { x ∈ I | x ∈ A}
Difference of sets
B \ A = { x | x ∈ B and x ∉ A}
Cartesian product
A × B = {( x, y ) | x ∈ A and y ∈ B}
( 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
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
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 )
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 − ∪ {0} ∪ Z = .{ .., − 3, − 2, − 1, 0, 1, 2, 3,... }
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Roots of complex numbers
Irrational numbers:
1
Nonerepeating and nonterminating integers
Real numbers:
Union of rational and irrational numbers
1
θ + 2k π
θ + 2k π 

 r ( cos θ + sin θ )  n = r n  cos
+ sin

n
n


From this the n nth roots can be obtained by putting k = 0,
1, 2, . . ., n - 1
Complex numbers:
C = { x + iy | x ∈ R and y ∈ R}
4. Factoring and product
N ⊂Z ⊂Q⊂R⊂C
Factoring Formulas
a 2 − b 2 = ( a − b )( a + b )
(
= (a + b)(a
)
− ab + b )
3. Complex numbers
a 3 − b3 = ( a − b ) a 2 + ab + b 2
Definitions:
a3 + b3
A complex nuber is written as a + bi where a and b are
real numbers an i, called the imaginary unit, has the
2
property that i =-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
2
2
a 4 − b 4 = ( a − b)( a + b)( a 2 + b 2 )
(
a 5 − b5 = ( a − b ) a 4 + a 3b + a 2 b 2 + ab3 + b 4
Product Formulas
( a + b) 2 = a 2 + 2ab + b 2
( a − b) 2 = a 2 − 2ab + b 2
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + b)3 = a3 + 3a 2b + 3ab2 + b3
Subtraction of complex numbers
(a − b)3 = a3 − 3a 2b + 3ab 2 − b3
(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  bc − ad 
=
⋅
=
+
i
c + di c + di c − di c 2 + d 2  c 2 + d 2 
(a + b)
4
= a 4 + 4a 3 b + 6 a 2 b 2 + 4ab3 + b 4
( a − b )4 = a 4 − 4a3b + 6a 2 b2 − 4ab3 + b4
(a + b + c)2 = a 2 + b2 + c 2 + 2ab + 2ac + 2bc
(a + b + c + ...) 2 = a 2 + b 2 + c 2 + ...2(ab + ac + bc + ...)
Polar form of complex numbers
x + iy = r ( cosθ + i sinθ )
r − modulus, θ − amplitude
Multiplication and division in polar form
 r1 ( cos θ1 + i sin θ1 ) ⋅  r2 ( cos θ 2 + i sin θ 2 )  =
= r1r2 cos (θ1 + θ 2 ) + i sin (θ1 + θ 2 )
r1 ( cosθ1 + sinθ1 )
r
⋅ = 1 cos (θ1 − θ2 ) + sin (θ1 − θ2 ) 
r2 ( cosθ2 + sin θ2 ) r2
De Moivre’s theorem
n
 r ( cos θ + sin θ )  = r n ( cos nθ + sin nθ )
)
5. Algebric equations
Quadric Eqation: ax2 + bx + c = 0
Solutions (roots):
x1,2
−b ± b2 − 4ac
=
2a
2
if D=b -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
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Cubic Eqation: x3 + a1 x 2 + a2 x + a3 = 0
Let
3a2 − a12
Q=
,
9
9a1a2 − 27a3 − 2a13
R=
54
S = 3 R + Q3 + R2 ,
T = 3 R − Q3 + R2
then solutions are:
1
x1 = S + T − a1
3
1
1
1
x2 = − ( S + T ) − a1 + i 3 ( S − T )
2
3
2
1
1
1
x3 = − ( S + T ) − a1 − i 3 ( S − T )
2
3
2
3
3
if D = Q + R 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: x4 + a1x3 + a2 x2 + a3x + a4 = 0
Let y1 be a real root of the cubic equation
(
)
y3 − a2 y2 + ( a1a3 − 4a4 ) y + 4a2 a4 − a32 − a12 a4 = 0
Solution are the 4 roots of
z2 +
1
1
a1 ± a12 − 4a2 + 4y1 z + y1 ± y12 − 4a4 = 0
2
2
(
) (
)