Complex Numbersj^{2} = 1  j^{3} = j  j^{4} = 1  z = a + bj  z = r(sin θ + jsinθ)  z = re^{jθ}  tan^{1} b/a = θ  cos^{1} a/r = θ  sin^{1} b/r = θ  (a + bj)* = a  bj  z=r=sqrt(a^{2} + b^{2})  z^{x} = z^{x}  arg(z)^{x} = x arg (z)  arg(z) = θ + 2kπ  (cos θ + jsin θ)^{k}  = cos kθ + jsin kθ  = (e^{jθ})^{k} = e^{jkθ}  < DeMoivre's Theorum 
* means conjugate
j = i = sqrt(1) = imaginary unit
Find roots example:
z^{2} = 4j
Convert to exponential form first:
z^{2} = 4e^{jÏ€/2}
z^{2} = r^{2} = sqrt(0^{2} + 4^{2}) = 4
z = r = 2
k = (0, 1 ...n where n = expon' of z) = 0, 1
arg(z^{2}) = 2 arg(z) = Ï€/2 + 2kÏ€
arg(z) = Ï€/4 + kÏ€
Substitute values of k (0, 1) for z = ze^{jarg(z)} = 2e^{jÏ€/4}, 2e^{j3Ï€/4}   Discrete Probability & Sets & WhateverProbability
1. P(x) = ^{n}Cx . p^{x} . (1p)^{nx}
2. P(x) = (^{X}Ck)((^{NX})C(nk))/^{N}Cn
Set Theory
A = B when A subset of B & B subset of A
A  B = A n B'
A u (A n B) = A
A n (A u B) = A
A u A' = U
A n A' = nullset or {}
Power set of S is the set of ALL SUBSETS of S e.g. S = {1,2} , P(S) = { {}, {1}, {2}, {1,2}}
A = n, P(A) = 2^{n}
Sets A and B are disjoint iff A n B = {}
Cardinality of union: A u B = A + B  A n B
Proof by induction:
Show that when p(k) is true, p(k + 1) follows. 
1. Binomial Distribution
n = trials, x = successes, p = probability of success
2. Hypergeometric Distribution
N = deck size, n = draws, X = copies of card, k = successes
  Matrix ManipulationsA^{T}: Transpose of A  Switch Rows with Columns (R1 becomes C1, R2 becomes C2 etc.)
A = 1 . A
A^{1}: Inverse of A
A^{1} . I = I = A . I
A^{1}A=I
Augment Identity matrix to matrix and perform GuassJordon elimination on both to get change Identity matrix to the Inverse.
EROs:
Switch Rows
Scale Row (Multiply entire row)
Add multiple of different row to another
A matrix A is in row echelon form if
1. The nonzero rows in A lie above all zero rows (when there is at least a nonzero row and a zero row).
2. The first nonzero entry in a nonzero row (called a pivot) lies to the right of the pivot in the row immediately above it. 

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