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- Canada
- Wilfrid Laurier University
- Mathematics
- Mathematics 121
- Bauman
- Post-midterm notes

Cassy D.

Permutation

An ordered arrangement of objects.

n!

The number of permutations of n objects.

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r-combinations

Choose a set with r elements from a set with n elements.

nCr "n choose r"

= n!/((r!(n-r)!)

Properties of binomial coefficients

i. nC0=1

ii. nCn=1

iii. nC1=n

iv. nCk=nC(n-k)

v. (n+1)C(k+1)=((n+1)C(k+1))(nCk)

vi. nCk=((n-k+1)Ck)(nC(k-1))

Pascals identity

nCk=((n-1)C(k-1))+((n-1)Ck)

Symmetry

nCk=(nC(n-k))

The binomial theorem

Let a+b be real numbers. Then for all neZ n≤z: (a+b)^{n}=Σ(nCk)a^{n-k}b^{k}

General term

(nCk)a^{n-k}b^{k}

Dual E* of E

The equation obtained by replacing each occurence of:

∪ with ∩

∩ with ∪

U with Ø

Ø with U

Principle of duality

States that if any equation E is an identity then its dual E* is also an identity.

Finite Set

Contains exactly m distinct element where m is a non-negative integer.

The number of elements in A is denoted by |A|.

Otherwise, a set is infinite.

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|A∪B|

= |A| + |B| - |A∩B|

Class

a "set of sets"

Power set

For a set S, the class of all subsets of S.

Ordered pair

Denoted by (a,b), where a is called the first element and b is the second element.

(a,b) = (c,d) iff a=c and b=d

Cartesian product

For sets A and B, the set of ordered pairs (a,b) so a∈A and b∈B.

Written as A×B={(a,b):a∈A,b∈B}

Cartesian plane

ℝ^{2}

Function

A rule which assigns to each element of set A exactly one element of B.

Image (of x under f)

If x∈A, then the other unique element of B which f assigns to x is denoted by f(x).

Domain and Range

The set A is the domain of a function and B is the range.

One-to-one

When different elements of A are always assigned different images by f.

Onto

When every element of B is the image of at least one element of A

Permutation

Any arrangement of objects where order matters.

Combination

Any arrangement of objects where order doesn't matter.

P(n,r) or nPr

= n!/(k-r)!

nCr

= n!/(r!(n-r)!)

Pascal's Triangle

For all integers n>k>0, (nCk)=((n-1)C(k-1)+(n-1)Ck)

The Binomial Theorem

Any algebraic expression which is the sum of two terms is called a binomial.

Let a and b be real numbers. Then for any integer n≥0, (a+b)^{n}=n∑k=0(nCk)a^{n-k}b^{k}.

Real part of z.

a, in z=(a,b)∈C.

Imaginary part of z.

b, in z=(a,b)∈C.

Purely real number

A complex number with 0 imaginary part.

Purely imaginary number

A complex number with 0 real part.

Additive identity.

z∈C, if for all w∈C, w+z=w.

(0,0) in C.

Multiplicative identity

z∈C, iff for all w∈C, zw=w.

(1,0) in C.

Additive inverse

Let (a,b)∈C. Then z∈C is an additive inverse iff (a,b)+z=(0,0) and is denoted by -(a,b).

Multiplicative inverse

Let (a,b)∈C with (a,b)≠0. Then w∈C is a multiplicative inverse of (a,b) iff (a,b)w=(1,0) and is denoted by (a,b)^{-1}

Quotient

z^{1}/z^{2} is the complex number z_{1}×z_{2}^{-1.}

Complex conjugate

Of (a,b), it is (a,-b) and is written as:

____

(a,b)=(a-b)

Modulus

Of w, it is denoted by |w|=√(a^{2}+b^{2}).

Properties of the modulus (let w,z∈C)

i. |w|=0, iff w=(0,0)

ii. |w|=|wconj|

iii. |w|=(wwconj)^{1/2}

iv. |wz|=|w|×|z|

v. Triangle inequality, |w+z|≤|w|+|z|

vi. |wz^{-1}|=|w|/|z| whenever z≠=(0,0)

vii.|z^{-1}|=|1/2|=1/|2| whenever z≠(0,0)

Standard form

a+ib(≡a+bi)

Argand Diagram

A graph of the complex number (a,b)

Notes:

1 The complex conjugate numbers w and conj.w are reflections of each other in the real axis

2 The numbers w and -w are reflections of each other on the origin

3 The modulus |w| is the distance from the origin to the to the point P(a,b) or length of the vector representing w

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More notes on Argand Diagrams:

4 The standard form w=a+ib is often called the Cartesian or rectangular form of the complex number because a and b are the Cartesian coordinates of the point P

5 Because of the correspondance with the Cartesian coordinates x and y, we often write z=x+iy when discussing argand complex numbers

Radian

The angle between arms when the length of an arc is one unit.

Congruent

When two triangles can superimposed on one another only through translations and rotations

Similar

When two triangles have corresponding angles, but their sides are not.

Cosine

The x-coordinate

Sine

The y-coordinate

Tan

=sinθ/cosθ

=y/x, x≠0

Trig Identities

sin(θ+π)=-sinθ

sin(π/2-θ)=cosθ

cos(θ+π)=-cosθ

cos(π/2-θ)=sinθ

[Sin and cos are periodic of period 2π]

sin(-θ)=-sinθ

cos(-θ)=-cosθ

Co-functions

The name for the cos and sin functions.

Odd function

The type of functions sin is.

Even function

The type of function cos is.

Pythagorean Identity

cos^{2}θ+sin^{2}θ=1

Sum of angles

cos(α+β)=cosαcosβ-sinαsinβ

sin(α+β)=sinαcosβ+sinβcosα

Half angle and double angle formulas

sin2θ=2sinθcosθ

cos2θ=cos^{2}θ-sin^{2}θ

sin^{2}θ=(1-cos2θ)/2

cos^{2}θ=(1+cos2θ)/2

SOHCAHTOA

sinx=opp/hyp

cosx=adj/hyp

tanx=opp/adj

Law of sines

(sinA/a)=(sinB/b)=(sinC/c)

Law of cosines

a^{2}=b^{2}+c^{2}-2bc(cosA)

cosA=(b^{2}+c^{2}-a^{2})/2bc

The polar form

For (a,b)s a+bi∈C.

r(cosθ+isinθ)=rcisθ

θ is the arguement of the complex number.

The number r≥0 is the modulus of (a,b).

cosθ+isinθ=cisθ is the phase factor.

Reference angle

θ', the angle between the x-axis and the vector (a,b). Can be found using cos(θ')=|a/r| or sin(θ')=|b/r|. θ is then found:

2nd Quadrant; θ=π-θ'

3rd Quadrant; θ=π+θ'

4th Quadrant; θ=2π-θ'

Polar form multiplication

z_{1}z_{2}=r_{1}r_{2}cis(θ_{1}+θ_{2)}_{}_{}

Polar form division

z_{2}/z_{1}=r_{2}/r_{1} cis(θ_{2}-θ_{1)}_{}_{}

Polar form exponentiation

z^{2}=r^{2} cis(2θ)

De Moivre's Theorem

Let z=r cisθ be any non-zero complex number. Let n∈Z, then z^{n}=r^{n} cis(nθ).

[See proof on page 102]

Roots of unity

The n^{th} roots of w=1

Square root

W, iff w×w=conj.z

Addition with complex numbers

(a+bi)+(c+di)=(a+c)+(b+d)i

(a,b)+(c,d)=(a+c,b+d)

Subtraction with complex numbers

(a+bi)-(c+di)=(a-c)+(b-d)i

(a,b)-(c,d)=(a-c,b-d)

Multiplication with complex numbers

(a+bi) (c+di)

=ac+adi+bci+bdi^{2}

=ac+adi+bci-bd

=(ac-bd)+(ad+bc)i

(a,b)×(c,d)

=(ac-db,ad+bc)

^{}

Multiplying by a real

a(c+di)=ac+adi

Properties of complex number operations

Commutative

i. zii. z_{1}z_{2}=z_{2}z_{1}

Associative

iii. (z_{1}+z_{2})+z_{3}=z_{1}+(z_{2}+z_{3})

iv. (z_{1}z_{2})z_{3}=z_{1}(z_{2}z_{3})

Distributive

v. (z_{1}+z_{2})z_{3}=z_{1}z_{3}+z_{2}z_{3}

vi. z_{1}(z_{2}+z_{3})=z_{1}z_{2}+z_{1}z_{3}

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