Arithmetics_160205_01

© Art Traynor 2011
Mathematics
Definition
Mathematics
Wiki: “ Mathematics ”
1564 – 1642
Galileo Galilei
Grand Duchy of Tuscany
( Duchy of Florence )
City of Pisa
Mathematics – A Language
“ The universe cannot be read until we have learned the language and
become familiar with the characters in which it is written. It is written
in mathematical language…without which means it is humanly
impossible to comprehend a single word.
Without these, one is wandering about in a dark labyrinth. ”

© Art Traynor 2011
Mathematics
Definition
Algebra – A Mathematical Grammar
Mathematics
A formalized system ( a language ) for the transmission of
information encoded by number
Algebra
A system of construction by which
mathematical expressions are well-formed
Expression
Symbol Operation Relation
Designate expression
elements or Operands
Transformations capable of
rendering an expression
into a relation
A mathematical structure
between operands
represented by a well-formed
expression
A well-formed symbolic representation of operands, of discrete arity, upon which one
or more operations can structure a Relation
1. Identifies the explanans
by non-tautological
correspondences
Definition
2. Isolates the explanans
as a proper subset from
its constituent
correspondences
3. Terminology
a. Maximal parsimony
b. Maximal syntactic
generality
4. Examples
a. Trivial
b. Superficial
Mathematics

© Art Traynor 2011
Mathematics
Disciplines
Algebra
One of the disciplines within the field of Mathematics
Mathematics
Others are Arithmetic, Geometry,
Number Theory, & Analysis

The study of expressions of symbols ( sets ) and the
well-formed rules by which they might be consistently
manipulated.

Algebra
Elementary Algebra
Abstract Algebra
A class of Structure defined by the object Set and
its Operations

Linear Algebra
Mathematics

© Art Traynor 2011
Mathematics
Definitions
Expression
into a relation
A mathematical structure
between operands
represented by a well-formed
expression
A well-formed symbolic representation of operands, of discrete arity, upon which one
or more operations may structure a Relation
Expression – A Mathematical Sentence
Proposition
A declarative expression
asserting a fact, the truth
value of which can be
ascertained
Formula
A concise symbolic
expression positing a relationVariablesConstants
An alphabetic character
representing a number the
value of which is arbitrary,
unspecified, or unknown
Operands ( Terms )
A transformation
invariant scalar quantity
Mathematics

© Art Traynor 2011
Mathematics
Definitions
Expression
into a relation
A mathematical structure between operands represented
by a well-formed expression
Proposition
asserting a fact, the truth
value of which can be
ascertained
Formula
A concise symbolic
expression positing a relation
VariablesConstants
An alphabetic character
representing a number the
value of which is arbitrary,
unspecified, or unknown
Operands ( Terms )
A transformation
invariant scalar quantity
Equation
A formula stating an
equivalency class relation
Inequality
A formula stating a relation
among operand cardinalities
Function
A Relation between a Set of inputs and a
Set of permissible outputs whereby each
input is assigned to exactly one output
Univariate: an equation containing
only one variable
Multivariate: an equation containing
more than one variable
(e.g. Polynomial)
Mathematics

© Art Traynor 2011
Mathematics
Definitions
Expression
Proposition Formula
VariablesConstants
Operands ( Terms )
Equation
A formula stating an
equivalency class relation
Linear Equation
An equation in which each term is either
a constant or the product of a constant
and (a) variable[s] of the first order
Mathematics

© Art Traynor 2011
Mathematics
Expression
Mathematical Expression
A Mathematical Expression is a precursive finite composition
to a Mathematical Statement or Proposition
( e.g. Equation) consisting of:

a finite combination of Symbols
possessing discrete Arity
Expression
A well-formed symbolic
representation of operands, of
discrete arity, upon which one
or more operations can
structure a Relation
that is Well-Formed
Mathematics

© Art Traynor 2011
Mathematics
Arity
Arity
Expression
The enumeration of discrete symbolic elements ( Operands )
comprising a Mathematical Expression is defined as
its Arity

The Arity of an Expression is represented by
a non-negative integer index variable ( ℤ + or ℕ ),
conventionally “ n ”

A Constant ( Airty n = 0 , index ℕ )or Nullary represents
a term that accepts no Argument

A Unary or Monomial has Airty n = 1
VariablesConstants
Operands
Expression
A relation can not be defined for
Expressions of arity less than
two: n < 2
A Binary or Binomial has Airty n = 2
All expressions possessing Airty n > 1 are Polynomial
also n-ary, Multary, Multiary, or Polyadic


© Art Traynor 2011
Mathematics
Arity
Arity
Expression
VariablesConstants
Operands
Expression
A relation can not be defined for
Expressions of arity less than
two: n < 2
Nullary
Unary
n = 0
n = 1 Monomial
Binary n = 2 Binomial
Ternary n = 3 Trinomial
1-ary
2-ary
3-ary
Quaternary n = 4 Quadranomial4-ary
Quinary n = 5 5-ary
Senary n = 6 6-ary
Septenary n = 7 7-ary
Octary n = 8 8-ary
Nonary n = 9 9-ary
n-ary

© Art Traynor 2011
Mathematics
Equation
Equation
Expression
An Equation is a statement or Proposition
( aka Formula ) purporting to express
an equivalency relation between two Expressions :

Expression
Proposition
asserting a fact whose truth
value can be ascertained
Equation
A symbolic formula, in
the form of a proposition,
expressing an equality
relationship
Formula
A concise symbolic
expression positing a
relationship between
quantities
VariablesConstants
Operands
Symbols
Operations
The Equation is composed of Operand terms and
one or more discrete Transformations ( Operations )
which can render the statement true

© Art Traynor 2011
Mathematics
Equation

Expression
Proposition
Equation
relationship
Formula
A concise symbolic
quantities
Polynomial
Polynomial
An Equation with LOC set consisting of
the arithmetic Transformations
( excluding negative exponentiation )
LOC ( Pn ) = { + , – , x bn ∀ n ≥ 0 , ÷ }
A Term of a Polynomial Equation is a compound
construction composed of a coefficient and variable
in at least one unknown
Polynomial
Equation

© Art Traynor 2011
Mathematics
Equation

Expression
Proposition
Equation
relationship
Formula
A concise symbolic
quantities
Polynomial
Polynomial
Σ an xi
n
i = 0
P( x ) = an xn + an – 1 xn – 1 +…+ ak+1 xk+1 + ak xk +…+ a1 x1 + a0 x0
Variable
Coefficient
Polynomial Term
Polynomial
Equation

© Art Traynor 2011
Mathematics
Linear Equation
Linear Equation
Equation
An Equation consisting of:
Operands that are either
Any Variables are restricted to the First Order n = 1
Linear Equation
An equation in which each term
is either a constant or the
product of a constant and (a)
variable[s] of the first order
Expression
Proposition
Equation
Formula
n Constant(s) or
n A product of Constant(s) and
one or more Variable(s)
The Linear character of the Equation derives from the
geometry of its graph which is a line in the R2 plane

As a Relation the Arity of a Linear Equation must be
at least two, or n ≥ 2 , or a Binomial or greater Polynomial


© Art Traynor 2011
Mathematics
Linear Equation
Equation
Standard Form ( Polynomial )
 Ax + By = C
 Ax1 + By1 = C
For the equation to describe a line ( no curvature )
the variable indices must equal one

 ai xi + ai+1 xi+1 …+ an – 1 xn –1 + an xn = b
 ai xi
1 + ai+1 x 1 …+ an – 1 x 1 + a1 x 1 = bi+1 n – 1 n n
ℝ
2
: a1 x + a2 y = b
ℝ
3
: a1 x + a2 y + a3 z = b
Blitzer, Section 3.2, (Pg. 226)
Section 1.1, (Pg. 2)
Test for Linearity
 A Linear Equation can be expressed in Standard Form
As a species of Polynomial , a Linear Equation
can be expressed in Standard Form
 Every Variable term must be of precise order n = 1

© Art Traynor 2011
Mathematics
Operand
Arity
Operand
the object of a mathematical operation,
a quantity on which an operation is performed
 Arithmetic: a +b = c
Within an expression or set
 “a” and “b” are Operands
 The number of Operands of an Operator is known as its Arity
n Nullary = no Operands
n Unary = one Operand
n Binary = two Operands
n Ternary = three Operands…etc.
In other words…
Operands
“Belong To”
their Operators

© Art Traynor 2011
Mathematics
Linear Equation
Equation
Standard Form
 Ax + By = C
Section 3.2, (Pg. 226)
 Ax1 + By1 = C
For the equation to describe a line ( no curvature )
the variable indices must equal one

 ai xi + ai+1 xi+1 …+ an – 1 xn –1 + an xn = b
 ai xi
1 + ai+1 x 1 …+ an – 1 x 1 + a1 x 1 = bi+1 n – 1 n n
ℝ
2
: a1 x + a2 y = b
ℝ
3
: a1 x + a2 y + a3 z = b

© Art Traynor 2011
Mathematics
Conventions
Notation
Well Formed Mathematical Expression
 Infix Notation
The most common form of notation whereby the symbols indicating
a Mathematical Operation are juxtaposed so as to conjunctively
demarcate the individual Terms of a Mathematical Expression
 Capital Sigma ( Summation ) Notation
The capital Greek letter sigma is an alternate form to infix
notation by which to indicate the operation of addition on a
Sequence of Terms
Σi = 1
n
ai = ai + ai+1 +…+ ak – 1 + ak + ak+1 +…+ an – 1 + an + an+1 …
a +b = c

© Art Traynor 2011
Mathematics
Operand
Addition
Operand
 Arithmetic-Addition: a +b = c
n Arithmetic Operands are more precisely denoted as Addends, Summands , or Terms
n The first arithmetic Operand (in the order of operation) can be yet further denoted
as an Augend, though this nomenclature has been deprecated due to the broad
application of the commutative property of addition
 “+” is the Operator indicating addition as the operation to performed
“c” is the Sum of the Operands

© Art Traynor 2011
Mathematics
Operand
 Arithmetic: a +b = c
 The number of Operands of an Operator is known as its Arity
n Nullary = no Operands
n Unary = one Operand
n Binary = two Operands
n Ternary = three Operands…etc.
In other words…
Operands
“Belong To”
their Operators
Operand
Arity

© Art Traynor 2011
Mathematics
Operand
 Arithmetic-Subtraction: a – b = c
n The first arithmetic-subtractive Operand ( “ a ” or the term which precedes the
minus sign in the order of operation) can be yet further denoted as the Minuend
 “ – ” is the Operator indicating subtraction as the operation to performed
“c” is the Difference of the Operands (Minuend less Subtrahend)
n The second arithmetic-subtractive Operand ( “ b ” or the term which follows the
minus sign in the order of operation) can be yet further denoted as the Subtrahend
 As subtraction is neither commutative or associative, a more versatile method to compute
a difference entails substituting the complement of the Subtrahend into the expression
and changing the operation to an arithmetic addition: a + ( – b ) = c
Operand
Subtraction

© Art Traynor 2011
Mathematics
Summation
Arithmetics
Summation
 The operation indicating the summation of a sequence of terms can
be represented by the Greek capital letter sigma Σ , and is denoted
by the following:
Swok Section 5.3, (Pg. 257)
Sequence: A collection of real numbers that is in one-to-one
correspondence with ℤ+ ( the set of positive integers ),
the set { a1 , a2 , a3 , a4 , a5 , a6 }
 Swok Section 11.1, (Pg. 520)
Term: A symbol representing the elements of the sequence,
the “ a ” symbol in: ai = a1 + a2 + a3 + a4 + a5 + a6

Limit: A value or expression supplying the initial value ( lower limit ) and
terminating value ( upper limit ) for the summation operation

Index: An incremented variable representing successive integer values,
the “ i ” symbol in: ai = a1 + ak – 1 + ak + ak+1 …+ an – 1 + an

Σ1
6
Σi = 1
n = 6
Σ 1
6
aiΣ1
6
Σ 1
6
ai Σi = 1
n = 6
ai

© Art Traynor 2011
Mathematics
Summation
Summation
 The operation indicating the summation of a sequence of terms can
be represented by the Greek capital letter sigma Σ , and is denoted
by the following:
Sequence: the set { a1 , a2 , a3 , a4 , a5 , a6 } Swok Section 11.1, (Pg. 520)
Term: the “ a ” symbol in: ai = a1 + a2 + a3 + a4 + a5 + a6
Limit: A value or expression supplying the initial value ( lower limit ) and
terminating value ( upper limit ) for the summation operation

Index: the “ i ” symbol in: ai = a1 + ak – 1 + ak + ak+1 …+ an – 1 + an
Σ1
6
Σi = 1
n = 6
Σ 1
6
aiΣ1
6
Σ 1
6
ai Σi = 1
n = 6
ai
Yielding:
Σi = 1
n
ai = a1 + ak – 1 + ak + ak+1 …+ an – 1 + an
Arithmetics

© Art Traynor 2011
Mathematics
Summation
Summation – Translation of the Index Variable
 The incrementation variable can be turned into an expression
permitting the summation of a Sequence ( i.e. a Series ) to translate
to an alternate initiating value or incrementation interval
Σ1
Σi = 1
n =
Σ 1 aiΣ1
Σ 1 ai Σi = 1
n =
ai
Yielding:
Σi = 1
ai = a1 + ai + 1 +…+ an – 1 + an+…
∞
∞
∞ ∞ ∞
∞
→ → Σi = k + 1
n =
ai
∞
∞
Σi = k + 1
ai = a2 + ak + ( i +1 ) +…+ ak + ( n – 1 ) + ak + n +…
∞
Arithmetics

© Art Traynor 2011
Mathematics
Summation
Properties Of Summation
Swok, Section 5.3,
Theorem 5.10, pg 258Σi = 1
n
c = nc Summation of a Constant
Σi = 1
n
( ai + bi ) = Σi = 1
n
ai + Σi = 1
n
bi Summation of a Sum Swok, Section 5.3,
Theorem 5.11, pg 258
Σi = 1
n
( ai – bi ) = Σi = 1
n
ai – Σi = 1
n
bi Summation of a Difference
Σi = 1
n
cai = c (Σai )i = 1
n
for every real
number “ c ”
Scalar Multiple
of a Summation
Swok, Section 5.3,
Swok, Section 5.3,
Arithmetics

© Art Traynor 2011
Mathematics
Σi = 1
n
k2 = 12 + 22 +…+ n2 =
n ( n + 1 ) ( 2n + 1 )
2
Summation
Properties Of Summation ( Closed Forms )
Closed Form of
First Degree Term
Summation
Swok, Section 5.3,
Theorem 5.12 (ii),
pg 259
Swok, Section 5.3,
Theorem 5.12 (iii),
pg 259
Σi = 1
n
k = 1 + 2 + …+ n =
n ( n + 1 )
2
Swok, Section 5.3,
Theorem 5.12 (i),
pg 259
Σi = 1
n
k3 = 13 + 23 +…+ n3 =
n ( n + 1 )
2
2
Closed Form of
Second Degree Term
Summation
Closed Form of
Third Degree Term
Summation
Arithmetics

© Art Traynor 2011
Mathematics
Transformation
Multiplication
Operand
 Multiplicative: a(b + c)
 “a” and “(b + c)” and respectively the Operands, as are “b” and “c”
n “(b+ c)” is more precisely denoted as an Expression consisting of two Terms
 There are two Operators
n The “+” within the two Term Operand Expression “(b+ c)”
indicates addition as the operation
n The “( )” indicates multiplication as the operation to be performed
on the two Term Operand Expression “b + c”
 The distributive property dictates the “distribution” of the common Factor “a” between
the two Term Operand Expression “b + c” yielding “ab + ac” or alternatively
“ac + ab” by further application of the commutative property of addition

© Art Traynor 2011
Mathematics
Operand
 “a” and “(b + c)” are the principal Operands, in the order of operations
 “b” and “c” are the secondary, or subsidiary Operands, in the order of operations
the two Term Operand Expression “b + c” yielding “ab + ac”
 “ac + ab” is also a valid ordering of the distributed common Factor by
further application of the commutative property of addition
Multiplication
Transformation

© Art Traynor 2011
Mathematics
Operand
 “a” and “(b + c)” respectively are Operands, (as are “b” and “c” )
 There are two Operators
n The “+” within the two Term Operand Expression “(b+ c)”
indicates addition as the operation
n The “( )” indicates multiplication as the operation to be performed
on the two Term Operand Expression “b + c”
the two Term Operand Expression “b + c” yielding “ab + ac” or alternatively
“ac + ab” by further application of the commutative property of addition
Multiplication
Transformation

© Art Traynor 2011
Mathematics
Operand

 “a” and “b ” are Operands
Multiplicative: a x b = ai + ai+1 …+ ab-1 + ab = Σi = 1
b
ai
 “a” however is more precisely denoted as a Multiplicand or Factor
(the number to be multiplied), evaluating the expression left-to-right
 “b” however is more precisely denoted as a Multiplier
(the number of multiples), , evaluating the expression left-to-right
n In an algebraic expression such as 3ab2 the multiplier “3” however
is more precisely denoted as a Coefficient
Multiplicand & Multiplier are not
always unambiguously classified
due to the commutative property of
multiplication
The operand “ onto which ” the
scaling operation is to be
performed ( the operand to be
scaled)??...yes, I think so.
Multiplication
Transformation

© Art Traynor 2011
Mathematics
Transformation
Multiplication
Multiplication
whereby one object or quantity
is scaled by another

A mathematic operation
Repeated Addition: a x b = ai + ai+1 …+ ab-1 + ab = Σi = 1
b
The Multiplication operation can be restated/expanded as a summation of identical terms
 Geometric Interpretation: area

 Infix Notation: a x b = y, or Cross Product where “x” indicates the multiplication operation
and “ = ” denotes equivalence (vector multiplication)
Notation
 Dot Product: a · b = y, where “ ·” indicates the multiplication operation (scalar multiplication)
 Asterisk: a * b = y, where “*” indicates the multiplication operation
ai
 Juxtaposition: ab = y implies the multiplication operation by proximity

© Art Traynor 2011
Mathematics
Multiplication as Summation
Multiplication
Multiplication as Summation
Wiki: Product (Mathematics)
Σi = 1
s
Σj = 1
r
r · s = r = s
If s = r then exponentiation is effected:
Σi = 1
r
r · r = r = r2
A matrix of “ r ” rows and “ s ” columns is represented
in summation notation as either:
It’s not clear how we can produce
higher indices than two utilizing
summation notation??

© Art Traynor 2011
Mathematics
Product of Sequence Summations
Multiplication
Product of Sums
Rosen, Section 2.4, pg 165
Wolfram Mathworld, Double Series
Σi = 1
m
Σj = 1
n
xi yj
Σi = 1
m
yjΣj = 1
n
xi
 The inner sum ( LHS ) is expanded first
( x1 + x2 +…+ xm –1 + xm )
Σi = 1
m
Σj = 1
n
xi yj
The summation operators ( Capital Sigmas )
are juxtaposed ( indicating that a product is
to be evaluated ) and the LHS argument
remains in the LHS position of the
composed Summation product
( x1 + x2 +…+ xm –1 + xm )Σj = 1
n
yj

© Art Traynor 2011
Mathematics
Wolfram Mathworld: “Double Series”
Σi = 1
m
Σj = 1
n
xi yj
Σi = 1
m
yjΣj = 1
n
xi
 Then the outer sum ( RHS ) is expanded and distributed over each of the
LHS sums ( as a product with each of the ith through mth inner sum terms )
( x1 + x2 +…+ xm –1 + xm )
Σi = 1
m
Σj = 1
n
xi yj
Σj = 1
n
yj
( x1 + x2 +…+ xm –1 + xm ) y1 + ( x1 + x2 +…+ xm –1 + xm ) y2 +…
( x1 + x2 +…+ xm –1 + xm ) yn –1 + ( x1 + x2 +…+ xm –1 + xm ) yn
Multiplication
Product of Sums

© Art Traynor 2011
Mathematics
Wolfram Mathworld, Double Series
Σi = 1
m
Σj = 1
n
xi yj
Σi = 1
m
yjΣj = 1
n
xi
( x1 + x2 +…+ xm –1 + xm )
Σi = 1
m
Σj = 1
n
xi yj
Σj = 1
n
yj
( x1 + x2 +…+ xm –1 + xm ) y1 + ( x1 + x2 +…+ xm –1 + xm ) y2 +…
( x1 + x2 +…+ xm –1 + xm ) yn –1 + ( x1 + x2 +…+ xm –1 + xm ) yn
x1 y1 + x2 y1 +…+ xm –1 y1 + xm y1 + x1 y2 + x2 y2 +…+ xm –1 y2 + xm y2 +…
x1 yn –1 + x2 yn –1 +…+ xm –1 yn –1 + xm yn –1 + x1 yn + x2 yn +…+ xm –1 yn + xm yn
Multiplication
Product of Sums

© Art Traynor 2011
Mathematics
Product of a Sequence ( Capital Pi Notation )
Multiplication
Sequence Product
n
k
( ( n – k ) + 1 )
1 If k = 0
k!
If 0 < k
Πk = 1
n
ai = ai x ai+1 x … x ak – 1 x ak x ak+1 x … x an – 1 x anΠi = 1
n
Q: How does the Capital Pi operator
differ from repeated multiplication
( i.e. Exponentiation )?
Wiki : “Multiplication”
A: Exponentiation is a product of ( a
sequence of ) identical elements,
or a “sequence” of a constant
term
i = 1 x 2 x … x n = n!Πi = 1
n
Example ( Canonical ) :
Example ( Trivial ) :
2 = 2 x 2 x … x 2 x 2 x 2 x … x 2 x 2 ( n-times ) = 2n
Πi = 1
n
Example ( Superficial ) :
Example ( Combinatorics – n choose k ) :
n = 4

© Art Traynor 2011
Mathematics
Product of a Sequence ( Capital Pi Notation )
Multiplication
Sequence Product
n
k
( ( n – k ) + 1 )
1 If k = 0
k!
If 0 < kΠk = 1
n
Example ( Combinatorics – n choose k ) :
n = 4n
k
= 4
= 0
1
0!
n
k
= 4
= 1
( ( 4 – 1 ) + 1 )
1!
n
k
= 4
= 2
( ( 4 – 1 ) + 1 ) · ( ( 4 – 2 ) + 1 )
2!
→
( ( 3 ) + 1 )
1
→
4
1
→ 4
1
1
→ → 1
4 · (( 2 ) + 1 )
1 · 2
→ → → 6
4 · 3
2
→
12
2
n
k
= 4
= 3
4 · 3 · ( ( 4 – 3 ) + 1 )
3!
12 · (( 1 ) + 1 )
1 · 2· 3
→ → → 4
12 · 2
6
→
24
6
n
k
= 4
= 4
4 · 3 · 2 ( ( 4 – 4 ) + 1 )
4!
24 · (( 0 ) + 1 )
1 · 2· 3 · 4
→ → → 1
24 · 1
24
→
24
24

© Art Traynor 2011
Mathematics
Transformation
Division
Division
whereby one object or quantity is partitioned by another
A mathematic (arithmetical) operation,

 Obelus Notation: If a = b x c, then a ÷ b = c, where “ ÷”, or the Obelus, indicates the
division operation and “ = ” denotes equivalence
Notation (Short Division)
constituting an inverse operation to that of multiplication,
n “a” is the Dividend, or that quantity to be partitioned
n “b” is the Divisor, or the count of partitions to be formed
n “c” is the Quotient, or size of the resultant groups formed by the operation
 Repeated Subtraction…
a
b( )

© Art Traynor 2011
Mathematics
Transformation
Division
Division

 Vinculum Notation: = c, where “ – ” the fraction bar or Vinculum, indicates the division
operation (“a” is divided by “b”) and “ = ” denotes equivalence
Notation (Short Division)
n “a” is the Numerator, or that quantity to be partitioned
n “b” is the Denominator, or the count of partitions to be formed
n “c” is the Quotient, or size of the resultant groups formed by the operation
a
b
 Other Conventions
n a/b – with a slash indicating division and the Numerator preceding the Denominator
n
a/b – with a Solidus indicating the division operation and the Numerator elevated over
the Denominator

© Art Traynor 2011
Mathematics
Transformation
Division
Division


Notation (Long Division)
ab
c
n “ a ” is the Dividend, or that quantity to be partitioned
n “ b ” is the Divisor, or the count of partitions to be formed
n “ c ” is the Quotient, or size of the resultant groups formed by the operation
 Vinculum Notation: = c , where “ – ” the fraction bar or Vinculum, indicates
the division operation ( “ a ” is divided by “ b ” ) and “ = ” denotes equivalence
n “ a ” is the Numerator , or that quantity to be partitioned
n “ b ” is the Denominator , or the count of partitions to be formed
n “ c ” is the Quotient , or size of the resultant groups formed by the operation
a
b

© Art Traynor 2011
Mathematics
Definitions
Fractions
Fractions
A fraction is a mathematic expression
whereby one number or collection of terms (numerator),
represents an equal partition of another number or collection of terms (denominator)
 The fraction represents a ratio: “ n : d ” (the part ‘ n ’ to the whole ‘ d ’ )
 The division operation is implied: n ÷ d
 The fraction expresses an enumeration of equal parts (numerator) while indicating the
denomination of those parts which constitute a unity or whole of the parts (denominator)
Rational Numbers
The set of all numbers which can be expressed in the form n/d where d ≠ 0
is the set of rational numbers, denoted Q (standing for quotient)
Forms
A common fraction (a.k.a. Vulgar Fraction, or Simple Fraction)
is a rational number where the integers can be written as n/d (solidus form) or (vinculum
form or “ fraction bar ” form)
n
d

© Art Traynor 2011
Mathematics
Definitions
Fractions
Fraction Forms
A common fraction can be further characterized as either proper or improper
 Proper Fraction
 The numerator is less than the denominator: n < d
 The absolute value of the fraction is less than one: < 1| |n
d
 Improper Fraction
 The numerator is less than the denominator: n > d
 The absolute value of the fraction is greater than one: > 1| |n
d
 Complex or Compound Fraction
 Where either the numerator or the denominator terms (or both) is a fraction itself
a
b
c
d
= ÷ = x
a
b
c
d
a
b
d
c

© Art Traynor 2011
Mathematics
Definitions
Fractions
Fractions – Lowest Common Denominator
Examples
 1 + y
2
– x
+ ·y
– x
1
1
y
– x
+ y2
x2
1
1
1 · y2 is LCD
+ y2
x2
y2
y2
y2
y2 + x2

© Art Traynor 2011
Mathematics
Definitions
Order of Operations
Order of Operations
Which dictate the hierarchy or sequence of operations
 Multiplication precedes Addition: a + b x c
Otherwise know as Rules of Precedence
 The multiplication operator “x” has Operands “b” and “c” respectively
 The addition operator “+” has Operands “a” and “b x c” respectively
 Order Preservation
 where a > 0, if b > c, then ab > ac
 where a < 0, if b > c, then ab < ac (Multiplication by a negative number reverses order)

© Art Traynor 2011
Mathematics
Properties
Associative
Associative Property
Within an expression
 Associate = “Group”
 Addition: ( a + b ) + c = a + ( b + c )
 Multiplication: a( bc ) = ( ab ) c
as long as the sequence of the operands is not changed
containing two or more occurrences in a row of the same operator
the order in which the operations are performed does not matter
Changes Order of Operations
as per “PEM-DAS”, Parentheses
are the principal or first operation
Parenthesis are the “first to fight”
Always entails parentheses
 Associative Property: Governs Order of Operations
( a + b ) + c = a + ( b + c )
1
2 2
1

© Art Traynor 2011
Mathematics
Commutative
Commutative Property
Within an expression
 Commute = “Move Around”
 Addition: a + b = b + a
 Multiplication: a · b = b · a
containing two or more occurrences in a row of the same operator
changing the order of the operands does not change the result
Properties
Re-Orders Terms
Does Not Change
Order of Operations – PEM-DAS
 Commutative Property: Governs Order of Terms

© Art Traynor 2011
Mathematics
Distributive
Distributive Property
Within an additive expression
 Distributive = “Distribute”
 Left-Distributive: a(b + c) = ab + ac
 Right Distributive: (b + c)a = ba + ca
containing a common factor
changing the sequence over which the multiplicand is distributed does not change the result
 Addition distributes over Multiplication – Left & Right Distributive
 A Commutative expression exhibits Logical Equivalency – Left & Right Distributive
Properties

© Art Traynor 2011
Mathematics
Identity
Identity Element
 Identity = “to make, represent to be, or regard or treat as the same or identical”
 Left-Identity: e(a) = a
 Right Identity: (a)e = a
an Identity or Neutral Element is one which
when combined with other terms or elements,
 Left & Right (symmetric) Identity
does not change other terms or elements nor alter the result of the expression.
 Addition: Identity Element = 0; 0 + a = a
 Multiplication: Identity Element =1; 1(a) = a
Element
Think not of “ 1 ” as a mere integer but more as representing
“ Unity ” in the functional, definitive sense of its status as the
identity element of multiplication: anything scaled by one is
returned unto itself; even the reductive might of zero is hapless
to superordinate the power of Unity.

© Art Traynor 2011
Mathematics
Inverse Element
 Inverse = “reversed in position, order, direction, or tendency”
 Left-Inverse: a is the Left Inverse of b (Left Invertible)
 Right-Inverse: b is the Right Inverse of a (Right Invertible)
an Element which
when combined with other terms or elements,
 Left & Right (symmetric) Inverse: if x is both a Left & Right Inverse of y,
yields the Identity Element
 Addition: Negation/Additive Inverse; a + (-a) = 0
 Multiplication: Reciprocal/Multiplicative Inverse; (a)( 1/a ) = 1
For (a)(b) = e, where e is an Identity Element (e.g. 1, Multiplicative) then
then x is a Two-Sided (symmetric) Inverse of y
Inverse
Element

© Art Traynor 2011
Mathematics
Zero Element
 Multiplication: a x 0 = 0; any number multiplied by zero yields zero,
the Zero Property of Multiplication
Zero
Element

© Art Traynor 2011
Mathematics
Negation Element
 Multiplication: (-1) x a = -a; negative one times any number yields the opposite of the number
Negation
Element

© Art Traynor 2011
Mathematics
Equality
Integer Properties
Axioms of Integer Equality
Reflexive
Symmetric
a = a
“a” is ‘reflected’ onto itself
O’Leary, Section 5.1,
Pg. 171
if a = b then b = a
Transitiveif a = b and b = c then a = c
Additive Equality
Multiplicative Equality
if a = b then a + c = b + c
Also see:
Blitzer, Section 2.1,
pg. 115
These Equality Axioms are
(for purposes of proof)
those that enable us to
algebraically manipulate
equivalent expressions.
if a = b then ac = bc

© Art Traynor 2011
Mathematics
Axioms of Integer Operation
Integer Properties
Integer Axioms Under Addition & Multiplication O’Leary, Section 5.1,
Pg. 172
Additive Identitya + 0 = a
Zero ( 0 ) is the Identity
Element for the operation
of Addition
Additive Inverse ( Negation )
a + ( – a ) = 0 Rosen, Appendix 1,
Pg. 2
( ∃ a ℤ ) ( x + a ) = 0

© Art Traynor 2011
Mathematics
Equality
Equality Property (Identity Relation)
 Reflexive Relation: a = a (reflecting onto itself)
 Symmetric Relation: if a = b then b = a (the equality need not be ordered)
 Transitive Relation: if a = b and b = c then a = c (“moving across”)
Equality is an archetype of Equivalence which includes three constituent relations:
Integer Properties

© Art Traynor 2011
Mathematics
Discrete Structures
Sets
Properties of Relations
A relation R on a set A is transitive
If whenever ( a, b )  R and
Transitive Property
( b, c )  R then
( a, c )  R for all a, b, c  R
"a "b "c ((( a, b ) R  ( b, c ) R )  ( a, c ) R )
A relation on a set A is an equivalence relation if it is
Equivalence Relation
reflexive, symmetric, transitive

© Art Traynor 2011
Mathematics
Definitions
Inequality
Inequality
between objects which are not identical
A mathematic Relation
 “Not Equal To”: a ≠ b
 Does not permit inference of relative magnitude (scalar) between elements “a” and “b” unless
they are constituents of an ordered set (e.g. Integers or Real Numbers)
 Strict Inequalities
 a b ; a is Greater Than b ( GT )
 Inequalities (Non-Strict)
 a ≤ b ; a is Less Than or Equal to b
 a ≥ b ; a is Greater Than or Equal to b
 Inequalities (by orders of magnitude)
 a « b; a is much less than b
 a » b; a is much greater than b

© Art Traynor 2011
Mathematics
Properties
Inequality
Inequality
 Transitive Property of Inequality (also true of non-strict inequality)
 If a > b and b > c , then a > c
 If a b and b = c, then a > c
 If a b > cn
Chained Notation Variant: a < b < cn
TPIE-CNV

© Art Traynor 2011
Mathematics
Properties
Inequality
Inequality
 Special Case: Multi-Directional Inequality Expressions ( MDIE )
Expressions such as a c ≤ d are scarcely well-
formed but that they can be decomposed into their constituent
inequalities and the string evaluated as a Logical Conjunction
n
Example Problems Dot Com :
“ Inequality ”
a c ≤ d → ( a c ) ⋀ ( c ≤ d )
Which is more adroitly manipulated if restated with uni-directional inequality
operators” ( less-than , number line monotonic increasing format “ < ” preferred )
a c ≤ d → ( a < b ) ⋀ ( c < b ) ⋀ ( c ≤ d )

© Art Traynor 2011
Mathematics
Properties
Inequality
Inequality
Expressions such as a c ≤ dn
“ Inequality ”
a c ≤ d → ( a < b ) ⋀ ( c < b ) ⋀ ( c ≤ d )
( c ≤ d ) ≡ c ⇒ d
( a < b ) ≡ a ⇒ b
( c < b ) ≡ c ⇒ b
Suppose these inequalities could
be symbolically re-expressed as
a series of implications
p ⇒ q
This re-imagining of the Terms
reveals that the Consequents fail
to allow us to imply anything about
the Antecedents (as the Converse
of an Implication is not logically
equivalent to the Implication
expression from which it is derived

© Art Traynor 2011
Mathematics
Properties
Inequality
Inequality
Expressions such as a c ≤ dn
“ Inequality ”
a c ≤ d → ( a < b ) ⋀ ( c < b ) ⋀ ( c ≤ d )
( c ≤ d )
( a < b )
( c < b )
( a < c ) ⋁ ( a = c ) ⋁ ( c < a )
( b ≤ d ) ⋁ ( b = d ) ⋁ ( d ≤ b )
Possible < and/or ≤ Relations

© Art Traynor 2011
Mathematics
Properties
Inequality
Inequality
Expressions such as a c ≤ d
are scarcely well-formed but that they can be
decomposed into their constituent inequalities and
the string evaluated as a Logical Conjunction
n
“ Inequality ”
a c ≤ d → ( a c ) ⋀ ( c ≤ d )
Which is more adroitly manipulated if restated with uni-directional inequality
operators” ( less-than , number line monotonic increasing format “ < ” preferred )
a c ≤ d → ( a < b ) ⋀ ( c < b ) ⋀ ( c ≤ d )

© Art Traynor 2011
Mathematics
Properties
Inequality
Inequality
 Converse Property of Inequality (also true of non-strict inequality)
 a < b, then b > a
 a < b, then b > a

© Art Traynor 2011
Mathematics
Properties
Inequality
Inequality
 Additive Inverse
 If a < b, then –a > –b
 If a > b, then –a < –b
 Multiplicative Inverse (where a & b ≠ 0, and are both +a, +b or –a, –b)
 If a b then 1/a > 1/b
 Multiplicative Inverse (where one of a & b > 0 and the other is < 0)
 If a b then 1/a > 1/b

© Art Traynor 2011
Mathematics
Definitions
Inequality
Properties Of Inequalities
Blitzer, Section 2.7, pg 186If a < b, then a + c 0, then ac < bc
If a < b, then a – c 0, then bc
a
cIf a < b, and c < 0, then >
b
c
Negative Multiplication
Property of Inequality
Blitzer, Section 2.7, pg 186
Swok, Section 11.1, pg 525n ln | a |
n
< ln b → n >
ln b
ln | a |
APIE
PMPIE
NMPIE
Special Case: NMPOI
applied to
Modulus Divisor

© Art Traynor 2011
Mathematics
Properties
Inequality
Inequality
 Chained Notation
 a < b < c can be restated as a < b and b < c by the Transitivity Property of Inequality,
by which it also follows that a < c, hence also:
Example: 4x < 2x + 1 ≤ 3x + 2
n a < b + e < c can be restated as a – e < b < c – e
(– 1) + 4x < 2x +1 – 1
4x – 1 < 2x
22
(– x) + 2x – ½ < x – x
½ + x – ½ < 0 + ½
x < ½
(– 2x) + 2x +1 ≤ 3x +2 – 2x
(– 2) + 1 ≤ x +2 – 2
– 1 ≤ x
x ≥ – 1
– 1 ≤ x < ½

© Art Traynor 2011
Mathematics
Properties
Absolute Value
Absolute Value (Modulus)
The absolute value or Modulus |a | of a real number “ a ” is defined as follows:
 Properties – for b > 0
 |a | b if and only if either a > b or a < – b
|x | =
a if a ≥ 0
– a < 0
 |a | = b if and only if a = b or a = – b
Swok, Section 1.1, Pg. 3
Lay, Section 3.11, Pg. 100
 |a | ≥ 0
 |ab | = |a | · |b |
 |a + b | ≤ |a | + |b |
Triangle Inequality
Modulus Equivalence
of LT Inequality (MEOLTIE)

© Art Traynor 2011
Mathematics
Properties Of Absolute Value
Square Root Equivalence
Non-Negativity
Properties
Absolute Value
|a | = √ a 2
| a | ≥ b
| a | = 0 ⟺ a = 0 Positive Definiteness
| ab | = | a || b | Multiplicative

© Art Traynor 2011
Mathematics
Absolute Value Inequalities
Properties
Absolute Value
 Strict Inequality ( SI )
For a Modulus Expression ( ME ) e.g. | 2x + 3 |
( a set of terms for which an Absolute Value might be ascertained )
constituted of one or more operand terms
and related by strict inequality to a scalar or other expression ,
the following equivalent expressions pertain:
 Less Than
 More Than

© Art Traynor 2011
Mathematics
Properties
Absolute Value
A propositional ME, the terms of which adhere to the form
“ | { an xn } | ” composed with a “Less Than” operator positing the
ME to be of lesser evaluated magnitude than a scalar or other expression
– e.g. | { an xn } | < b , shares a representational equivalence
with the chained notation expression: – b < an xn < b
 Less Than – the property whereby the evaluated magnitude of a
propositional expression stands in decreased monotonic order relative to an
antipodal scalar or expression denoted by the relation of strict inequality.
Chained Notation Form (CNF)
| a | = – b < a < b
Modulus Equivalence of LT Inequality (MEOLTIE)

© Art Traynor 2011
Mathematics
Properties
Absolute Value
“ | { an xn } | ” composed with a “Less Than” operator
positing the ME to be of lesser evaluated magnitude than a
scalar or other expression – e.g. | { an xn } | < b ,
shares a representational equivalence with the chained
notation expression: – b < an xn < b
 Less Than – the property whereby the evaluated magnitude of a propositional
expression stands in decreased monotonic order relative to an antipodal scalar
or expression denoted by the relation of strict inequality.
Example:
| 2x + 2 | < 6
– 6 < 2x + 2 < 6
The chained notation, conjunctive
bifurcation reveals that the “ less
than ” relation is logically equivalent
to an “ AND ” statement, and requires
simultaneous (non-singular)
satisfaction of both bifurcated
propositions to constitute a valid
solution set.
∀ ai |ai |≤ M
– M ≤ ai ≤ M
0
– M – ai ≤ 0 ≤ M – ai
– M – a i M – a i
M– M
|:

© Art Traynor 2011
Mathematics
Properties
Absolute Value
“ | { an xn } | ” composed with a “More Than” operator positing the
ME to be of greater evaluated magnitude than a scalar or other expression
– e.g. | { an xn } | > b , shares a representational equivalence
with the bifurcated disjoint expression: an xn < – b or an xn > b
 Greater Than – the property whereby the evaluated magnitude of a
propositional expression stands in increased monotonic order relative to an
Chained Notation Form (CNF)

© Art Traynor 2011
Mathematics
Properties
Absolute Value
“ | { an xn } | ” composed with a “More Than” operator positing the ME
to be of greater evaluated magnitude than a scalar or other expression
– e.g. | { an xn } | > b , shares a representational equivalence
with the bifurcated disjoint expression: an xn < – b or an xn > b
 Greater Than – the property whereby the evaluated magnitude of a
propositional expression stands in increased monotonic order relative to an
Example:
| 2x – 3 | > 5
2x – 3 < – 5
The disjunctive bifurcation reveals
that the “ greater than ” relation
is logically equivalent to an “ OR ”
statement, and a solution that
satisfies either proposition will
constitute a valid solution set.
2x – 3 > 5
or

© Art Traynor 2011
Mathematics
Properties
Absolute Value
Challenges:
 Less Than
 More Than
| x – 2 | > – 3
| 2x + 2 | < – 1
There is an immediate problem
with each of these expressions,
namely that a Magnitude ( which
is the general resultant to which a
Modulus argument is evaluated )
cannot be negative
In both cases, the solution set can be
more readily apprehended by
eliminating the negative antipode by
factoring each side of the inequality
by negative unity (thus reversing the
inequality)

© Art Traynor 2011
Mathematics
Properties
Absolute Value
Challenges:
 Less Than
 More Than
| x – 2 | > – 3
| 2x + 2 | < – 1
| 2x + 2 |
– 1
>
– 1
– 1
| x – 2 |
– 1
>
– 3
– 1

© Art Traynor 2011
Mathematics
Properties
Absolute Value
Challenges:
 Less Than
 More Than
– | x – 2 | > 3
– | 2x + 2 | < 1
| 2x + 2 |
– 1
>
– 1
– 1
| x – 2 |
– 1
<
– 3
– 1
It is readily apparent that any
real number will satisfy this
expression S = ℝ
It is readily apparent that no
real number will satisfy this
expression S = Ø

© Art Traynor 2011
Mathematics
Definitions
Order of Operations
Order of Operations
 P = parentheses
Please Excuse My Dear Aunt Sues
 E = exponents
 M = multiplication
 D = Division
 A = Addition
 S = Subtraction
Proceeding within the expression from left-to-right

Arithmetics_160205_01

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