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Arithmetics_160205_01

1. Mathematics is defined as a formalized language for transmitting information encoded by numbers. Key figures like Galileo emphasized that the universe is written in mathematical language. 2. Algebra is a mathematical grammar that defines well-formed expressions using symbols, operations, and relations. It is the study of expressions of symbols and rules for their manipulation. 3. Key concepts in algebra include expressions, operands, operations, equations, polynomials, and linear equations. An equation expresses an equality relationship between two expressions using variables, constants, and operations.

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© 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 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 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 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 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 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 Symbol Operation Relation Expression – A Mathematical Sentence 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 A declarative expression 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 An Equation is a statement or Proposition ( aka Formula ) purporting to express an equivalency relation between two Expressions :  Expression Proposition A declarative expression 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 Polynomial The Equation is composed of Operand terms and one or more discrete Transformations ( Operations ) which can render the statement true 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 An Equation is a statement or Proposition ( aka Formula ) purporting to express an equivalency relation between two Expressions :  Expression Proposition A declarative expression 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 Polynomial The Equation is composed of Operand terms and one or more discrete Transformations ( Operations ) which can render the statement true 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 the object of a mathematical operation, a quantity on which an operation is performed  Arithmetic-Addition: a +b = c Within an expression or set  “a” and “b” are Operands 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 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 Operand Arity
© Art Traynor 2011 Mathematics Operand the object of a mathematical operation, a quantity on which an operation is performed  Arithmetic-Subtraction: a – b = c Within an expression or set  “a” and “b” are Operands 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: Swok Section 5.3, (Pg. 257) 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 Swok Section 11.2, (Pg. 539) 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, Theorem 5.11, pg 258 Swok, Section 5.3, Theorem 5.11, pg 258 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 the object of a mathematical operation, a quantity on which an operation is performed  Multiplicative: a(b + c) Within an expression or set  “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 the object of a mathematical operation, a quantity on which an operation is performed  Multiplicative: a(b + c) Within an expression or set  “a” and “(b + c)” are the principal Operands, in the order of operations n “(b+ c)” is more precisely denoted as an Expression consisting of two Terms  “b” and “c” are the secondary, or subsidiary Operands, in the order of operations  The distributive property dictates the “distribution” of the common Factor “a” between 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 the object of a mathematical operation, a quantity on which an operation is performed  Multiplicative: a(b + c) Within an expression or set  “a” and “(b + c)” respectively are 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 Multiplication Transformation
© Art Traynor 2011 Mathematics Operand the object of a mathematical operation, a quantity on which an operation is performed  Within an expression or set  “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 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  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 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 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 Product of Sequence Summations
© Art Traynor 2011 Mathematics 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 ( 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 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 Product of Sequence Summations
© 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 whereby one object or quantity is partitioned by another A mathematic (arithmetical) operation,   Vinculum Notation: = c, where “ – ” the fraction bar or Vinculum, indicates the division operation (“a” is divided by “b”) and “ = ” denotes equivalence Notation (Short Division) constituting an inverse operation to that of multiplication, 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 whereby one object or quantity is partitioned by another A mathematic (arithmetical) operation,   Notation (Long Division) constituting an inverse operation to that of multiplication, 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 Within an expression or set  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 Within an expression or set  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 Less Than b ( LT )  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 between objects which are not identical A mathematic Relation  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 and b = c, then a > c  If a < b and b = c, then a < c Swok, Section 1.1, pg 2 TPIE Wiki: “Inequality (Mathematics)Chained Notation Variant: a > b > cn Chained Notation Variant: a < b < cn TPIE-CNV
© Art Traynor 2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  Transitive Property of Inequality (also true of non-strict inequality)  Special Case: Multi-Directional Inequality Expressions ( MDIE ) Expressions such as a < b > 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 < b > c ≤ d → ( a < b ) ⋀ ( b > c ) ⋀ ( c ≤ d ) Which is more adroitly manipulated if restated with uni-directional inequality operators” ( less-than , number line monotonic increasing format “ < ” preferred ) a < b > c ≤ d → ( a < b ) ⋀ ( c < b ) ⋀ ( c ≤ d )
© Art Traynor 2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  Transitive Property of Inequality (also true of non-strict inequality)  Special Case: Multi-Directional Inequality Expressions ( MDIE ) Expressions such as a < b > c ≤ dn Example Problems Dot Com : “ Inequality ” a < b > 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 between objects which are not identical A mathematic Relation  Transitive Property of Inequality (also true of non-strict inequality)  Special Case: Multi-Directional Inequality Expressions ( MDIE ) Expressions such as a < b > c ≤ dn Example Problems Dot Com : “ Inequality ” a < b > 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 between objects which are not identical A mathematic Relation  Transitive Property of Inequality (also true of non-strict inequality)  Special Case: Multi-Directional Inequality Expressions ( MDIE ) Expressions such as a < b > 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 < b > c ≤ d → ( a < b ) ⋀ ( b > c ) ⋀ ( c ≤ d ) Which is more adroitly manipulated if restated with uni-directional inequality operators” ( less-than , number line monotonic increasing format “ < ” preferred ) a < b > c ≤ d → ( a < b ) ⋀ ( c < b ) ⋀ ( c ≤ d )
© Art Traynor 2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  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 between objects which are not identical A mathematic Relation  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  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  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 < b + c If a < b, and c > 0, then ac < bc If a < b, then a – c < b – c Addition Property of Inequality a cIf a < b, and c > 0, then < b c Positive Multiplication Property of Inequality Blitzer, Section 2.7, pg 186 If a < b, and c < 0, then ac > 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 between objects which are not identical A mathematic Relation  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 – b < a < b  |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 Lay, Section 3.11, Pg. 100  |ab | = |a | · |b |  |a + b | ≤ |a | + |b | Swok, Section 1.1, Pg. 3 Lay, Section 3.11, Pg. 100 Swok, Section 1.1, Pg. 3 Swok, Section 1.1, Pg. 3 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 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: 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 Absolute Value Inequalities Properties Absolute Value  Strict Inequality ( SI ) 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. 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 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: A propositional ME, the terms of which adhere to the form “ | { 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 antipodal scalar or expression denoted by the relation of strict inequality. Chained Notation Form (CNF)
© Art Traynor 2011 Mathematics Absolute Value Inequalities Properties Absolute Value  Strict Inequality ( SI ) A propositional ME, the terms of which adhere to the form “ | { 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 antipodal scalar or expression denoted by the relation of strict inequality. 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 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: 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 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: 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 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: 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

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Arithmetics_160205_01

  • 1.
    © Art Traynor2011 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. ”
  • 2.
    © Art Traynor2011 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
  • 3.
    © Art Traynor2011 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
  • 4.
    © Art Traynor2011 Mathematics Definitions 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 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
  • 5.
    © Art Traynor2011 Mathematics Definitions 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 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 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
  • 6.
    © Art Traynor2011 Mathematics Definitions Expression Symbol Operation Relation Expression – A Mathematical Sentence 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
  • 7.
    © Art Traynor2011 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
  • 8.
    © Art Traynor2011 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 
  • 9.
    © Art Traynor2011 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
  • 10.
    © Art Traynor2011 Mathematics Equation Equation Expression An Equation is a statement or Proposition ( aka Formula ) purporting to express an equivalency relation between two Expressions :  Expression Proposition A declarative expression 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
  • 11.
    © Art Traynor2011 Mathematics Equation An Equation is a statement or Proposition ( aka Formula ) purporting to express an equivalency relation between two Expressions :  Expression Proposition A declarative expression 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 Polynomial The Equation is composed of Operand terms and one or more discrete Transformations ( Operations ) which can render the statement true 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
  • 12.
    © Art Traynor2011 Mathematics Equation An Equation is a statement or Proposition ( aka Formula ) purporting to express an equivalency relation between two Expressions :  Expression Proposition A declarative expression 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 Polynomial The Equation is composed of Operand terms and one or more discrete Transformations ( Operations ) which can render the statement true 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
  • 13.
    © Art Traynor2011 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 
  • 14.
    © Art Traynor2011 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
  • 15.
    © Art Traynor2011 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
  • 16.
    © Art Traynor2011 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
  • 17.
    © Art Traynor2011 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
  • 18.
    © Art Traynor2011 Mathematics Operand Addition Operand the object of a mathematical operation, a quantity on which an operation is performed  Arithmetic-Addition: a +b = c Within an expression or set  “a” and “b” are Operands 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
  • 19.
    © Art Traynor2011 Mathematics 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 Operand Arity
  • 20.
    © Art Traynor2011 Mathematics Operand the object of a mathematical operation, a quantity on which an operation is performed  Arithmetic-Subtraction: a – b = c Within an expression or set  “a” and “b” are Operands 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
  • 21.
    © Art Traynor2011 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
  • 22.
    © Art Traynor2011 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: Swok Section 5.3, (Pg. 257) 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
  • 23.
    © Art Traynor2011 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 Swok Section 11.2, (Pg. 539) 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
  • 24.
    © Art Traynor2011 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, Theorem 5.11, pg 258 Swok, Section 5.3, Theorem 5.11, pg 258 Arithmetics
  • 25.
    © Art Traynor2011 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
  • 26.
    © Art Traynor2011 Mathematics Transformation Multiplication Operand the object of a mathematical operation, a quantity on which an operation is performed  Multiplicative: a(b + c) Within an expression or set  “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
  • 27.
    © Art Traynor2011 Mathematics Operand the object of a mathematical operation, a quantity on which an operation is performed  Multiplicative: a(b + c) Within an expression or set  “a” and “(b + c)” are the principal Operands, in the order of operations n “(b+ c)” is more precisely denoted as an Expression consisting of two Terms  “b” and “c” are the secondary, or subsidiary Operands, in the order of operations  The distributive property dictates the “distribution” of the common Factor “a” between 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
  • 28.
    © Art Traynor2011 Mathematics Operand the object of a mathematical operation, a quantity on which an operation is performed  Multiplicative: a(b + c) Within an expression or set  “a” and “(b + c)” respectively are 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 Multiplication Transformation
  • 29.
    © Art Traynor2011 Mathematics Operand the object of a mathematical operation, a quantity on which an operation is performed  Within an expression or set  “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
  • 30.
    © Art Traynor2011 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
  • 31.
    © Art Traynor2011 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??
  • 32.
    © Art Traynor2011 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
  • 33.
    © Art Traynor2011 Mathematics 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  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 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 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 Product of Sequence Summations
  • 34.
    © Art Traynor2011 Mathematics 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 ( 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 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 Product of Sequence Summations
  • 35.
    © Art Traynor2011 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
  • 36.
    © Art Traynor2011 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
  • 37.
    © Art Traynor2011 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( )
  • 38.
    © Art Traynor2011 Mathematics Transformation Division Division whereby one object or quantity is partitioned by another A mathematic (arithmetical) operation,   Vinculum Notation: = c, where “ – ” the fraction bar or Vinculum, indicates the division operation (“a” is divided by “b”) and “ = ” denotes equivalence Notation (Short Division) constituting an inverse operation to that of multiplication, 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
  • 39.
    © Art Traynor2011 Mathematics Transformation Division Division whereby one object or quantity is partitioned by another A mathematic (arithmetical) operation,   Notation (Long Division) constituting an inverse operation to that of multiplication, 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
  • 40.
    © Art Traynor2011 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
  • 41.
    © Art Traynor2011 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
  • 42.
    © Art Traynor2011 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
  • 43.
    © Art Traynor2011 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)
  • 44.
    © Art Traynor2011 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
  • 45.
    © Art Traynor2011 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
  • 46.
    © Art Traynor2011 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
  • 47.
    © Art Traynor2011 Mathematics Identity Identity Element Within an expression or set  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.
  • 48.
    © Art Traynor2011 Mathematics Inverse Element Within an expression or set  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
  • 49.
    © Art Traynor2011 Mathematics Zero Element  Multiplication: a x 0 = 0; any number multiplied by zero yields zero, the Zero Property of Multiplication Zero Element
  • 50.
    © Art Traynor2011 Mathematics Negation Element  Multiplication: (-1) x a = -a; negative one times any number yields the opposite of the number Negation Element
  • 51.
    © Art Traynor2011 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
  • 52.
    © Art Traynor2011 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
  • 53.
    © Art Traynor2011 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
  • 54.
    © Art Traynor2011 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
  • 55.
    © Art Traynor2011 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 Less Than b ( LT )  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
  • 56.
    © Art Traynor2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  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 and b = c, then a > c  If a < b and b = c, then a < c Swok, Section 1.1, pg 2 TPIE Wiki: “Inequality (Mathematics)Chained Notation Variant: a > b > cn Chained Notation Variant: a < b < cn TPIE-CNV
  • 57.
    © Art Traynor2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  Transitive Property of Inequality (also true of non-strict inequality)  Special Case: Multi-Directional Inequality Expressions ( MDIE ) Expressions such as a < b > 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 < b > c ≤ d → ( a < b ) ⋀ ( b > c ) ⋀ ( c ≤ d ) Which is more adroitly manipulated if restated with uni-directional inequality operators” ( less-than , number line monotonic increasing format “ < ” preferred ) a < b > c ≤ d → ( a < b ) ⋀ ( c < b ) ⋀ ( c ≤ d )
  • 58.
    © Art Traynor2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  Transitive Property of Inequality (also true of non-strict inequality)  Special Case: Multi-Directional Inequality Expressions ( MDIE ) Expressions such as a < b > c ≤ dn Example Problems Dot Com : “ Inequality ” a < b > 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
  • 59.
    © Art Traynor2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  Transitive Property of Inequality (also true of non-strict inequality)  Special Case: Multi-Directional Inequality Expressions ( MDIE ) Expressions such as a < b > c ≤ dn Example Problems Dot Com : “ Inequality ” a < b > 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
  • 60.
    © Art Traynor2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  Transitive Property of Inequality (also true of non-strict inequality)  Special Case: Multi-Directional Inequality Expressions ( MDIE ) Expressions such as a < b > 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 < b > c ≤ d → ( a < b ) ⋀ ( b > c ) ⋀ ( c ≤ d ) Which is more adroitly manipulated if restated with uni-directional inequality operators” ( less-than , number line monotonic increasing format “ < ” preferred ) a < b > c ≤ d → ( a < b ) ⋀ ( c < b ) ⋀ ( c ≤ d )
  • 61.
    © Art Traynor2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  Converse Property of Inequality (also true of non-strict inequality)  a < b, then b > a  a < b, then b > a
  • 62.
    © Art Traynor2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  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  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  If a > b then 1/a > 1/b
  • 63.
    © Art Traynor2011 Mathematics Definitions Inequality Properties Of Inequalities Blitzer, Section 2.7, pg 186If a < b, then a + c < b + c If a < b, and c > 0, then ac < bc If a < b, then a – c < b – c Addition Property of Inequality a cIf a < b, and c > 0, then < b c Positive Multiplication Property of Inequality Blitzer, Section 2.7, pg 186 If a < b, and c < 0, then ac > 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
  • 64.
    © Art Traynor2011 Mathematics Properties Inequality Inequality between objects which are not identical A mathematic Relation  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
  • 65.
    © Art Traynor2011 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 – b < a < b  |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 Lay, Section 3.11, Pg. 100  |ab | = |a | · |b |  |a + b | ≤ |a | + |b | Swok, Section 1.1, Pg. 3 Lay, Section 3.11, Pg. 100 Swok, Section 1.1, Pg. 3 Swok, Section 1.1, Pg. 3 Triangle Inequality Modulus Equivalence of LT Inequality (MEOLTIE)
  • 66.
    © Art Traynor2011 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
  • 67.
    © Art Traynor2011 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
  • 68.
    © Art Traynor2011 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: 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)
  • 69.
    © Art Traynor2011 Mathematics Absolute Value Inequalities Properties Absolute Value  Strict Inequality ( SI ) 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. 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 |:
  • 70.
    © Art Traynor2011 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: A propositional ME, the terms of which adhere to the form “ | { 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 antipodal scalar or expression denoted by the relation of strict inequality. Chained Notation Form (CNF)
  • 71.
    © Art Traynor2011 Mathematics Absolute Value Inequalities Properties Absolute Value  Strict Inequality ( SI ) A propositional ME, the terms of which adhere to the form “ | { 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 antipodal scalar or expression denoted by the relation of strict inequality. 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
  • 72.
    © Art Traynor2011 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: 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)
  • 73.
    © Art Traynor2011 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: Challenges:  Less Than  More Than | x – 2 | > – 3 | 2x + 2 | < – 1 | 2x + 2 | – 1 > – 1 – 1 | x – 2 | – 1 > – 3 – 1
  • 74.
    © Art Traynor2011 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: 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
  • 75.
    © Art Traynor2011 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