Chapter 2: Harmonic Systems The Humanization of Math: The Ten Processes of Binary Logical Math: Dual-sign Math Systems. Addition as Discrimination.Negation as a Shift in Perspective. Division as Parallelism, a Shift in Perspective, as well as the Simplification of Units of Measurement Through Dominance.Multiplication as Polyglotism. Summation of Squares as Making Distinctions Among Varied Self-Amplified Values. Summation of Reciprocals as Making Distinctions Among Varied Values Whose Orientation of Magnitude has been Switched from One Form of Infinity to Another. Multiplation of Terms Involving Integer Exponentiation.Multiplation of Terms Involving Exponents of Unit Fractions(1/a). Summation of Terms Involving Integer Exponentiation.Summation of Terms Involving Exponents of Unit Fractions(1/a). Harmony as the Use of All Ten Processes Within a Unified System: The Construction of Simple Polynomials in One and Multiple Unknowns.The Unique Polynomial Qualities of an Algebraic Number.Integrated Approximate Solutions of Simple Polynomials in One Unknown by Way of: Exponentiation and Division. Three Side-bars: Exponentiation Within the Context of Approximating Factors of a Polynomial Gives Much Needed Identity Towards Each Factor.Parallelism as a Function of Harmony---Topic: Parallel Series.Itegrated Parallel Polynomial Functions in Multiple Unknowns Approximating a Polynomial Function in One Unknown.How the Ten Math Functions Are Used. Overview of Some Binary Logic Aesthetic Classes * Division as Parallelism, a Shift in Perspective, as well as the Simplification of Units of Measurement Through Dominance.Multiplication as Polyglotism. a * b = c * d >>>>>polyglot; parallelism is on the order of groups of values, "ab" as a group parallels "cd". a / c = d / b >>>>>"a" parallels "d" and "c" parallels "b" because of distinctions made by both the divisions involved and the equal sign. Multiplication of a*b=c*d destroys all individual distinctions among variables. Dominance is in terms of the denominator in both instances: "a" (as a unit of measuring something) is redefined in terms of "c" as the unit of choice. "a" and "c" are a functional pair which are being related to another pair. "a" is a prominent actor being molded by "c"'s guidance. "a" is the one standing out in front of the audience, while "c" is imbueing all of its character into "a" such that "c" will not have to get up on stage itself---all of its acting will be done by "a". The perspec- tive is from "c". a*b=c*d has no individual perspective, but an overview of the "big picture". Rather than looking at the relations among members, it is looking at relations among groups. "a" and "d" are on topside, while "a" and "c" are on the left. Thus, there are two pairs of distinctions marking the positions of four variables. If "a" and "b" are cross-multiplied and made equivalent to "c" and "d"'s cross-multiplication, then their individual positions of distinction are lost by associating opposites together as siblings: a / c = d / b "a":left-top is grouped with "b":right-bottom reulting in, a * b. "d":right-top is grouped with "c":left-bottom resulting in, c * d. Left's associating with right and top with bottom mutually cancels opposites of similar type: horizontal and vertical positions. The Construction of Simple Polynomials in Multiple and One Unknowns Problems are best solved if they are described both in terms of what is known as well as what is unknown. Then the relationship between the two can be sought. But there are different ways of stating the facts. It can make all the difference. Examples: x, y = unknown values x = 5 y = 6 x = 5 y = 6 x - 5 = 0 y - 6 = 0 x * y = 5 * 6 (x - 5) * (y - 6) = 0 xy = 30 xy - 6x - 5y + 30 = 0 In the previous two illustrations, the contrast in multiplying two numbers together vary greatly. In the left example, a simple multiplication among two numbers, 5 and 6, along with their compatriots, x and y, yield a simple answer of 30. But wait, why is the right side so complicated? If we threw the 30 back over to the right of the equal sign, you would think every- thing should be the same----or is it? xy - 6x - 5y = -30 Oops! Where did that negative come from? Maybe we should substitute the original values for x and y and see what happens: (5 * 6) - (6 * 5) - (5 * 6) = -30 (30 - 30) - 30 = -30 0 - 30 = -30 -30 = -30 Why go to all this trouble? What would happen if we tried to substitute other values for x and y in both examples?: x = 3 y = 10 xy = 30 xy - 6x - 5y + 30 = 0 3 * 10 = 30 (3 * 10) - (6 * 3) - (5 * 10) + 30 = 0 30 = 30 (30 - 18) - 50 + 30 = 0 (12 - 50) + 30 = 0 -38 + 30 = 0 -8 <> 0 See the difference? The left example is correct for allowing any pairs of factors to be multiplied together, provided they result in the number 30. But the right example is more specific. It only allows 5 and 6 to be substituted into the equation. Even though both illustrations looked alike, they turned out being different. In this text, multiplication figures in as a common theme. But to what extent can we integrate its use as a system for aesthetic use? Does the system harmonize with itself? There is a more subtle change that can be made in the right example. By economizing on the number of different variables used, a different logic state- ment can be made about how to solve this: x = 5 and y = 6 x = 5 or x = 6 x - 5 = 0 and y - 6 = 0 x - 5 = 0 or x - 6 = 0 (x - 5) * (y - 6) = 0 (x - 5) * (x - 6) = 0 xy - 6x - 5y + 30 = 0 x^2 - 11x + 30 = 0 Both are substituted at once. Only one substitution is allowed at a time. (5*6)-(6*5)-(5*6)+30 = 0 (5*5)-(11*5)+30=0 or (6*6)-(11*6)+30=0 30 + (-30 - 30) +30 = 0 (25 - 55) +30=0 or (36 - 66) +30=0 30 - 60 +30 = 0 -30 +30=0 or -30 +30=0 -60 + (30 + 30) = 0 0 = 0 or 0 = 0 -60 + 60 = 0 0 = 0 All three statements are correct, but how are they different? The left example allows, requires, all two solutions to be substituted into the equation for it to equal zero. The right two examples only allow one solution at a time to be inserted into the equation, although either choice will work. It is actually easier to solve the right two equations using factoring formulas or approxima- tion techniques. In the left equation we would have to guess and guessing could take forever in some cases. It also economizes terms. By uniting terms of similar character, the single unknowns: -6x and -5y become -11x, and the state- ment becomes more thematically organized as a cascade of descending term groups: double unknowns, single unknown, no unknown. The coefficients associated with each term become very useful tools for approximating the solutions of the equation without much ado. The Unique Polynomial Qualities of an Algebraic Number This next paragraph assumes uniqueness to numbers in that each algebraic (non-transcendetal) number belongs to one and only one polynomial. (A transcen- dental number cannot be depicted by a mere polynomial, but instead must be represented by an infinite process; this book deals mostly with the infinite processes of approximating algebraic numbers: numbers that can serve as factors for polynomials). Each polynomial cannot be factored down any further without altering the integer coefficients into irrational ones making its set of factors incomplete. Irrational algebraic numbers are like pieces to a jigsaw puzzle. Each piece cannot be replaced by any other number. The puzzle requires a perfect and unique fit. Herein lies the uniqueness to irrational numbers: they each belong exclusively to one and only one set of numbers. The set may contain any number of elements from one to infinity. The size of a set's elements is not unlike dimensioning space: the whole polynomial shapes the values of all its factors simultaneously. Although it may appear that each element is reacting with its neighbor, in reality the polynomial is molding its factors as if they were progeny. Examples: x - 1 = 0 1 is a linear algebraic number in that its polynomial is of x = 1 linear degree: one factor. 2x + 1 = 0 -0.5 is also a linear algebraic number. 2x = -1 x = -1/2 x^2 - 3x + 2 = 0 >>>> is techniquely a quadratic polynomial, but qualita- (x - 1)*(x - 2) = 0 tively is a composite of two linear polynomials. x - 1 = 0 x - 2 = 0 x = 1 or x = 2 x^2 - x - 1 = 0 >>>> is techniquely and qualitatively a quadratic polynomial. (x-1.618...)*(x+0.618...)=0 x=1.618... or x=-0.618... >>> both equal >>> (1 +- <5>) / 2 x=(1+<5>)/2 or x=(1-<5>)/2 So far, beautiful numbers have been generally depicted. Power numbers are a little different. They can always be represented as equivalant to an algebraic statement of shelled radicals containing simple integers. There are never any arithmetic values outside of the outermost radical shell, only geometric values and/or signs are allowed. For example, {cube root} 1 + {2} >>>> would not be a power value, but <5>/3 >>>> would be. For another example: ...-{+-<3> + 1/2}... The shelling can go on indefinitely, but for this example let us stop here and find its polynomial expression: (-{+-<3> + 1/2} = x)^3 +-<3> - 1/2 = x^3 (+-<3> = x^3 + 1/2)^2 (3 = x^6 + x^3 + 1/4) - 3 (0 = x^6 + x^3 - 11/4) * 4 0 = 4x^6 + 4x^3 - 11 So x takes on six values, four of them are complex: x = -{<3> + 1/2} = -1.3068769... = = = -{-<3> + 1/2} = 1.07203635... = = Notice too the double sign possibility for every even powered root and a single sign option for odd powered roots. On the outside, double signs merely change the sign without affecting value. On the inside, they affect value and possibly sign. Phrased another way, the exponents of the terms of a power polynomial progress from the x^0 term exclusively in a geometric manner. Example: 0 = a*x^0 + b*x^z + c*x^zy + d*x^zyw + ... In this example, w would be the radical of the outermost shell, y would be in the middle, and z would be on the inside of the algebraic expression: {....}^1/w [....]^1/y (....)^1/z x = ...{e + [f + (g)]}... So a power polynomial might look something like this: 0 = 3 + 2x^3 - 5x^6 + x^18 + 4x^90 - 2x^630.... 0 = 3 + 2*x^3 - 5*x^(3*2) + 1*x^(3*2*3) + 4*x^(3*2*3*5) - 2*x^(3*2*3*5*7).... Integrated Approximate Solutions of Simple Polynomials in One Unknown Aesthetics is defined by number theory as being linear approximations of algebraic numbers and the uses which they are put to. These algebraic numbers are elements to sets of factors of unique and simple polynomials in one unknown. Each set may be composed of any positive number of elements. It represents the scope of potential aesthetics and becomes operative in creation as a harmonic system when multiple criteria of timeless values converge to support that system. In a much larger sense, aesthetics is the interplay of a self-looping style of mathematics and comes in several degrees of sophistication. Each plane of sophistry satisfies the needs of that plane by being composed of elements of thought(logistic math functions) from that plane. The whole grand mess is then macro-organized to incorporate the whole system of cosmic being as a grand aesthetic architecture. I may be using aesthetics as a term rather loosely at the moment, but I mean to imply the existence of mathematical systems for harmonizing relational number values at every level of creation----from the top on down. x - 1 = 0 x = 1 2x - 3 = 0 2x = 3 x = 3/2 y =: x y - 1 = 0 and x^2 - 2 = 0 y = 1 x^2 = 2 y = 1 =: x = <2> Or: 2y - 3 = 0 and x^2 - 2 = 0 2y = 3 x^2 = 2 y = 3/2 =: x = <2> Or: 5y - 7 = 0 and x^2 - 2 = 0 5y = 7 x^2 = 2 y = 7/5 =: x = <2> etc. 2x - 3 =: x^2 - 2 2x - 3 = y x^2 - 2 = z 2x = y + 3 x^2 = z + 2 Overview of Aesthetic Classes Aesthetics of Real Numbers General Classification by Quality Ideal (Real and Non-Real in a mixed state) Golden Beauty Power Hybrid Geometric Light Modeled Classification by Type of Accuracy Degree of Polynomial Linear(Simple) Quadratic(Straightforward) Cubic(Transformational) Quartic+(Approximate) under what degrees do these go?: 2nd+?: Stabilizing(Converging) 3rd+?: The Gap(The Hole: Diverging) Dynamic(Fractal) Cyclic(Partly Fractal) 4th+?: Approximate Unclassified General Aesthetics of Non-Real Numbers General Beauty = Complex Numbers General Power = Imaginary Numbers Unclassified Aesthetics of Non-Real Numbers Speculated Aesthetics (Transcendental Numbers, etc.) Higher Order (Logic) Aesthetics Although golden polynomials are a special case subset of the hybrid class, it is the golden formula that forms the basis of hybrid formulas. Golden polynomials are pure bred.