Mathematics
My Tropes
Mathematics is the field committed to represent numbers as a scientific language. There is variety of works which their scope ranges from entertaining to specialized. While mathematical tropes are over-looked for their explanatory potential, still they have not been updated since they were accepted, as is. We tend to focus on their workability properties in sciences instead of their validity. What if a historical definition should be found obsolete to the study of linear algebra? In this modest article, I will explore the definition of a second-degree polynomial under the lens of simplification. I will state why it is new, important, and better than what comes before.
Argument
Firstly, the Relation between the Mean and the Spread is defined:
X1 or X2 = (X1+X2)/2 ± (X1-X2)/2.
After substitution of the equation set
(The Definition of the Polynomial Jacobian) for the quadratic parameters, I validated that the square root expression in the quadratic formula simplifies, with knowledge of the sign-memory is being embedded into the parameters. Failure at observing this simplification results from ignoring this equation set.
Secondly, whenever we acknowledge the equation set for the quadratic parameters A, B, and C, then we realize that quadratic parameter C is immaterial:
A = C/(X1*X2);
B = -C*(X1+X2)/(X1*X2);
C = SIGN(X1-X2)*SIGN(X1*X2)*ABS(ARBITRARY).
One conceptual mistake in the quadratic formula
{-B/(2*A)±(SQRT(B²-4*A*C))/(2*A)} is to allowing for simplification of quadratic parameter C. If we simplified quadratic parameter C, then we could not distinguish the sign of the sum of quadratic parameters A, B, and C.
Thirdly, providing we know either a solution for X1 or X2 and the sum of quadratic parameters A+B+C = N, then the
Relation between the Mean and the Spread is defining an equality:
A+B+C = N = C*((X1-1)*(X2-1))/(X1*X2);
X1 = (X2–1)/(X2–1±N*X2);
X2 = (X1–1)/(X1–1±N*X1).
The validation process acts upon retrieving the right pair of solutions with respect to the sum of quadratic parameters N and either a solution X1 or X2. For instance, simplifying the quadratic parameter C on both sides will cause the variable X to be proportionally related to itself through a sign-change operator (±). Consequently, there are two pairs of solutions, but only one is respecting both initial conditions for N and either X1 or X2. Quickly done, for example, this calculation process informs of whether or not X1 = 5, when
N = 10: it happens even if we wanted X1 to equal 5, it is not respecting the condition N = 10, for X2 must equal 5 instead, so X1 is either equal to -0.08696 or 0.074074.
Fourthly, summing on quadratic parameters implies that their global units are compatible with the variable X. For this motive, one should assess of the immaterial property of quadratic parameter C, for revealing the proportionality of both quadratic parameters A and B with C. Hence the sum of quadratic parameters is immaterial, while we look at the equation set. Because of the function SIGN() is being described as immaterial. Inevitably, the dimensionless features of quadratic parameters in the Cartesian plane only justifies the second-degree polynomial curve family, for an arbitrary height. It has always been said, in promotion for science, that we should investigate end-points as if they are informing of trajectories. Hereby, specialized knowledge of the Relation between the Mean and Spread allows for calculation of an arbitrary height to which the solutions should be pre-defined, by the initial context.
Furthermore, we are taking account of the essays of solving the second-degree polynomials, with a new light. I am glad to writing my first essay on the topic. It is the right methodology for sketching these curves. As a matter of fact, specialized knowledge allows for performing graphical analyticity upon three mathematic features: the influx-point, at the apex of the height (mean), and the location of the end-points (spread). This study enables calculators to recovering the second-degree polynomial equation which it characterizes the analytical points.
Fifthly, the fundamental equation is defined A*X²+B*X+C = 0, historically. It is important to update our knowledge of this polynomial form because of the quadratic parameters have an image domain which it is bound by two zeroes. Because of early scholars’ confidence in the realm of philosophical mathematics, they included a non-specified domain to the mutual relationships of parameters, in the second-degree polynomials (through the algebraic formulation). This conceptual mistake leads to describing fragile variables which its null value is undefined within the solution set boundaries. In this context, writing the second-degree polynomial in this form A*X²+B*X+C = 0 ought to imply it has a solution set in the Cartesian plane.
By understanding the Relation between the Mean and the Spread, one will realize the general expression requires an update upon its syntax representation. My experimental model enables me to get a hint at the Relation between the Mean and the Spread through the lens of equation transformation processes.
Experiments with numbers
In the development stage, my experimental framework aimed at examining the distribution behavior of a function defined by two lines of equality, with making a relationship comparison on the classic functional operators: scale, power, divider, and root. I observed a measure of two outputs from providing an only input. In this context, Y was the input, and X1 and X2 were the outputs. With this experiment, I was able to conceptualize intersecting points on a ladder of intervals, in a column, in the calculus spreadsheet grid.
The nature of the linear equation Y= X2/X1 suggested me that the validity of the solutions calculated with it was contradicting to the domain of the original equations. For example, the full expressions that were created with the lines of equality informs of a functional gain from X1 to X2, for
X2 = X1^Y, by the laws of exponents.
X1 = Y^(1/(Y-1));
X2 = Y^(Y/(Y-1));
Y= X2/X1.
Consequently, there are minus sign Y values for which either solution
X1 or X2 is defined though Y= X2/X1 suggests that for every Y and one solution, then the other solution is defined. With regards to the specialized knowledge being applied to products, my experimental framework allows for my sketching the curve that binds together the two solutions: Y was not any vertical variable, and the spread of X1 and X2 was not dependent on irregular patterns. It became important to noticing that Y was an immaterial operator, and that the functional gain was homogeneous, for every pair of solutions. This phenomenon is explained by the functional weight of the operators with respect to the rates of change, as X is increasing.
Therefore, with applying the set of equations for the quadratic parameters, then it was possible to functionalize Y in terms of two real and measurable solutions. Meantime, the involved mathematical height has to be arbitrary. From this perspective, I foresee the use of the second-degree polynomials for describing the Relation between the Mean and the Spread for two solutions and one height.
Crafting novel exercises for presenting the algebraic complement.
Given the distance equation S = SQRT((Y2-Y1)²+((X2-X1)²). Hereby, everyone should recognize the algebraic expression for a circle upon which P(X,Y) is always equal to one (S ≡ 1, and R = S). The distance equation becomes coupled for both coordinates X and Y, for
X = ±SQRT(R²/(1+(b)²));
Y = X*(b).
I introduced the variable (b) which it is defined as a slope towards P(X,Y) from a circle origin, in the Cartesian plane. In this algebraic form, X = COS(); Y = SIN(); and the arguments are not representing angles, but any singularity slope bound by the distance equation. Indeed, I defined a linear expression which suggests that (b)² cannot be equal to (-1), due to the #DIV/0! calculation error code. Thus, this coordinate system can be assumed of being bi-dimensional for the process of embedding into a second-degree polynomial expression the circle domain.
My exercise should assign each root to a coordinate: X1 = X; X2 = Y. I want to find the point on the circle which its slope (b) equals twice the product of the coordinates b =2*X*Y. In this exercise, b = 1, and X = SQRT(2)/2;
Y = SQRT(2)/2. Therefore, the Relation between the Mean and the Spread enables to represent P(X,Y) with regards to quadratic parameters:
A = 2;
B = -2*SQRT(2);
C = 1 (the sign of zero (0) is positive, and it is equal to one (1)).
Conclusion
In this short article, I presented the tropes which they helped me to grasp the Definition of the Polynomial Jacobian. Of course, the finding details are showing variety and richness of forms and formulas for unidimensional polynomials. However, the experiments with numbers really made me notice the symmetrical formulation of the variable X, with respect to N. For this motive, I greatly value this calculation framework. I searched for an extension to this concept, and I am glad to have been developing valuable understanding of the functional relationship to the global parameters.
