MATH MAY BE DIFFERENT?
(1) Numbers are neutral. There are no positive numbers, no negative numbers.
Numerical expressions, not the numbers considered alone, enter in the
calculations, doing so under the rules of arithmetic, algebra, geometry, or
science.
(2) One must learn to multiply. The order of the factors changes the
product and it's not true that “less for less gives more”. I can multiply one
measurement by another measurement, but I cannot multiply a count by another
count.
(3) Equations are horizontal tournaments involving "x", but
without "y". True equations are of the first degree and have a single
unknown. The strange equation of the second degree is an exception, for it has,
in two different contexts, a false version with two roots, and a true but playful
version with two unknowns. By the way, there are no pre-established or for free
equations.
For ten very recent months I have been going through the foundations of
mathematics in total dedication, for the reasons I will present in the
continuation of this Introduction. It has been the first time I got body and
soul into the study of mathematics, from which I had been away for more than
fifty years. This time I did it without any strings attached or prejudices
whatsoever.
I've always suspected that number is a neutral tool: there are no
"positive" or "negative" numbers, let alone
"imaginary" numbers. This is an understanding I had never had the
courage to face, because it is in full contradiction with the mathematics I
learned in the books. Moreover, an attempt to prove this bold hypothesis would
be a task for experts in the field, not for me, since I'm not even a
mathematician.
This time, however, I acted differently and took the hypothesis that
number really does not have signal, and in the light of this understanding I
devoted myself to the task of examining numerical expressions, counts,
algebraic sums, multiplications, polynomials, and equations.
The results I have gathered therefrom are impactful and I have, though
with humility, the ambition of seeing them examined by some math scholar.
There are, in fact, two types of counts, namely counts of modules, that
is, counts of “things”, and abstract counts, that is, counts of steps or
positions, used in different mathematical objectives.
The counts, not the numbers, are positive or negative. A negative count
is the image of a positive count.
The images of the counts, in their interrelationships, can explain the
metaphor contained in the expression “less for less gives more”, which can also
be explained geometrically.
Only the multiplier can multiply, that is, the order of the factors is
unique and cannot be changed.
Equations involving counts are necessarily of the first degree, with a
single unknown. Equations of isolated numbers can be of any degree, but with
more than one unknown, so far I only know the equation of the second degree.
I also realized that there is a misconception in mathematics for
considering a zero-equated polynomial as an equation. A polynomial is a
variable algebraic sum, function of “x”, which calculates an amount for each
arbitrarily chosen value of “x”; equalized to zero, it indicates the point
where the algebraic sum becomes null. In turn, an equation is the equalization
of two algebraic sums, bound to be equal, and serves to calculate the value of
an unknown count, not to determine the location of a point.
That is, there are equations, which calculate counts, and false
equations, which locate points. Equations with script, concrete, and equations
without script, abstract.
Many news and discrepancies, which I will try to explain in this book.
I must now reveal the reason for my belated interest in the foundations
of mathematics. It was indeed an attempt, somewhat pretentious and above all
amateur, to rescue the studies of Professor Astyages Brasil da Silva, who died
in September 2017. He was a friend from college profoundly knowledgeable in
descriptive geometry, who repeatedly told me that he had constructed a new
mathematics and used to make comments that showed his discontent with what he
used to call the "current mathematical model."
As far as I know and understand, no one is aware of this alternative
math, which the master would reveal only in exchange for some sponsorship or
contract, which he never managed to obtain.
He argued, for example, that he had a new concept of negative numbers,
that "less for less gives less", that "an equation of degree “n”
can have more than “n” roots or less than “n” roots", that (-5)² = - 25,
that the image of “x” is “x-2x”.
I started the research by studying the equations of the first degree.
These were exhaustive attempts for months at a time. I didn't get to the wanted model whose
foundations I couldn't reach in the slightest.
Engineer Sandoval Amui decided, like me, to research the fundamentals of
mathematics taking as starting point Professor Astyages comments, which I have
passed on to him. In this endeavor, Amui, more interested in geometry, ended up
developing important and innovative studies of the Pythagorean Theorem, for
which presented a remarkable generalization, and built a convincing
demonstration of Fermat's Conjecture, as presented in the book “Circumference,
Pythagoras and Fermat,” published in November 2019 by Catalivros Publishing
House.
But he didn't either get any clues from the Astyages model.
Anyway, the effort earned me the insights I mentioned above. I shot what
I saw, I hit what I didn't see. I decided to publish the book “Reflections of
an accidental mathematician” in July 2019, with the sole intention of seeing
these understandings examined, even though I felt the book would arouse no
enthusiasm or interest.
After all, I am an assumed and unknown non-mathematician.
The indifference with which the book was received confirmed the
foreboding: I didn't get no feedback or comment, either for good or for evil. I
recognize, however, that it could hardly have been otherwise, for the book was
not primed by the organization, nor was it didactically optimized.
I chose not to give up and
here I return with this second book, which I wrote for the purpose of
reiterating and broadening the same understandings, this time following a
didactic script and with a text that I suppose more appropriate. I also correct
some imperfections present in the previous book and add a full explanation on the
equation of second degree, after introducing the concept of “false equations”.
This new book has only a few
pages, but it may not be attractive to reading because it is mainly
concentrated on concepts and definitions.
I believe in everything I wrote hereinafter, as my new perception of
math. I know I may be wrong, maybe very wrong. In any case, my only intention
is to contribute in some way to the progress of mathematics.
Remo Mannarino
CHAPTER I - THE UNDERSTANDINGS PRESENTED IN
THIS BOOK
The number is a neutral tool, neither positive nor negative.
The number, taken alone, gives a magnitude, but how to use a numeric
expression depends on the nature of what is being quantified and on the
mathematical application it is intended for.
The entire numerical expression enters in the calculations, not the
number by itself, and it does so by mathematical authorization, whether from
arithmetic, algebra, geometry, or science.
"Positive number" and "negative number" are not
numbers, but numerical expressions of counts. It makes no sense, for example,
to square a “negative number” or extract the square root of a “negative
number”.
What matters is the numerical information, not the number by itself. A
number, in isolation, is like a verb in the dictionary, cloistered there,
without subject and without complements, informing an action without telling
who, where, how or when.
About measuring, counting and
multiplying.
To measure (as much as to weigh or to gauge) is to compare. A measurement
(or weighing) is made in compliance with geometric and scientific rules.
Counting, on its turn, is made for arithmetic and algebra.
Only a special count, a “number of times”, can multiply another count,
either of modules or abstract.
Can a numeric expression be squared? It depends. For example, a geometric
measure can always; the mass of a body can sometimes; a count of modules can
never.
About numeric expressions
Mathematics is done with numeric expressions, not with numbers by
themselves. See in the four mathematic operations below that the numerical
expressions, not just the numbers, participate in the calculations, as data or
in the results:
3 x 3 = 9 (an abstract count is multiplied)
3 x 3 books = 9 books (a module count is multiplied)
3 m x 3 m = 9 m² (multiplication of two
measurements)
3 m/s x 3 s = 9 m (multiplication of two quantities
of physics)
About polynomial and equation
The polynomial is not exactly an equality, but a mathematical definition:
an algebraic sum of abstract counts, which has a variable count "x", that is, an "x"
that can assume arbitrarily chosen values.
The equation is an equality imposed to two algebraic sums, which enclose
an unknown count “x”. Thus understood, the equation has a single value for “x”,
except for the exceptional case, the only one I know, of the equation of second
degree, which I discuss in Chapter IX.
Polynomials equalized to zero do not give rise to equations. To solve
such equalities only serves to indicate the point or points at which the
polynomial nulls itself (that is, where a figure crosses the axis of x). They
are false equations, meaningless and almost always insoluble, except for the
case of the straight line and the exceptional case of the trinomial of second degree.
About Bhaskara’s Formula
Bhaskara's formula, which allows us to calculate the roots of a quadratic
equation, is the mathematical reciprocal of the square trinomial. When it comes
to a parable, you can choose the path:
given “x”, the trinomial calculates “y”;
given “y”, Bhaskara's formula calculates the corresponding pair of “x”
values.
Bhaskara's formula applies to all points in the parable. And nowhere
outside of it! Be careful, therefore, not to create "imaginary
numbers"…
CHAPTER II - NUMBER CONCEPT
Number is a neutral and multiple up to infinity tool that serves to
express counts and measurements in general. Counts can be either of modules or
abstract.
Explaining neutral and multiple up to infinity tool
The number is neutral, a tool with no sign.
The number is multiple up to infinity: there are infinite bins, each with
a number, starting at 1.
There are no positive or negative numbers in the infinite bins.
Explaining count of modules
The count of modules is the algebraic sum, unit per unit, of a given
number of modules, ie “things” (100 oranges). Subtracting a count is the same
as adding the image of that count.
A count of modules multiplied by a “number of times” results in another count
of modules.
(5 times 100 oranges = 500 oranges)
An algebraic sum of counts of modules is also a count of modules.
(500 oranges + 100 oranges = 600 oranges)
(500 oranges - 100 oranges = 400 oranges)
Because it is an algebraic sum, a count of modules can be either positive
or negative. If negative, the minus sign is required in the numerical
expression. (- 500 oranges).
The module count results in a number followed by the module to which it
refers: for example, a bank account, which counts dollars, may be either a
surplus (+ 100 dollars) or a deficit (- 100 dollars).
The plus (+) and minus (-) signs are not attributes of numbers, but of
counts.
Explaining abstract count
While a count of modules specifies what has been counted, the abstract
count quantifies an advance (or indent), as in the case of building a figure,
or a "number of times" required for multiplication.
Because it is an algebraic sum, the abstract count also can be either
positive (traditionally and improperly called “positive number”) or negative
(traditionally and improperly called "negative
The plus (+) and minus (-) signs are not attributes of numbers, but of
counts.
Abstract counts are used: as multipliers of counts or of measurements and in the study of functions
and figures.
Explaining measurement
Measurements, weightings and verifications are made by comparison with
arbitrarily chosen physical units such as the meter, the kilogram and the
second. The result is a numeric expression, whose number indicates how many
times the unit was required in the measurement.
Correlation with sports
The number is the nucleus of the numerical expression, be it of a
frequency, a position, a count of modules, a geometric measurement or a
property of physics.
May I suggest in this respect a correlation between mathematics and
sports, many of them practiced with ball. Without a ball there is no game, and
each sport has a different ball: soccer ball, tennis ball, volleyball,
basketball.
The rules of the game are different in each sport.
The mathematical correspondent of the ball is the numerical expression,
without which there is no mathematics either. The numeric expressions with
their mathematical games are as follows:
(1) Abstract counts of frequency (“number of times”): their numbers are
used in the counting multiplications.
2) Abstract counts of positions: their numbers are used in function and
figure studies.
(3) Module count: their numbers are used in the arithmetic of everyday life and in the equations involving number of modules ("things").
(4) Measurements: their numbers are used in geometry and physics.
(5) Physical properties’ mensuration: their numbers are used in science.
Each mathematical game has its numerical expression and its rules for using the number it contains. For example, counting, measuring and weighting are operations that result in different numerical expressions, such as: (a) 8; (b) 8 oranges; (c) 8 meters; (d) 8 kilograms.
Can we square these expressions?
To answer the question, in each case it is necessary to link the number with what is being quantified:
(a) 8: The answer is yes, because 8 is an isolated number: 8 x 8 = 64
We are talking here about isolated numbers.
b) 8 oranges: The answer is no, because there are no oranges squared.
Module counts may not be subjected to any potentiation or root extraction.
We are talking here about module counts.
(c) 8 meters: The answer is yes, because in geometry there are also
square meters and cubic meters. The answer is yes: 8 meters x 8 meters = 64
square meters.
We are talking here about geometry.
As we will see in Chapter VI, the meter is a measure (a unit); the square
meter, another unit, different from the first; and the cubic meter, a third
unit, different from the previous ones. One measurement (in meters) can be
multiplied by another measurement (in meters) to obtain an area (in square
meters) and by an area (in square meters) to obtain a volume (in cubic meters).
(d) 8 kilograms: The answer is “it
depends” because the kilogram is the unit of a physical property.
We are talking here about science.
In this case, the square is possible when so established in a formula
postulated by physics, and not possible in formulas that do not so determine.
As we shall see in Chapter VII, by scientific authorization a physical
quantity may be multiplied by the same physical property or any other physical
property, provided that the applicable formula so determines, always observing
that the units are engaged in operations, from which arises a new unit for the
result.
May I mention three examples of applying numeric expressions:
(1) In the balance sheet, which is an algebraic exercise, the numbers are
given in dollars and the answer (profit or loss) is obtained in dollars.
(2) In calculating a volume, which is a geometry exercise, the numbers are given in meters and the answer is given in cubic meters.
(3) In calculating the force acting on a body, its mass is given in kilograms, its acceleration is given in meters per second squared, and the answer is obtained in newtons.
In all three cases, the calculations are made with the numbers, but under the aegis of the respective numerical expressions, which means, in each case, knowing, for example, whether or not I can do multiplications or how should I express the result of the operation. All done with the care of observing what happens to the units.
In other words, we must respect the rules of each game,
CHAPTER III – MULTIPLICATION OF COUNTS
Multiplication of counts is a mechanism that allows adding, in an algebraic
sum, the same count, positive or negative, by a “number of times” called a
multiplier.
The multiplier, although be a counting, is always neutral. In fact, a
count of times is special: it never goes back! There is no negative “number of
times”!
Nor does it make sense for a module count to multiply a multiplier!
It is clear, therefore, that the order of factors changes the product!
Three classrooms of twenty students give sixty students. But twenty students in
three classrooms do not give sixty rooms!
Also, one module count cannot multiply another module count! I can't
multiply a profit on the balance sheet by the score of a game or elevate a debt
to its square! It sounds like a truism, and yet this observation is fundamental
in mathematics!
That is why the equations are of the first degree!
Multiplication with two negative factors
The question of the product with two negative factors can only come up
within polynomials and algebraic sums, where multiplication with a seemingly
negative multiplier covers-up the subtraction of a negative product.
It is possible to admit that the count, whether abstract or modulus, is a
linear operation on the axis of abscissas, from a zero point. Thus, a “negative
number”, that is, a negative count, is the image of a “positive number”, that
is, a positive count:
(- x) = - (+ x);
A "positive number", in turn, is the image of a "negative
number":
(+ x) = - (- x).
Thus, a negative sign corresponds to
the subtraction of a positive count,
resulting in a negative count. Two negative signs require the subtraction of a negative count, resulting in a positive count.
resulting in a negative count. Two negative signs require the subtraction of a negative count, resulting in a positive count.
That said, let's see how to proceed in case of two negative factors.
Calculate (-7). (-5)
Only one of the factors can be negative, as we know. Thus:
(-7). (-5) = - (7). (-5) = - (-35) = + 35
One of the negative signs indicates that the product is negative (-35).
The other indicates that the wanted result is the image of this negative
product; therefore, the result is a positive number (+ 35).
The same reasoning should be repeated, with alternating signals, for
higher powers.
Calculate x³ for x = - 7
x³ = (-7). (-7). (-7) = - - (7) x (7) (-7) = - 343
In a mathematical development, a count can be represented by a letter:
"a", "b", etc. "to have literal expressions of the
type (a + b + 85), (7 - a) and (A x B).
When multiplying two literal expressions, you can face a multiplication
with two negative factors, as in the examples above. This multiplication must
obey the rationale previously presented:
(-A) x (-B) = - (A) x (-B) = - (-AB) = + (AB).
(-A) x (-B) = - (A) x (-B) = - (-AB) = + (AB).
Let us calculate (a-5) x (b-6)
(a-5) x (b-6) = ab-6a-5b + (-5) x
(-6) =
ab-6a-5b - (5) x (-6) =
ab-6a-5b - (-30) =
ab-6a-5b + 30
Therefore, (a-5) x (b-6) = ab-6a-5b + 30
It is not about (- 5) x (- 6), but the image of (5) x (-6) or the image of (6) x (-5) = image of (-30) = + 30.
Note: As we know, multiplications of two literal expressions with each
other are only possible if they refer to measurements or numbers considered in
isolation.
Resolution of
the previous exercise with geometry features
Let us calculate (x- a). (x - b)
Multiplying (x- a) by (x-b) is equivalent to calculating the area A1 of a rectangle of sides (x-
a) and (x-b), which we can do with the aid of geometry.
Let be the square of side x, therefore of area A = x², as in the figure
below.
Let us break one side of this square into (x - a) and "a", and
the adjacent side into (x - b) and "b". See in the figure the square
is thus divided into four rectangles with areas A1, A2, A3 and A4, so that:
A = A1 + A2 + A3 + A4
A = x²
A1 = (x - a). (x - b) = desired product
A2 = ax - ab
A3 = bx - ab
A4 = ab
A1 = A - A2 - A3- A4
A1 = x² - (ax - ab) - (bx - ab) - ab
A1 = x² - ax - bx + ab + ab – ab
Therefore
(x-a). (x - b) = x² - ax- bx + ab
Product (ab) is positive as the result of an algebraic sum of areas
required to be equal to x², not because “less multiplied by less gives more”,
although this expression can (and should) be used metaphorically.
CHAPTER IV – POLYNOMIALS
The polynomial is an algebraic sum where an abstract count, “x”, called
variable, varies arbitrarily. For each value of variable “x”, we have the value
of the polynomial or the point of the figure that corresponds to it.
The polynomial is a mathematical definition, not an equality.
We will see later that, unlike, an equation is a confrontation of two
algebraic sums of counts required to be equal and in the general case involving
a single desired count, “x”, called the unknown.
A polynomial equalized to zero becomes a false equation. The root of a
polynomial is the abscissa of a point that nullifies the polynomial. Since the
correspondent figure can make revolutions around the axis of x, the false
equation can have more than one root or no root at all. The roots of a higher
degree polynomial can be obtained graphically or by successive approximations.
However, by way of exception, the second-degree trinomial equalized to zero may
have its roots determined mathematically by the Bhaskara’s formula or as set
forth in Chapter IX.
An algebraic binomial corresponds to the straight line; if we multiply
one straight line by other straight line we will have a second degree
trinomial; if we multiply this one by another straight line, a polynomial of
the third degree; and so on, until having a nth degree polynomial. Equating
this polynomial to zero results in a false equation, which is nullified
whenever one of the “n” factors is nulled, that is, for any root of the “n”
lines that generated it. Trying to solve this equation for finding the “n” roots
is a pointless, backward-looking mathematical exercise, something like drying
ice. I'll be back to this point.
CHAPTER V - MATH OF THE
COUNT OF MODULES
Paulo Silva has 3 apartments, a bank debt of 9,300 dollars and 1,600 Petrobras bonds. This information contains three counts, which are not abstract counts but module counts, namely apartments, dollars, and actions. “Things”, I would say.
Count of modules results from algebraic sums of modules or of modules
grouped by using a multiplier.
The multiplier is the count of "a number of times" and we know
it can never be negative.
Image
As a linear operation originating from zero point, a negative module
count is the image of a positive module count in relation to the zero point:
Image of (+ x) = (- x)
Image of (- x) = - (-x) = (+ x)
Multiplication
A module count can never be used as a multiplier.
None of the Paulo Silva module counts mentioned above can be squared, as there are no squared apartments, squared dollars or squared titles!
Consequence: every equation involving modules is first degree and has a single root.
CHAPTER VI - GEOMETRY
Geometry behaves in its relation to measurements as if it were a branch
of science, imposing its properties and generating derived units. Indeed, the
geometry of the measurements does not work with modules, but with three units,
one in length, measured directly, and two derivatives (eg, meter, square meter
and cubic meter, respectively).
Direct measurements quantify straight lines (in meters), products of two lines quantify areas (in square meters), and products of one line by one area quantify volumes (in cubic meters).
Direct measurements quantify straight lines (in meters), products of two lines quantify areas (in square meters), and products of one line by one area quantify volumes (in cubic meters).
Furthermore, a measurement cannot be elevated to any power greater than three, nor to any fractional power.
Moreover, there are no equations in geometry, whose rare equalities are demonstrated by recurrence to theorems, such as the Pythagorean theorem:
“The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).”
This is an equality imposed
by a geometry theorem, which cannot be confused with an equation, which is a
confrontation of to two algebraic sums especially constructed to be equal.
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CHAPTER VII - SCIENCE
Science,
like the geometry of measurements, defines what must be quantified, imposes the
way to make measurements and comparisons, creates units, and proposes formulas
where to use the numerical expressions.
Formulas
are mathematical definitions imposed by science and its laws.
The
formulas indicate the mathematical calculations to be performed, involving
numbers and units of the numerical expression. The units undergo modifications
resulting from the interactions required by the formulas. For example, a
resultant force expressed in newtons may arise from expressions given in meters
and kilograms, as in the Universal Law of Gravitation:
F = G (m1 m2)/R²
“In
the universe everything happens as if two bodies attract each other with force
proportional to their masses and inversely proportional to the square of the
distance that separates their centers of gravity.”
The
measure of the force of gravity (F),in newtons, results from the product of a gravitational
coefficient G, given in newton.m²/kg², by
the quotient of the product of the attracting masses (m1 and m2), given in
kilograms, by the square of the distance that separates them, given in meters.
Two
masses, in kilograms, one distance, in meters, a gravitational coefficient with
its units: the formula gives a response in newtons.
CHAPTER VIII - EQUATIONS
Equation is an equality imposed on two algebraic sums, of which there is
an unknown term, designated by the letter “x” and called unknown.
Solving the equation is to determine the unknown.
If the unknown is a count of modules, the equation is necessarily of the
first degree because one count of modules cannot be multiplied by another count
of modules or raised to any power. Thus, the equation involving modules is
always of the first degree and only have one root, the unknown “x” to be
discovered.
Venetian method
The equation has as its paradigm the “Venetian method”, also called the
“method of double entries”, confronting two algebraic sums of counts
constructed to be equal. This is what private companies do to determine if
there was profit or loss in the activities of the year.
What legitimizes the equation is the guarantee of equality of the two
algebraic sums, which can be methodological, as in the double entries of
accounting and of its balance sheets, or established by confronting counts
forced to be equal, as in the exercise
below: the calculation of the age at which the mathematician Diophantus died as
from the information inscribed on his
grave.
“His childhood lasted a sixth of his life;
maturity came a dozen years later;
a seventh beyond, he contracted nuptials;
his son was born five years later;
this son, who was weak and sick, died with half his father's life;
and the distressed father survived only four more years.”
The message above suggests the following equation:
x / 6 + x / 12 + x / 7 + 5 + x / 2 + 4 = x
Preparing the equation, we have
9x = 756
x = 84 years old (wanted age)
False equations
A false equation arises when an algebraic sum is arbitrarily equalized to zero. To solve the false equation is to find the points where said algebraic sum becomes equal to zero, an idle exercise and with no prospect of success, except for the case of the straight line and the special case of the trinomial of second degree, as discussed in Chapter IX.
A polynomial is an algebraic sum and the roots of a polynomial equalized to zero indicate the abscissas of points where the polynomial nullifies itself.
If the figure makes 17 revolutions around the X axis, we will have 17 roots with no meaning beyond the fact that they are nullifying the figure 17 times. Solving a false equation is to discover these points, usually a frivolous exercise, with the bias of being insoluble, with the exception of the straight line case and the curious case of the second degree equation, the roots of which can be found by using the formula of Bhaskara.
Another case of existent roots occurs when a product of n straight lines is equalized to zero. This product, which is of degree “n”, becomes nullified when one of the factors is null, which means that it is a false equation, with “n” roots, which are of course the roots of the straight lines. There is no interest in finding them.
Example
Write the (false) third degree equation that corresponds to the
multiplication of the following three binomials:
(y = x - 17), whose root is 17;
(y = 2- x), whose root is 2;
(y = x + 1), whose root is -1.
Solution:
Product: (x-17) (2- x) (x + 1) = x³ - 18x² + 15x + 34 (polynomial)
False equation: x³ - 18x² + 15x + 34 = 0 (polynomial equalized to zero)
Clearly the above false equation has three roots, (17); (2); and (-1), each one nullifying one of the multiplication factors that originated it. Solving such an equation is rowing backwards, a futile ice-drying exercise.
The above example showed a false equation, carefully constructed to have three roots. Imagine the reader the waste of time of someone trying to find the roots, if any, of a randomly constructed false equation!
Second degree equation
We will see in Chapter IX the case of the equation of the second degree,
which has one exceptionality: it can be considered false or true.
In fact, a trinomial of second degree equalized to zero gives rise to a
false equation of second degree, an equation of abstract counts. We will see that this equation coincides with a true equation of
numbers: that is, the same equation will be false or true, depending on the
purpose of the mathematical operator. It is like in quantum physics, where
light can be wave or particle, alternating itself according to the experience
conducted by the observer.
Its two unknowns (when the equation is true) or its two roots (when the
equation is false) can be found using Bhaskara's formula.
Playful equations
Playful equations are equations that involve isolated numbers, not module or abstract counts.
For example, the equation that solves the following question is playful: find three consecutive numbers whose sum is 153.
Why call these equations playful? Because interest in these equations is only recreational, without application in real life. What is the use of finding numbers that mean nothing? Playful interest: 51,52 and 53.
An equation involving isolated numbers can be of any degree provided that it results from an equality imposed on two algebraic sums. However, it is not easy to construct true equalities of algebraic sums with unknowns higher than one. If you can build it, another problem will be to solve it.
Equations in science
Science imposes its formulas and its case-to-case special units.
Formulas are scientific definitions, not equations. For example,
Einstein's famous equation, e = m.c², is a scientific proposition, and not
really an equation, that is, it is not a confrontation of two equalities
constructed to obtain an unknown quantity. I dare not say that science has no
equations, but understand that the Law of Universal Gravitation, the Clapeyron
Equation, Newton's Second Law, Coulomb's Law, as well as Einstein's Field
Equations, are scientific impositions, not equations.
CHAPTER IX - THE EQUATION THAT MAY BE FALSE
OR TRUE
Quantum physics teaches us that light can behave like a wave or like a
particle, depending on the experience being conducted by the observer.
For example, ultraviolet light falling on a metal plate behaves like a particle and collides with the metal electrons, thus producing the photoelectric effect. Isaac Newton stated: light is a particle, for only one particle displaces another particle, like a billiard ball crashing into another.
If instead the light falls on a screen with two nearby slits, the light passes through both slits at the same time. That is why Christiaan Huygens concluded that light is a wave, for only one wave could accomplish this feat of passing at the same time in two different places.
Who was right?
Both, Einstein decided more than two hundred years later, because light at the same time is particle and wave. That is, nature has two models, of which, perceiving one, the observer is prevented from perceiving the other.
Two models and only one aspect! An order explained and an order implied!
Something like this happens in mathematics, with the equation of the
second degree, which can be false or true. One equation with two mathematical
meanings!
The false equation of the second degree
We have seen that a polynomial equalized to zero gives rise to a false
equation. An example is the trinomial of the second degree, y = ax² + bx + c,
corresponding to a parable, which, equalized to zero, gives rise to a false
equation with two roots. Such roots are the points where the parable crosses
the “x” axis. It is an equation of counts, which means that x² = x *.x, where x
* is a multiplier and x is the unknown, with the same numerical value. This is
the only counting equation I know that is not of the first degree, with the
important observation that it is a false equation, designed to locate points.
The parable has two symmetrical branches that begin at the vertex “V”, of
abscissa Xv = - b /2a, which is the point of the least ordinate of the trinomial.
Any point in the left branch of the parable, as point C in the figure, obviously has
the same ordinate of its symmetrical point in the right branch. See that the
roots, “x1” and “x2”, are equally two symmetrical points of the parable.
Equating the trinomial to zero, we have a false equation whose roots, “x1” and “x2”, can be
calculated using the Bhaskara’s formula.
But Bhaskara's formula does not apply only to the points where Y = 0. Applies to all points of the parable! For each “Y” value of the parable, such as point “C”, the two corresponding values of “x” can be calculated using the formula.
As follows:
Given any value of “Y”, just equate the trinomial to it, which changes the value of “c” to “ c´ ”, where c’ = c – y*, and apply Bhaskara to obtain the two values, “x1” and “x2”, corresponding to the given “y” and its symmetrical.
Bhaskara’s formula for y = y*
x1 and x2 = (- b ± √ (b² - 4ac'))/2a, where c’ = c – y*
That is, the millennial Bhaskara’s formula, which mathematicians use to extract the roots of the quadratic equation, is actually the reciprocal of the trinomial, that is, given “x”, the trinomial calculates “y”; given y, Bhaskara calculates its two corresponding values of “x1” and “x2”.
The dependent variable becomes independent, and vice versa!
Obviously, when y = 0, “x1” and “x2”, become the roots of the trinomial.
Bhaskara does not work when “Y” is less than “Yv”, because in that case “Y” does not belong in the parable: in this case Bhaskara leads to the square root of a negative number, which is not possible in mathematics. A value of “y” less than “yv” only exists imaginatively.
Exercise
Calculate the two “x” values of the parable defined by the square
trinomial (y = x² - 5x + 6), which correspond to the following ordinates:
y* = 0;
y* = 2;
y* = 10;
y* = - 3
First case y*= 0
x² - 5x + 6 = 0
c´ = c - 0 = 6
x1 and x2 = (5 ± √ (5² - 4x6))/2: 2 and 3
These are the roots of the trinomials, that is, the two points where the parable nullifies itself.
Second case y* = 2
x² - 5x + 6 = 2
c´ = 4
x² - 5x + 4 = 0
x1 and x2 = (5 ± √ (5² - 4 x 4)/2: 1 and 4
These are two symmetrical points of the parable.
Third case y* = 10
x² - 5x + 6 = 10
c´ = - 4
x² - 5x - 4 = 0
x1 and x2 = (5 ± √ (5² + 4 x 4))/2: - 0.7 and 5.7
These are two symmetrical points of the parable.
Fourth case y* = -3
x² - 5x + 6 = - 3
c´ = + 9
x² - 5x + 9 = 0
x1 and x2 = (5 ± √ (5² - 4x9))/2 = (5 ± √ (-11))/2 (impossible operation).
It is impossible to extract the square root from a negative number (-11), what means that y* = -3 is not ordinate of any point of the given parable.
In fact,
Trinomial: y = x² - 5x + 6
First derivative of the trinomial: y´ = 2x- 5
2xmin - 5 = 0
xv = xmin = 2.5,
yv = (2.5² - 5 x 2.5 + 6) = - 0.25.
Therefore, no point in the parable has a Y less than - 0.25. y* = - 3 do not belong in the parable, as we have already suspected.
The true equation of
the second degree
Let's see now what happens when we try to solve the following proposition: calculate two numbers whose sum is 5 and whose product is 6.
This statement leads us to multiply x by (5-x) and equalize the result to 6, that is, to construct the following equation:
(x) . (5 -x) = 6
or
5x - x² = 6 (an equation of the second degree).
Preparing the equation (5x - x² = 6) to apply the Bhaskara’s formula, we
have:
x² - 5x + 6 = 0,
This is a playful equation, because it was built with numbers, not with counts, which allows (x) to multiply (5-x). It is, moreover, a true equation because it corresponds to an imposed equality. Not to a polynomial equal to zero.
The same false equation of the zeroed trinomial!
One single equation with two distinct mathematical functions: for it is one thing to search out of the infinite points of a parable the two points that nullify it; another, quite another, is to calculate two numbers whose sum is 5 and the product is 6.
In both cases of course the two numbers sought are 2 and 3, obtained using the Bhaskara formula. They are, at the same time, the two unknowns of the equation and the two roots that annul the trinomial, indicating the two points where the parable meets the axis of the abscissas in a Cartesian graph.
How to calculate unknowns (or roots) without using Bhaskara formula
To solve the quadratic equation without using
Bhaskara, we have first to calculate the minimum x of the parable (xmin), for what
it is necessary to derive the trinomial. Xmin is the point from which the
two roots are equidistant:
xmin = - b/2a
x1 = xmin - d
x2 = xmin + d
Just replace x with (xmin + d) in the equation to get the value of “d”
directly. The same value of "d" can be obtained by substituting in
equation x for (xmin - d).
With the values of “xmin” and “d”, the roots are obtained.
Example:
Solve the equation x² - 5x + 6 = 0
xmin = - b/2a = 2.5
One can calculate d with either x1 or x2.
A - We initially chose to replace x with x2 = 2.5 + d
x2² - 5x2 + 6 = 0
x2 = 2.5 + d
(2.5 + d) ² - 5 (2.5+ d) + 6 = 0
(2.5² + 5d + d²) - 12.5 - 5d + 6 = 0
6.25 + d² - 6.5 = 0
d² = 0.25
d = 0.5
Therefore:
x1 = 2.5 - d = 2
x2= 2.5 + d = 3
B – Let us repeat the development, but replacing x with x1 =
2.5 - d
(2.5 - d) ² - 5 (2.5 - d) + 6 = 0
(2.5² - 5d + d²) - 12.5 + 5d + 6 = 0
6.25 + d² - 6.5 = 0
d² = 0.25
d = 0.5
Therefore:
x1 = 2.5 - d = 2
x2 = 2.5 + d = 3
An interesting exercise is to examine the developments in A and B above
and observe how mathematics adjusts its plots to correspond to the fact that,
with respect to xmin, the deviation of x2 and the deviation of x1 are equal and symmetrical (- d + d , respectively).
In A:
(2.5² + 5d + d²) - 12.5 - 5d + 6 = 0, or 6.25 + d² - 6.5 = 0
In B:
(2.5² - 5d + d²) - 12.5 + 5d + 6 = 0, or 6.25 + d² - 6.5 = 0
Two paths and a single result: 6.25 + d² - 6.5 = 0.
(Powerful math ...)
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