New Calculus at a glance (Powerpoint Presentation):
/uploads/5/6/7/4/5674177/the_new_calculus_-_presentation.ppt
Quote this page as follows:
The New Calculus (c) John Gabriel 2010 [ http://thenewcalculus.weebly.com ]
Companion Sites:
Educators: http://whatyouhadtoknowbutyoureducators.weebly.com
German language: http://hanspeterguettinger.weebly.com
Wordpress Blog: http://mathphile.wordpress.com
For a quick introduction to the New Calculus:
The following documents are simple and a teaching approach is used:
/uploads/5/6/7/4/5674177/newcalculus-abstract-part-1.pdf
/uploads/5/6/7/4/5674177/newcalculus-abstract-part-2.pdf
/uploads/5/6/7/4/5674177/new_calculus_-_a_short_course.pdf
New Calculus on a single page! /uploads/5/6/7/4/5674177/spnc.jpg
From the previous graphic (spnc.jpg), one can see that the area between the curve f ' (x) and the x-axis is given by the product of the rectangular width and height corresponding to the same. The height is equivalent to the average value of f ' (x). The width is simply the length of the interval, that is, m+n. Observe that the value of the summation index k, does not influence the average value which is the same for any integer k>0. The identity SMA (Sub-interval Mean Average) is new in calculus.
Standard Calculus evolved from the flawed ideas of Newton, Leibniz and Cauchy. Until my New Calculus was developed, a rigorous formulation of calculus was unknown or unrealized (to or by humans of course). Rather than admit they had no answers, academics conjured up theories based on half truths and self-deception. For almost 300 years academics were unable to resolve the problem of finding the gradient of a tangent line to a curve using a rigorous method. Instead they chose to base the entire theory of calculus on the ill-defined notion of limit (/uploads/5/6/7/4/5674177/magnitude_and_number.pdf)
When one repeats an untruth sufficient times, one begins to believe it. I have lost count of how many times I have read or heard the untrue phrases "The calculus was placed on a firm foundation." or "The calculus was placed on a rigorous footing." or "The calculus was made more rigorous."
Aside from being false, this belief that calculus was made rigorous has generated much flawed theory which should never have been thought of, never mind printed. For example, the idea of an infinitesimal is a myth because it is an ill-defined concept. Whatever one imagines is real, if and only if what one imagines is well-defined.
Epsilonics theory was an attempt to avoid the use of infinitesimal ideas. One important reason epsilonics fails, is that it is based on the concept of arbitrarily small distances (meaningless nonsense) where there is no clear boundary or distinction between given magnitudes (aka real numbers) and so called infinitesimal "numbers". Any academic who supports infinitesimal theory, is unfit to be called a mathematician.
A typical dialogue between myself and other academics:
Academic: For every epsilon greater than zero, there exists a delta greater than zero,
such that for all x close to c, 0<|x-c|<delta implies |f(x)-L|<epsilon.
Gabriel: How do you find the limit L when you do not know its value?
Academic: You can approximate L by using some small h and observing what happens
to the finite difference ratio.
Gabriel: L is never an exact value. How can you be sure it will always satisfy |f(x)-L|<epsilon?
Academic: You can determine L exactly through the first principles method, that is, using
the limit definition: f'(x) = (lim as h approaches 0) { f(x+h) - f(x) } / h
Gabriel: The first principles method requires that L exists because it is a limit definition. By
forming the symbolic difference quotient, one automatically assumes that L exists.
The first principles method assumes the limit exists; it does not prove a limit exists,
but merely provides a flawed guide or method of how to find the limit. A fast way
to see this, is to recognize that the limit (lim) appears on the right hand side of the
first principles method. f'(x) is an L-value (an assigned value). Although defined as:
f'(x) = (lim as h approaches 0) { f(x+h) - f(x) } / h , academics have misinterpreted
the definition as: (lim as h approaches 0) { f(x+h) - f(x) } / h = f'(x) = L
Still incorrect, but it should not surprise one that these idiots got it backward, they
have been using flawed methods and vague understanding since Newton and Leibniz.
In one part of the method you treat h different from 0 and in another part you treat h
as you would 0. Does h undergo a change in its nature - perhaps due to certain
quantum fluctuations? (joke)
Academic: (Confounded...)
Does any of the previous circular logic make sense to you?
The following applet demonstrates the stark differences between Cauchy's jury-rigged calculus and the rigorous formulation of the New Calculus. Observe how the limit L or f'(2) becomes indeterminate as the finite difference ratio in Cauchy's Kludge approaches the form 0/0. Also note how the gradient is always correct in the New Calculus but never correct using standard calculus. Be sure to drag the slider in the following applet so that dst = 0.
Cauchy's Kludge
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Although I first called this new idea the Secant Theorem (because it has been proved), it is actually also an axiom. One might equally well have called it the Secant-Tangent Axiom. Ask yourself why it is that academics have been so ignorant and incompetent all these centuries.
The idea of limit was born from the inability of mathematicians to realize what I have finally accomplished in the New Calculus - a task not completed by any one else since the time of Archimedes.
What better method exists to find the slope of a tangent than by means of a parallel secant? The New Calculus is founded on this idea. This beautiful axiom/theorem makes it possible to develop the entire calculus without the use of limits or other ill-defined concepts such as infinitesimals. If you are astute, then you will notice that the introductory graphic on this page summarizes all calculus. Well, you might have to study a bit...
The next applet demonstrates the secant theorem. Slide the point m or c (by using slider tangentpoint) and observe how the gradient is always correct. The derivative f'(c)=E(c)+Q(m,n) is also displayed. Check its values manually to satisfy yourself the secant method is true. Remember m must be less than or equal to c and n must be greater than or equal to c. m and n are the distances on either side of point c. Moving point m to the right of c will yield unpredictable results. Note that f'(c)=E(c) where E(c) is an expression for the derivative in terms of c. Q(m,n) is always equal to 0 in the secant method. Hold down CTRL; press and release + while the applet is not in focus in order to view the whole applet; then click on the applet to give it focus.
If at any time, the values of m or n are undefined, this is due to the limitations of the GeoGebra software. Observe that you can find an (m;n) distance pair for any distance m in the interval [c-m;c] regardless of whether c-m>n-c or c-m<n-c. When the software fails to work, you can determine the (m,n) pair from the auxiliary equation. One of the powerful features of the New Calculus is its potential use in computer graphics. If the creators of Geogebra knew the New Calculus, the following applet would not suffer from undefined values.
Secant Theorem
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For more information on why epsilon-delta theory is flawed, be sure to read: /uploads/5/6/7/4/5674177/limits.pdf
The flawed theory of epsilonics is discussed in more detail further on in this page.
Cauchy's Kludge (/uploads/5/6/7/4/5674177/cauchykludge.pdf) demonstrates the errors of Cauchy's definition in about 6 pages. One of many contradictions and errors that arise from the Kludge is a theorem that states polynomial functions are generally everywhere differentiable. One can read about the fallacy regarding the differentiable cubic (/uploads/5/6/7/4/5674177/cubicfallacy.pdf) at the origin.
For the single variable New Calculus, a tangent line at any point P is constructed by knowing the relationship between the abscissa of P and the distances m and n from the abscissa to the endpoints of a parallel secant. It is my opinion that tangent lines were invented by Ancient Greeks to measure smoothness of curves when continuity is given, that is, a curve is smooth if exactly one tangent line can be constructed at every point except perhaps a point of inflection. Many modern academics do not know what is a tangent line because their definitions are ill-defined. Bad mathematics resulting from ill-defined concepts does not end with the tangent line definition. Consider that until my New Calculus, academics were completely in the dark about the nature of differentials. I now enlighten all those who are interested: /uploads/5/6/7/4/5674177/dydx_compared.pdf
Even though Cauchy's fake calculus works, consider that it has given rise to more false and unsound theory (infinitesimals, limits, real analysis, etc) and obstructed the progress of mathematics in this regard.
Differentiability:
The concept of differentiability is poorly defined in standard calculus, with incorrect results at points of inflection. One of the attributes of inflection or saddle points is that they are not differentiable. This is caused by a change in concavity. If a tangent line could be constructed at a point of inflection, then there would be no change in concavity. A tangent line (of finite length) by definition cannot cross a function's path. It intersects a path in exactly one point and crosses it nowhere. The file called cubicfallacy.pdf on this page explains much more. Read the following article to see what it means for a function to be differentiable at a given point in single variable calculus (not the same in new multi-variable calculus): /uploads/5/6/7/4/5674177/differentiability.pdf
To think of a function f being differentiable at a point c in standard calculus is in fact incorrect, because in order to be differentiable at c, implies that f is continuous and smooth on the interval or sub-interval containing c, that is, f is differentiable over the interval containing c, not only at c. Therefore, it is ignorant to state that if the Cauchy limit can be found, then a function is differentiable at a given point, to wit, the cubic (x^3) is not differentiable at the origin due to the fact that a finite tangent line with defined gradient cannot be constructed at the origin. In fact, it is just absurd to talk about a function being differentiable at a point. More importantly, a function f is differentiable over an interval, if and only if, one finite tangent line with defined gradient can be constructed at every point in the interval. In other words, the question to ask is: "Is the function differentiable over a given interval?" and not "Is a function differentiable at a given point?" Moreover, one assumes that a function is differentiable on a given interval and then finds a derivative at a point in the interval to confirm this is true or not. By calculating a derivative, one always assumes a function is differentiable. Therefore it is incorrect to think that one proves differentiability, rather one confirms or rejects that a function is differentiable over a given interval.
In the New Calculus, provided a derivative exists at a given point of a function, then it follows the function is differentiable over a finite interval containing the same point.
Cauchy did not realize that in order to form the difference quotient used in his kludgy limit definition, the function must be continuous and smooth, that is, differentiable over the interval for which the secant line gradients (difference quotients) are formed. It is absurd to think that one can find tangent line gradients to functions that are not differentiable over a given interval! Yet this is exactly the implication of Cauchy's kludgy limit. The unique finite tangent lines with defined gradients are the reason a function is differentiable over a given interval. Stupid academics did not realize this important fact for centuries until I made them aware. Calculus is about natural averages which are only possible with continuous and smooth functions.
Certain academics will shun the new calculus because they are lazy and do not like to exercise what little brains they have. A typical academic might claim the new calculus is difficult. For one who has learned the wrong methods all one's life and is resistant to change, this is true. However, I have found that students learn the New Calculus far easier than the flawed standard calculus. Their general mathematics ability also improves tenfold because the new calculus requires sound reasoning and thought processes which are rigorous.
Introduction:
I decided to call my reformulation of this important branch of mathematics: The New Calculus. In fact, it is not so much a new calculus, as much as it is a sound, well-defined and easy-to-learn calculus, without the use of limit theory or real analysis, of which neither existed in Newton's period. The secant method is at the core of single variable differentiation, just as the tangent disc method is at the core of 3 dimensional differentiation (*) in the new calculus.
Tangent objects exist for higher dimensions even though these can only be visualized up to and including 4 dimensions. Calculus, in one of its aspects, is the subject that describes how to calculate attributes of tangent objects (such as gradient, normal vector, etc), to functions that are both continuous and smooth. In another aspect, it is the branch of mathematics that through the use of natural averages, describes distances, tangent objects, areas, volumes and hyper-volumes in different dimensions. Calculus developed into a very complex subject when real analysis became widely used in education.
(*) A new kind of mathematics vastly different from standard multi-variable calculus but far simpler and easier to learn. Most mathematicians like to work from a set of axioms. For a simple, no nonsense list of well-defined mathematical axioms, the following will be very useful:
/uploads/5/6/7/4/5674177/mathematical_axioms.pdf
This web page is about introducing the New Calculus which will change the way mathematicians do business. Although single variable calculus is discussed, there is one example of how it can be extended to multi-variable calculus (/uploads/5/6/7/4/5674177/partial_derivatives.pdf).
The New Calculus will change the way students perceive calculus, which has nothing to do with limits, but due to Cauchy's wrong ideas, is now mistakably associated with limit theory.
If this is your first visit, you will encounter new knowledge. Whether you are a renowned mathematics professor or a high school graduate, prepare to be astounded.
The new calculus is the first rigorous formulation of calculus without the use of limits or real analysis. One can also say it is the first rigorous formulation of any sort ever, given Cauchy's limit based calculus is fake.
Newton and Leibniz were both credited for inventing calculus independently of each other. Although both these individuals were fine academics, the truth is neither of them invented calculus. Furthermore, neither individual was able to formulate a sound definition of either the integral or derivative. Moreover, neither academic was able to state the mean value theorem which is the fundamental theorem of calculus. The following document explains: /uploads/5/6/7/4/5674177/mvt-indivisibles.pdf
The Real Beginning:
The first three propositions mentioned in the section called Quadrature of the Parabola (The Works of Archimedes), were stated without proof. Archimedes claims these were proved in the Elements of the Conics, presumably a work by Euclid and Aristarchus, that is thought to be lost forever. Calculus began with these propositions. The geometric objects called curves and tangent lines were the source of the modern concepts such as continuity, smoothness and differentiability. These concepts are not well-defined in standard calculus but they are well-defined and easy to understand in the new calculus.
Proposition 1 (Quadrature of the parabola) states that if a straight line from a point V on a chord of a parabola is constructed parallel to the axis (or the axis itself) and meets a tangent line at some point P, then V is the midpoint of the chord. Had Archimedes thought of the tangent line concept in the same incorrect way as modern academics, the first three propositions would never have been thought of or published.
Ideas and concepts exist independently of the human mind. Those who chance to think of them are but flash moments in the history of time.
The Secant Theorem.
Given a function f, that has exactly one tangent line (with defined gradient) at each point in an interval [c-m,c+n] containing some point c, the gradient of the tangent line to the function at the point c is given by:
However, we use the following notation once it is understood that c is a point, while m and n are distances on either side of c such that c-m < c < c+n. c_x means the x coordinate of the point c.
m and n are related distances corresponding to the endpoints [c-m,f(c-m)] and [c+m,f(c+m] of a parallel secant to the tangent line at c.
For any function f with a tangent line (at x=c) having gradient k, the ordinate difference of the secant endpoints is always k(m+n) because of the gradient ratio. This implies the ordinate difference is always divisible by (m+n). Provided f is continuous and smooth over any interval (c-m,c+n), there are infinitely many secant ordinate pairs f(c+n) and f(c-m), such that any secant gradient ratio [f(c+n)-f(c-m)]/(m+n) produces k.
To see how the New Calculus works dynamically, click on the slider called dst in the applet that follows. Changing the value of dst (by dragging the black dot on the slider in either direction) repositions the red line which is parallel to the blue tangent line. You will notice that there is always a relationship between m, n and c. The tangent line is drawn at the point (c,0). You can click on the function f(x) in the Objects pane, and change it to see how the secant method works with other functions. If for some reason when you change the function, nothing seems to happen, your best bet is to reload/refresh the page and try again. There are some bugs in the Geogebra applet API. Experiment by changing f(x) and moving the sliders dst and c to see the effect on the secant and tangent line. Please bear in mind that if a discrepancy occurs between the software and the algebra, it could be that the limitations of the graphing software are inadvertently misleading.
Illustration of the Secant Method in the New Calculus
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Finding the relationship between m and n is not always an easy task and in some cases requires some ingenuity. It is always possible to determine a general derivative (explained shortly) given a primitive function that is differentiable.
Finding an auxiliary equation:
To find a relationship between m and n by means of an auxiliary equation is described further here: /uploads/5/6/7/4/5674177/appetizer.pdf
In order to convince oneself the secant theorem is true, form the quotient described by f ' (c) for a given differentiable function and simplify the result. Next, one must find a relationship between m and n. This can be done by equating the sum of all the terms in m and n [denoted by Q(m,n)] to zero. Choose a suitable m or n and then find the corresponding m or n. Finally, substitute the values of m and n into your simplified quotient result f ' (c) to obtain the gradient.
Note that an expression such as [sin(c+n)-sin(c-m)]/(m+n) is not an expression whose terms all contain m and/or n. Thus, equating [sin(c+n)-sin(c-m)]/(m+n) to zero would yield incorrect results. However, (m+n) is a factor of every term found in the ordinate difference formed by f(c+n)-f(c-m). It can be demonstrated that m+n is a factor of every term in the ordinate difference sin(c+n)-sin(c-m). So, what do you think the relationship between m and n might be in the case of the derivative where f(x)=sin(x) ?
All differentiable functions possess the special relationship where the sum of the terms in m and n equals to 0 after cancellation (known in the new calculus as the distance pair (0;0) ) in the difference quotient as will be explained shortly. Once again, it must be remembered that although we can disregard terms in m and n, this does not matter in the case of a straight line where m and n can take any values. A straight line is the only geometric object whose gradient does not depend on secant ordinates because straight lines do not have tangents.
For a given interval (m;n), all the parallel secants are part of a tangent space for the given (c, f(c)) which is the point of tangency. Each parallel secant has its own unique (m;n) pair. The tangent line owns the (0;0) pair. A special relationship exists between m, n and c for any continuous and smooth function. If only the pair (0;0) is possible, then a point of inflection exists.
Differences between the standard and new calculus:
If f(x) = x^2 (x squared), then the general derivative is given by f ' (x) = 2x and a numeric derivative is given by f ' (c) = 2c, where c is the x coordinate of the tangent point.
The standard definition (with limit), that is, [f(x+h)-f(x)] / h as h approaches 0, never represents the tangent line gradient until cancellation has taken place (even in this case it represents only the general derivative, never the numeric derivative), that is, k(h)/(h), where h/h is replaced by 1, h is replaced by 0 in any terms still containing h, and k is the gradient.
Newton's idea of considering a denominator in a finite difference ratio that decreases to an infinitesimal value is incorrect. In fact, it is incorrect to substitute zero for h (old calculus) or m+n (new calculus) in the terms of the finite difference quotient that contain them, until after cancellation, for otherwise the ratio does not represent any meaningful gradient. In the New Calculus, one of the terms which is the gradient, does not contain m or n. The remaining terms always sum to zero. By taking the limit as per standard calculus, one is concluding that the sequence:
{ [f(x1)-f(c)]/[x1-c] ; [f(x2)-f(c)]/[x2-c] ; [f(x3)-f(c)]/[x3-c] ; ... k }
converges to some real number k, which is the tangent line gradient. Although this is generally true, the sequence depends on the validity of the limit as a well-defined concept, which of course the limit is not. Furthermore, k derived in this way, is always a numeric derivative, never a general derivative. In fact the value of k is always approximate, unless k is a rational number.
Bishop Berkeley was correct in being skeptical. The denominator of a finite difference ratio which describes the consequent of a gradient can never be zero. Setting h=0 in the polynomial resulting from the difference quotient (that is, after cancellation) is entirely legal, although not correct, because as previously explained, the derivative produced is a general derivative as opposed to a numeric derivative, that is, f ' (x) for some value of x
On the other hand, the new calculus definition always represents the tangent line gradient (given appropriate values of m and n for each parallel secant in the interval considered) and does not rely on the ill-defined notion of limit - this in contrast to the standard definition which erroneously refers to a non-parallel secant gradient until after the difference ratio is reduced by cancellation. Furthermore, it is entirely correct in the New Calculus to set m=n=0 in the polynomial resulting from the difference quotient after cancellation.
The new calculus does not suffer from absurd results such as 0/0 in the finite difference ratio, the ill-defined concept of infinitesimals or any other confusion that arises naturally from Cauchy's ill-defined derivative.
One could say that standard calculus is fake because it is based on definitions that are ill-formed. If you understood the contents of the Cauchy Kludge pdf, then no doubt you will agree that Cauchy's definition is a perfect example of jury-rigging.
Simplifying Explanation of Cauchy's wrong ideas:
The main problem with Cauchy's definition is that h is infinitesimal (greater than zero but less than every other magnitude?) before cancellation (*) and zero after cancellation - neither of which are permitted for a numeric derivative. As explained in the pdf called Debunking wrong ideas about the derivative /uploads/5/6/7/4/5674177/debunking_wrong_ideas_about_derivative.pdf,
h cannot be infinitesimal before cancellation as it represents the horizontal component for the gradient of a non-parallel secant and h cannot be zero after cancellation because then the horizontal component of the gradient is zero and thus the finite difference ratio is meaningless. Standard calculus performs bogus arithmetic operations twice:
1. Assumes h is not zero and performs division by h.
2. Assumes h is very small (meaningless nonsense) and discards terms in h effectively treating h as if it were 0.
Moreover, the Cauchy definition relies on the empirical approach of Newton which is an indeterminate process directed at finding a numeric derivative rather than a logical method in finding a general derivative. For example, it does not matter how small the denominator of the ratio is made, because no infinitely small denominator exists that can be the horizontal component of the tangent line gradient. This means that although f '(x) represents the general derivative (after cancellation and h=0), it never represents the derivative when x=x in Newton's empirical approach! In fact, Cauchy's definition is incorrect if it is interpreted as valid for both a numeric and general derivative. Newton was stumped by this, which I think is one of the reasons he refrained from publishing his ideas sooner. The following graphic will help clarify these ideas:
The numeric derivative presumes prior knowledge of limits. The general derivative is only possible if the red secant lines are parallel to the green tangent line. If not, then one cannot set h=0 to find the general derivative because the difference quotient is that for a non-parallel secant. By setting h=0 in Cauchy's definition after the quotient is reduced (diagram on right in previous illustration), one arrives at f '(c), where x<c<c+h, and f ' (c) is not equal to f ' (x). This is discussed in Cauchy's kludge.
In the previous illustration, the diagram on the left shows how h can never be 0. To make sense of the difference quotient, one must use parallel secants (as in the diagram on the right), in which case h can be 0, and subsequently a general derivative can be found. However, the general derivative is not the same as Cauchy's numeric derivative. In my new calculus, one arrives at the same derivative (numeric or general) every time.
Can you see how Cauchy's definition is jury-rigged? You may have to study these facts several times before your understanding becomes clear because Cauchy's error is subtle. The diagram on the right illustrates why the kludge works for general derivatives, that is, f(x+h) and f(x) in the left diagram correspond to f(c+h) and f(x) in the right diagram respectively. These facts do not affect the use of Cauchy's definition, except for pedagogical purposes and numerous incorrect theorems. The last couple of centuries have shown that students and mathematics professors never acquire a clear understanding of the derivative (where limits are not required at all).
These inconsistencies are removed in my New Calculus. As a result, differentials are well-defined in the New Calculus. dy/dx always means the same thing in the New Calculus, whereas it has a serious identity crisis in the standard calculus, that is, it can be interpreted as a limit or as a rational expression depending on context. Cauchy would have been correct if he defined the derivative as follows from the diagram on the right:
f ' (c) = [ f(c+h) - f(c-x) ] : (c+h-x)
Cauchy's first error was in his conception of the limit. His second error was using the limit in an attempt to define the derivative, where it results in his jury-rigged definition. If there were any justification for using limits in calculus, it would probably be in respect to integration, however, even in this regard, limits are not required. Using my systematic approach in finding ante-derivatives (see end of this page for article on indefinite integration), it may now be possible to find any ante-derivative, although not a trivial process.
Leibniz's definition is really not much better than Newton's, even though it appears to be geometric and somewhat tidier. Leibniz would have defined the derivative as follows had he understood calculus as well as I :
df:dx = [ f(x+n) - f(x-m) ] : (m+n) [LD]
The following links explain more:
/uploads/5/6/7/4/5674177/meaning_of_dydx.pdf
/uploads/5/6/7/4/5674177/what_does_leibniz_notation_mean_exactly.pdf
/uploads/5/6/7/4/5674177/dydx_related_rates_example.pdf
The differentials df and dx are exactly equal symbolically (or proportional if numeric) to
[ f(x+n) - f(x-m) ] and (m+n) respectively where [LD] represents the gradient of a secant line parallel to the tangent line whose gradient is being determined. In the previous form [LD], df/dx is a symbolic fraction. df/dx becomes an exact fraction when the symbols (function placeholders and variables) are replaced with numbers. The difference between a symbolic and exact fraction is that a symbolic fraction's value is determined according to a given difference ratio whereas an exact fraction has known values. However, this difference is superficial as the fractions are used exactly the same way in algebra.
(*) Cancellation is the process of forming the separate individual quotients by considering the quotient of each term in the numerator with the denominator. The astute reader will notice that this process is assumed to be complete (according to Cauchy), before the finite difference is reduced later (through cancellation) in order to find the general derivative. Therefore, Cauchy's definition is not only fake, but is also fraudulent in terms of the simple properties of arithmetic.
The following file called newcalculus.pdf contains a few general examples. Divisibility_identities.pdf contains a proof that is used in newcalculus.pdf.
| newcalculus.pdf |
| divisibility_identities.pdf |
Back to the Beginning.
If you read and understood the previous files containing some general and actual examples, then I have no doubt you will want to know more. What happened between the time of Archimedes and Newton? It is a well-known fact that the foundations of integral calculus were laid by Archimedes. He used the method of exhaustion to approximate incommensurable magnitudes such as Pi and to calculate irregular areas. The mathematical objects that Archimedes used were the rational numbers.
Over 2,000 years later, these are still the same objects we use today. There is some new terminology (real number) and also some ill-defined concepts (limit, complex number). 'Real number' is terminology used to describe any magnitude that is incommensurable. Contrary to the opinion of most academics, a real number has yet to be well-defined. In fact, most academics don't know the difference between number and magnitude. Following article explains: /uploads/5/6/7/4/5674177/magnitude_and_number.pdf
A mathematician is like an artist: the objects arising from concepts in a mathematician's mind, are only as appealing as they are well-defined.
Zero enters the number club.
Many centuries passed after rational numbers had been introduced. Eastern mathematicians were researching the concept of zero magnitude that was initially rejected by the Greeks. At first, zero failed almost every requirement needed to qualify as a number. The main requirements were: (i) the mathematical object must be a magnitude (ii) the object must be measurable by the magnitude of an object different to itself, presumably the larger measured by the smaller as Euclid states in his masterpiece, The Elements.
In order for an object to qualify as a magnitude, it was considered imperative that such an object could be instantiated, if not physically, then at the very least it must be well-defined in one's imagination, which implies its physical instantiation is irrelevant. If what one imagines is well-defined, then that which is being imagined is as good as real!
Zero was not immediately accepted, for it could not even satisfy the first requirement, that is, it is not a magnitude. Although the first requirement was eventually waived, zero still failed to satisfy the second requirement, for no other object or magnitude can measure zero, except zero itself.
Zero manifested great potential in serving as a placeholder in radix systems and also in denoting equality between arithmetical operations, as opposed to the unit which denotes equality between numbers. For example, comparing p with p, that is, taking the difference (p - p), results in equality where p is some number, while a - b = c/d results in equality between the expressions a - b and c/d, that is, (a - b) - c/d = 0. It was eventually realized that granting zero membership had certain potential benefits. Provided there was a way to resolve this conflict, zero could be quite useful as a number.
What are the irrational numbers?
Given any integers c and d and some real number k, one can form averages k/c and 1/d. If no c and d exist such that k/c=1/d, then k is said to be irrational. In other words, no proper fraction p/q exists (where p and q are integers), such that p/q equals the fractional part of k, given that k is irrational.
This definition states that k is irrational if it cannot be written as c/d. The definition states what is not a rational number, but says nothing about what constitutes an irrational number. Consider the following analogy:
Humans are not human because they don't have tails. They are human because they think and reason. Chimpanzees do not have tails, yet they are not human. A quality or trait or characteristic, must indicate a feature of the object or idea being defined.
Would it have made any sense to say, Primates without tails are human ? Obviously not. The analogy here is that primates are magnitudes where humans are rational and chimpanzees are irrational.
A property of being well-defined is evident by defining a concept through an attribute it possesses, not by an attribute it lacks. This property is one of the core characteristics of an object that is well-defined. To state that irrational numbers are numbers that are not rational, does not say much, except that such magnitudes cannot be described by any known rational number. So, exactly what is an irrational number?
Unlike a rational number, an irrational number is an incommensurable magnitude that can neither be defined, nor represented in terms of comparison. It is possible only to represent incommensurable magnitudes through averages that are mere approximations. However, these same approximations never define incommensurable magnitudes which came to be known over time as irrational numbers.
In fact, irrational 'numbers' are not numbers; rather these are incommensurable magnitudes, which are now represented and used as approximations through sums of averages.
What are real numbers?
Real numbers came into existence as a result of irrational "numbers". Before incommensurable magnitudes (aka irrational numbers) were discovered by the ancient Greeks, all magnitudes were considered to be rational numbers, that is, all magnitudes were thought to be measurable. Therefore, incommensurable magnitudes, were simply renamed irrational numbers and lumped together with the rational numbers to form the new real numbers.
Given these facts, it is immediately evident that an irrational number is ill-defined. Now since real numbers are defined to be the union of a set which includes the set of rational numbers and the set of irrational numbers, it follows the real numbers also, are not well-defined.
It would thus be in order to say that when a real number is not rational, the same number represents some incommensurable magnitude, for example Pi, the square root of two or any other incommensurable magnitude.
The Limit.
It did not take long for mathematicians to become dissatisfied with their inability to well-define incommensurable magnitudes. Rather than concede these incommensurable magnitudes cannot be represented by numbers, those 'mathematicians' devised a new concept called the limit which they would use to redefine, not only all real numbers, but also attempt to provide rigour to the many problematic definitions in calculus. They failed, and what is taught in today's lecture rooms is the nonsense created by them.
Limits are not required in calculus or any branch of mathematics. Indeed, the arrival of the theory of limits is a major setback in the progress of mathematics. The theory is flawed for many reasons, but the most important reason is that limit theory was developed from the theory of sets, which is fundamentally in error.
The popular notion regarding the concept of limit being at the core of calculus is entirely false. In fact, as stated earlier, the inventors of calculus knew nothing about limits. The limit itself is poorly defined and takes on several different meanings, depending on context. For example, a limit can denote some value of an expression through cancellation of certain factors in the expression (*). A limit can also denote the upper bound of an infinite sum as in the case of a convergent sequence (also known as Cauchy sequence).
The most popular use of limits is for determination of gradients where they are not required at all. A gradient of a curve at a point is the gradient of the tangent line at that point, provided the tangent line exists and has a defined slope. The only geometric object that has a gradient or slope is a straight line. By gradient of a curve at a point, the implied meaning is the gradient of the tangent line to a curve at that point.
(*) This notion is extended to expressions where very small magnitudes (ill-defined nonsense) are discarded and the result is called a limit.
How did we get the idea of limit?
The great mathematician Archimedes was the first to formalize the ideas that eventually led to the unfortunate adoption of the limit by modern academics, and consequently also the theory that arose from this idea. The Archimedean Property states:
If x is any magnitude, then there exists a well defined natural number N such that x < N.
Note that I use the word magnitude - because a real number is not well-defined. In fact, Real Analysis, which was created from the nonsense of Georg Cantor’s misguided ideas, defines real numbers using the limit concept (Dedekind cuts or Cauchy sequences). Arbitrarily small distances between rational numbers are used to demonstrate convergence which leads to the notion of a limit. That which is being defined is used in its own definition! Refer to my article on magnitude and number under section called Debunking Real Analysis Myths.
The Archimedean property is misunderstood by most professors of mathematics. The property was not intended to be used as a prelude for the concept of a least upper bound of a set which is bounded above. Its purpose was to establish a method for measuring incommensurable magnitudes by means of approximation. For example, Archimedes knew that his method of exhaustion would never result in a value greater than some rational number that is larger than Pi. The property is stated more accurately as follows:
If x is any magnitude, then there exists a commensurable magnitude N such that x<N.
This last definition uses commensurable magnitude because natural numbers are an abstraction derived from ratios of commensurable magnitudes - a fact that every professor of mathematics does not know, unless such a professor learned it from me. What does the Archimedean property mean? The property simply states that given any commensurable or incommensurable magnitude, a commensurable magnitude exists which is greater. The gist is that given any magnitude X, there is a commensurable magnitude N, which is greater than the magnitude X.
The Archimedean property is best illustrated geometrically through the triangle inequality which states that the sum of any two sides of a triangle are greater than the remaining side. One can therefore let the magnitude be a side of the triangle and construct the other two sides such that they are commensurable, and thus the property is proved.
Anal retentive academics were not satisfied with Archimedes' ideas. They preferred to obfuscate matters by introducing non-issues that are unrelated to Archimedes' original ideas. What started out as simple and elegant ideas morphed into:
(i) a least upper bound concept for a given set called a limit.
(ii) the name given to a result of simplifying a given expression
through cancellation of its factors, is called a limit.
(iii) zero which assumes the role of a limit when the denominator
of a given fraction is assumed to increase or decrease without bound.
(iv) the name given to a result of simplifying a given expression by
disregarding terms whose values are decided by an indeterminately
large denominator. This is an extension of (iii).
The categorization of these various aspects based on this ill-defined concept of limit evolved into the ubiquitous calculus limit as it is known today.
Epsilonics (Epsilon-Delta) proofs.
Unfortunately, an entire semester is usually wasted learning epsilonic proofs. The fact that many students drop out of math courses because they think they do not have what it takes, is good reason to be alarmed. To those students who are contemplating dropping out, my advice to you is to simply tough it out. Your difficulty in understanding is mainly due to the fact that this theory is ill-defined. And then you have lecturers who are absolutely incompetent, which compounds the problem. If only they understood some mathematics... To help you understand what the non-noteworthy fuss is about, I have compiled a document called limits.pdf (link follows). Use this document in conjunction with the GeoGebra applet to understand the meaning. Whatever you do, don't quit your math studies because of this unimportant topic in Real Analysis. I expect that when my New Calculus eventually replaces standard calculus, this nonsense will be a thing of the past.
Stating a theory using numbers and symbols does not make it more rigorous. In most cases, unless the theory can be properly worded and more importantly well-defined, it usually ends up containing many flaws. Prior to learning standard calculus, students are required to spend much fruitless time trying to prove limits exist, when all they are doing is stating a limit exists using epsilonics. Finding a delta and epsilon, is analogous to saying that a triangle is a triangle because it has three angles. In fact it is insufficient to find a delta and epsilon - one has to find a relation between the two. Again, it is insufficient because of the reasons explained in limits.pdf. Whichever way one chooses to look at this epsilon-delta theory, it is ill-defined as the following article explains: /uploads/5/6/7/4/5674177/limits.pdf
The following applet calculates the exact epsilon-delta region for a given planar function f(x), which you can change by clicking on f(x) in the Free Objects region and entering your own function. The anchor button (rightmost button on toolbar) allows you to move the graph (move graphics view mode), to re-scale the axes and to zoom in and out. To re-scale an axis, click on any part of it whilst in move graphics view mode and whilst holding your mouse button down, drag in either direction (north-south or east-west) until you are satisfied and then release the mouse button. You may have to move the graph back into view (using anchor button) after rescaling the axes. To move the sliders you must be in pointer mode (leftmost button on toolbar). Move the sliders called delta and c to observe the effect on the limiting region.
In some cases it might appear as if there is no graph, but what this means is that you have to resize and re-centre the graph. Re-scaling the axes helps to see more granular details when the limiting region is very small.
Dynamic Illustration of Epsilon-Delta Limit theory on your choice of function.
Purpose of this applet: To demonstrate the uselessness of epsilonics in limit theory. |
Calculus without the use of limits.
Calculus is possible because of the properties of natural averages. Calculus is about natural averages, not limits. The New Calculus does not use limits. The following pdf provides a proof of the average value theorem and the fundamental theorem of calculus without the use of limits.
| proof_avtftc.pdf |
Newton and Leibniz would have loved to know this information. Both these academics were challenged to provide rigorous definitions of their calculus. Both failed.
Newton's greatest mathematical accomplishment and supreme work was his discovery of the finite difference interpolation polynomial. In terms of mathematics, all of Newton's remaining works are in my opinion, insignificant. Leibniz on the other hand, almost succeeded in redefining his definite integral correctly. He was on the right track by researching moments, but was unable to complete his task. Leibniz may not have been too surprised to learn from my redefinition of the integral, that all integrals are path or line integrals.
Most modern academics fail to understand that calculus only applies where natural averages are present, that is, continuous and smooth functions. For example, they have been known to apply calculus to conditional functions (aka piece-wise functions) where they remove discontinuities inadvertently or purposely, and then try to deduce conclusions about continuity and differentiability. Newton and Leibniz knew nothing about conditional functions.
To help you learn all about Newton's interpolation polynomial (and much more...), I have compiled the following publications:
How we got calculus: /uploads/5/6/7/4/5674177/howwegotcalculus.pdf
Newton's methods: /uploads/5/6/7/4/5674177/newton-methods.pdf
How we got calculus is a tour de force of Newton's most important work which I think is his interpolation polynomial. Only standard calculus is used in How we got calculus. Taylor's theorem can be derived from Newton's interpolation polynomial by using only the new calculus. Start with formula [T1] on page 15 of How we got calculus. Observe that my new calculus version of the Taylor polynomial is an equality (Gabriel polynomial), unlike Taylor's polynomial which is always an approximation. The main idea behind Taylor's theorem is ease of calculation even though convergence is a disadvantage in terms of acquiring sufficient accuracy.
Newton's methods were based largely on approximation because he lacked certain knowledge, that is, he probably did not know of the mean value theorem and certainly did not know of my New Calculus. The mean value theorem is hardly understood by most mathematics professors. There are many GeoGebra applets supposedly demonstrating the workings of the mean value theorem, but all of them are superficial and miss the most important aspect which is captured in the following GeoGebra applet: /uploads/5/6/7/4/5674177/mvtvisual.ggb
Roger Cotes is a mathematician who is not well-known. Yet the impact of Cotes' work has been significant. Without it there would have been very little progress in numeric integration and therefore the solving of differential equations. The following document shows how Cotes arrived at the general formula: /uploads/5/6/7/4/5674177/cotesintegration.pdf
A proof of L'Hopital's Rule using the New Calculus.
I am often asked how I would explain L'Hopital's rule without the use of limits. The following file contains an elegant proof that is less than one page. /uploads/5/6/7/4/5674177/lhopital.pdf
Finally, a rigorous calculus without the use of limits.
Almost 330 years later, I have presented a rigorous calculus and redefined not only the derivative, but also the definite integral without the use of limits. Newton and Cauchy were mistaken, but their wrong ideas have been corrected in the new calculus.
Conclusion:
I trust your visit to my site has been informative and also entertaining. These few ideas and secrets I have shared with you, are only a fraction of what is contained in my new calculus. One of the secrets I recently shared is the existence of a cumulative probability function for the normal density function. It would have been impossible or very difficult without the methods of the New Calculus.
Future of the new calculus:
1. The primary purpose of the new calculus is for academics to adopt a sound and rigorous branch of mathematics that one can learn and master in a short time. Standard calculus has proved to be difficult and hard to master in any reasonable period of time, one of the key problems being its use of the limit concept and real analysis.
2. Real calculus reform: there are no ambiguities, paradoxes or contradictions. Concepts are well-defined unlike the standard calculus which is fake.
3. The following key fields have been identified as important research areas in
the new calculus:
Solution of differential and partial differential equations, including the
use of numeric integration techniques.
Solution of area, volume and hyper-volume problems using well-defined
concepts such as natural averages.
Spatial representation of curves in n-dimensions (useful in computer
graphics).
Accurate curve fitting.
Systematic indefinite integration and differentiation.
New regression analysis (statistics).
Application to abstract algebra and discrete mathematics.
And much more...
4. Due to decreased complexity in the new calculus, research and understanding
is facilitated and thereby enhanced.
I have done substantial research in these aforementioned areas. With regards to numeric differentiation and integration the New Calculus has significant advantages over the n-Point central difference formulas and well-known quadrature/cubature algorithms. This information is currently withheld pending the publication of What you had to know... I am certain there are many more benefits to be realized by adopting the new calculus which the future is certain to reveal.
(C) John Gabriel, The New Calculus, 2010
All Rights Reserved
Recent Publications:
/uploads/5/6/7/4/5674177/indefinite_integral_-_systematic_method.pdf
Fools in academia:
These foolish academics need to be exposed, not because they are jealous and evil (they generally are), but rather because they are stupid beyond belief. It is a tragedy that today's most influential academics censor and denigrate those whose ideas they neither understand nor like. While fools are abundant throughout the earth, it is a crying shame that modern academia is the new Catholic church of knowledge. As the Catholics suppressed those who in the middle ages differed from their ideas, modern academia have established a biased knowledge repository in the form of academic journals.
If new knowledge is not printed in one of the recognized journals, then sheepish learners are trained to shun it, just as Catholics of the middle ages were warned of committing blasphemy. In the end, everyone is a loser.
Georg Cantor: Believed he was onto something with regards to countability of sets, in particular infinite sets. Cantor foolishly assumed that the set of real numbers is uncountable, when in fact the set is ill-defined for two reasons: it is infinite and its members are not well-defined. If the real numbers can be represented in base 10 (as Cantor assumes), then the set of real numbers is indeed countable because any radix system only represents rational numbers. In fact, the set of real "numbers" is uncountable because real numbers are not well-defined. See article on magnitude and number at the top of this page.
Guiseppe Peano/Abraham Fraenkel/Ernst Zermelo: These "mathematicians" formulated the ZF axioms which are nothing more than a juvenile attempt to form what they perceived as the foundations of arithmetic. Built on Cantor's dumb ideas, ZF is an attempt to define numbers without an understanding of ratio or measure.
David Hilbert: An ardent follower of Cantor whose efforts placed Cantorian ideas at the forefront of mathematics - a devastating action that set mathematics off course the last 120 years. Set theory is a failed attempt to understand numbers in terms of containment rather than measure. Containment disregards the aspect of comparison, whereas measure is defined by it. The New Mathematics (of set theory) espoused these wrong ideas and till this day mathematicians never fully grasp the concept of comparison, equality or inequality. In fact, most mathematics professors do not know the difference between a magnitude and number.
Kurt Godel: The father of the completeness/incompleteness theorems in logic. A little known fact is that Godel's own theorems disprove themselves. Only academic ignoramuses can be trusted to miss such a simple fact as this.
Bertrand Russell: Discovered paradoxes in set theory (unsurprising because set theory is ill-defined) and prepared the stage for flawed modern logic theory. Russell was an overrated logician whose debating skills earned him a place in the history of mathematics. He was a notorious smoker who never failed to appear in any photograph without a tobacco pipe in mouth or hand. Russell's Principia Mathematica is less worth than the paper used to print it.
Abraham Robinson: A fool of fools who designed non-standard analysis on Cauchy's kludgy idea from classic analysis, that is, the ill-defined concept of infinitesimal. The infinitesimals according to Robinson are a subset of the interval (0,1) with no least upper bound. Stupid academics claim that an infinitesimal is greater than 0 but less than every positive number. How they arrive at the plural is absurd, that is, there is no way of comparing infinitesimals with each other.
It is impossible to tell where infinitesimals end and the real numbers begin since the infinitesimal set has no least upper bound. Not one instance of an infinitesimal number can be demonstrated either in theory or practice. Rather an attempt is made to draw conclusions about more theorems using the same ill-defined concepts, unsurprisingly often with incorrect results. It is impossible to compare infinitesimals (measure them). Ironically, Robinson based his theory on the assumption that the real numbers are well-defined (and used the transfer principle to validate his non-standard theory) - I have proved this to be false, that is, the real numbers are not well-defined. In fact, real numbers do not exist. See my article on magnitude and number.
Robinson's useless theory survives because of the Jewish influence in academia. Wikipedia's Jewish sysops and admins tried to give this garbage theory of Robinson's more credibility by claiming Archimedes used infinitesimals, until they finally understood that Archimedes had no idea about any such nonsense, nor is his method of exhaustion in any way related to Robinson's absurd ideas.
Stephen Hawking: The obscurity of his theories are tantalizing to the community because they sound exotic and alluring. An honest historian will remember this fool as a speculating ignoramus. Most theoretical physicists are duller than dish water.
AMS (American Mathematical Society) and MAA (Mathematical Association of America): Both societies will gladly publish anti-mathematical nonsense on ill-defined calculus limits and non-existent infinitesimals. Unfortunately most sheepish journals follow their lead. That these societies are run by idiots, is beyond question.
More fools coming: I could spend my entire life refuting all these fools or continue working on my new ideas in mathematics. I choose to sound the warning bells and let those who ignore pay the price.
