What Is Calculus?

Calculus is a branch of mathematics that explores variables and how they change by looking at them in infinitely small pieces called infinitesimals. Calculus, as it is practiced today, was invented in the 17th century by British scientist Isaac Newton (1642 to 1726) and German scientist Gottfried Leibnitz (1646 to 1716), who independently developed the principles of calculus in the traditions of geometry and symbolic mathematics, respectively. 

While these two discoveries are most important to calculus as it is practiced today, they were not isolated incidents. At least two others are known: Archimedes (287 to 212 B.C.) in Ancient Greece and Bhāskara II (A.D. 1114 to 1185) in medieval India developed calculus ideas long before the 17th century. Tragically, the revolutionary nature of these discoveries either wasn't recognized or else was so buried in other new and difficult-to-understand ideas that they were nearly forgotten until modern times.

The word "calculus" has a modest origin, deriving from similar words such as "calculation" and "calculate," but all these words derive from a Latin (or perhaps even older) root meaning "pebble." In the ancient world, calculi were stone beads used to keep track of livestock and grain reserves (and today, calculi are small stones that form in the gallbladder, kidneys or other parts of the body).

The utility of infinitesimals

To understand what is meant by infinitesimal, consider the formula for the area of a circle: A=πr². The following demonstration is adapted from one given by Professor Steve Strogatz of Cornell, who points out that despite this formula's simplicity, it is impossible to derive without the utility of infinitesimals.

To start, we recognize that the circumference of a circle divided by its diameter (or twice the radius) is approximately 3.14, a ratio denoted as pi (π). With this information, we can write the formula for a circle's circumference: C=2πr. To determine a circle's area, we can start by cutting the circle into eight pie wedges and rearranging them to look like this:We see the short, straight edge is equal to the original circle's radius (r), and the long, wavy side is equal to half the circle's circumference (πr). If we repeat this with 16 pieces, it looks like this:

Again, we see the short, straight edge is equal to the original circle's radius (r), and the long, wavy side is equal to half the circle's circumference (πr), but the angle between the sides is closer to a right angle and the long side is less wavy. No matter how much we increase the number of pieces we cut the circle into, the short and long sides keep the same respective lengths, the angle between the sides gets progressively closer to a right angle, and the long side gets progressively less wavy.

Now, let's imagine we cut the pie into an infinite number of slices. In the language of mathematics, the slices are described as "infinitesimally thick," since the number of slices "is taken to the limit of infinity." At this limit, the sides still have lengths r and πr, but the angle between them is actually a right angle and the waviness of the long side has disappeared, meaning we now have a rectangle.

Calculating the area is now just the length × width: πr × r=πr². This case-in-point example illustrates the power of examining variables, such as the area of a circle, as a collection of infinitesimals.

Two halves of calculus

The study of calculus has two halves. The first half, called differential calculus, focuses on examining individual infinitesimals and what happens within that infinitely small piece. The second half, called integral calculus, focuses on adding an infinite number of infinitesimals together (as in the example above). That integrals and derivatives are the opposites of each other, is roughly what is referred to as the Fundamental Theorem of Calculus. To explore how this is, let's draw on an everyday example:

A ball is thrown straight into the air from an initial height of 3 feet and with an initial velocity of 19.6 feet per second (ft/sec).

If we graph the ball's vertical position over time, we get a familiar shape known as a parabola.

Differential calculus

At every point along this curve, the ball is changing velocity, so there's no timespan where the ball is traveling at a constant rate. We can, however, find the average velocity over any timespan. For example, to find the average velocity from 0.1 seconds to 0.4 seconds, we find the position of the ball at those two times and draw a line between them. This line will rise some amount compared with its width (how far it "runs"). This ratio, often referred to as slope, is quantified as rise ÷ run. On a position versus time graph, a slope represents a velocity. The line rises from 4.8 feet to 8.3 feet for a rise of 3.5 feet. Likewise, the line runs from 0.1 seconds to 0.4 seconds for a run of 0.3 seconds. The slope of this line is the ball's average velocity throughout this leg of the journey: rise ÷ run = 3.5 feet ÷ 0.3 seconds = 11.7 feet per second (ft/sec).

At 0.1 seconds, we see the curve is a bit steeper than the average we calculated, meaning the ball was moving a bit faster than 11.7 ft/sec. Likewise, at 0.4 seconds, the curve is a bit more level, meaning the ball was moving a bit slower than 11.7 ft/sec. That the velocity progressed from faster to slower means there had to be an instant at which the ball was actually traveling at 11.7 ft/sec. How might we determine the precise time of this instant?

Let's back up and observe that the span of 0.1 seconds to 0.4 seconds isn't the only timespan over which the ball had an average velocity of 11.7 ft/sec. So long as we maintain the line's slope, we can move it any place over this curve and the average velocity over the timespan between the two places the line intersects the curve will still be 11.7 ft/sec. If we move the line farther toward the edge of the parabola, the timespan decreases. When the timespan reaches zero, the points land on the same spot and the line is said to be tangent to (just barely resting against) the parabola. The timespan is described as having been "taken to the limit of zero."Here's where the notion of infinitesimals enters into play. Until this point, we've talked about velocity over a finite span of time, but now we're talking about a velocity at an instant; a timespan of infinitesimal length. Notice how we can't take the slope between two points that are infinitesimally far apart; we'd have rise ÷ run = 0 feet ÷ 0 seconds, which doesn't make any sense. To find the slope at any point along the curve, we instead find the slope of the tangent line. The results of six points are plotted below:

This graph is what's known as the original graph's derivative. In the language of mathematics and physics, it's said that "the derivative of an object's position with respect to time is that object's velocity."

Integral calculus

This process works in reverse, too. The opposite of a derivative is anintegral. Thus, "the integral of an object's velocity with respect to time is that object's position." We found derivatives by calculating slopes; we find integrals by calculating areas. On a velocity versus time graph, an area represents a length. The matter of finding areas under a graph is relatively simple when dealing with triangles and trapezoids, but when graphs are curves instead of straight lines, it is necessary to divide an area into an infinite number of rectangles with infinitesimal thickness (similar to how we added an infinite number of infinitesimal pie wedges to get a circle's area).To determine which of these curves will give us the original graph of position, we must also use some knowledge about the position of the ball at a certain time. Examples of this include the height from which it was thrown (the vertical position of the ball at time zero), or the time at which it hit the ground (the time where the vertical position was zero). This is referred to as an initial condition because we're usually concerned with predicting what happens after, though it's a bit of a misnomer, since an initial condition can also come from the middle or end of a graph.http://www.livescience.com/50777-calculus.html

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\begin{document}

\title{\bf  Announcement 300:  New challenges on the division by zero z/0=0\\

(2016.05.22)}

\author{{\it Institute of Reproducing Kernels}\\

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

%\date{\today}

\maketitle

{\bf Abstract: } In this announcement, for its importance we would like to state the

situation on the division by zero and propose basic new challenges.

\bigskip

\section{Introduction}

%\label{sect1}

By a {\bf natural extension} of the fractions

\begin{equation}

\frac{b}{a}

\end{equation}

for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0,

\end{equation}

incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the  case of real numbers.

 The division by zero has a long and mysterious story over the world (see, for example, Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628,  however,

  Sin-Ei Takahasi (\cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing the extensions of fractions and by showing the complete characterization for the property (1.2):

 \bigskip

 {\bf  Proposition 1. }{\it Let F be a function from  ${\bf C }\times {\bf C }$  to ${\bf C }$ satisfying

$$

F (b, a)F (c, d)= F (bc, ad) 

$$

for all

$$

a, b, c, d  \in {\bf C }

$$

and

$$

F (b, a) = \frac {b}{a },  \quad   a, b  \in  {\bf C }, a \ne 0.

$$

Then, we obtain, for any $b \in {\bf C } $

$$

F (b, 0) = 0.

$$

}

 Note that the complete proof of this proposition is simply given by  2 or 3 lines.

\medskip

We thus should consider, for any complex number $b$, as  (1.2);

that is, for the mapping

\begin{equation}

w = \frac{1}{z},

\end{equation}

the image of $z=0$ is $w=0$ ({\bf should be defined}). This fact seems to be a curious one in connection with our well-established popular image for the  point at infinity on the Riemann sphere. Therefore, the division by zero will give great impacts to complex analysis and to our ideas for the space and universe.

However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:

\medskip

1) by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose generalized inverse,

\medskip

2) by the intuitive meaning of the fractions (division) by H. Michiwaki,

\medskip

3) by the unique extension of the fractions by S. Takahasi,   as in the above,

\medskip

4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from  ${\bf C}$ onto ${\bf C}$,

\medskip

and

\medskip

5) by considering the values of functions with the mean values of functions.

\medskip

Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:

\medskip

\medskip

A) a field structure  containing the division by zero --- the Yamada field ${\bf Y}$,

\medskip

B)  by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,

\medskip

C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero,

\medskip

and

\medskip

D) by considering rotation of a right circular cone having some very interesting

phenomenon  from some practical and physical problem.

\medskip

In (\cite{mos}),  many division by zero results in Euclidean spaces are given and  the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.

\medskip

See  J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.

Meanwhile,  J. P.  Barukcic and I.  Barukcic (\cite{bb}) discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.

 Furthermore,  T. S. Reis and J.A.D.W. Anderson (\cite{ra,ra2}) extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.

 Meanwhile, we should refer to up-to-date information:

{\it Riemann Hypothesis Addendum - Breakthrough

Kurt Arbenz

https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum -   Breakthrough.}

\medskip

Here, we recall Albert Einstein's words on mathematics:

Blackholes are where God divided by zero.

I don't believe in mathematics.

George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as {\bf the biggest blunder of his life} [1]：

1. Gamow, G., My World Line (Viking, New York). p 44, 1970.

 For our ideas on the division by zero, see the survey style announcements 179,185,237,246,247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252,ann293}).

\section{On mathematics}

Apparently, the division by zero is a great missing in our mathematics and the result (1.2) is definitely determined as our basic mathematics, as we see from Proposition 1.  Note  its very general assumptions and  many fundamental evidences in our world in (\cite{kmsy,msy,mos}). The results will give great impacts  on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and  physical problems. See our announcements for the details.

The mysterious history of the division by zero over one thousand years is a great shame of  mathematicians and human race on the world history, like the Ptolemaic system (geocentric theory). The division by zero will become a typical  symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.

We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful,  and will give great impacts to our basic ideas on the universe.

\section{Albert Einstein's biggest blunder}

The division by zero is directly related to the Einstein's theory and various

physical problems

containing the division by zero.  Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.

\section{Computer systems}

The above Professors listed are wishing the contributions in order to avoid the zero division trouble in computers. Now,  we should arrange  new computer systems in order not to meet the division by zero trouble in computer systems.

\section{General  ideas on the universe}

The division by zero may be related to religion,  philosophy and the ideas on the universe, and it will creat a new world. Look the new world.

\bigskip

We are standing on a new  generation and in front of the new world, as in the discovery of the Americas.

 \bigskip

\bibliographystyle{plain}

\begin{thebibliography}{10}

\bibitem{bb}

J. P.  Barukcic and I.  Barukcic, Anti Aristotle—The Division of Zero by Zero. Journal of Applied Mathematics and Physics,  {\bf 4}(2016), 749-761.

doi: 10.4236/jamp.2016.44085.

\bibitem{bht}

J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,

Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).

\bibitem{cs}

L. P.  Castro and S. Saitoh,  Fractional functions and their representations,  Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.

\bibitem{kmsy}

M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,

Int. J. Appl. Math.  {\bf 27} (2014), no 2, pp. 191-198,  DOI: 10.12732/ijam.v27i2.9.

\bibitem{ms}

T. Matsuura and S. Saitoh,

Matrices and division by zero $z/0=0$,

Linear Algebra \& Matrix Theory (ALAMT)(to appear).

\bibitem{msy}

H. Michiwaki, S. Saitoh,  and  M.Yamada,

Reality of the division by zero $z/0=0$.  IJAPM  International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html

\bibitem{mos}

H.  Michiwaki, H. Okumura, and S. Saitoh,

Division by Zero $z/0 = 0$ in Euclidean Spaces.

 International Journal of Mathematics and Computation

 (in press).

\bibitem{ra}

T. S. Reis and J.A.D.W. Anderson,

Transdifferential and Transintegral Calculus,

Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I

WCECS 2014, 22-24 October, 2014, San Francisco, USA

\bibitem{ra2}

T. S. Reis and J.A.D.W. Anderson,

Transreal Calculus,

IAENG  International J. of Applied Math., {\bf 45}(2015):  IJAM 45 1 06.

\bibitem{s}

S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices,  Advances in Linear Algebra \& Matrix Theory.  {\bf 4}  (2014), no. 2,  87--95. http://www.scirp.org/journal/ALAMT/

\bibitem{ttk}

S.-E. Takahasi, M. Tsukada and Y. Kobayashi,  Classification of continuous fractional binary operations on the real and complex fields,  Tokyo Journal of Mathematics,   {\bf 38}(2015), no. 2, 369-380.

\bibitem{ann179}

Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.

\bibitem{ann185}

Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.

\bibitem{ann237}

Announcement 237 (2015.6.18):  A reality of the division by zero $z/0=0$ by  geometrical optics.

\bibitem{ann246}

Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.

\bibitem{ann247}

Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

\bibitem{ann250}

Announcement 250 (2015.10.20): What are numbers? -  the Yamada field containing the division by zero $z/0=0$.

\bibitem{ann252}

Announcement 252 (2015.11.1): Circles and

curvature - an interpretation by Mr.

Hiroshi Michiwaki of the division by

zero $r/0 = 0$.

\bibitem{ann281}

Announcement 281(2016.2.1): The importance of the division by zero $z/0=0$.

\bibitem{ann282}

Announcement 282(2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.

\bibitem{ann293}

Announcement 293(2016.3.27):  Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

\end{thebibliography}

\end{document}