Six physics equations that changed the course of history
Pivotal points in the past few centuries saw human innovation advance in leaps and bounds, and all thanks to physics. Cathal O'Connell explains the equations and how they transformed history.
Physics equations are forms of magic. They allow us to explain the past, such as why Halley’s comet visits every 76 years, and predict the future – as far as the ultimate fate of the Universe.
They place limits on the possible, as in the efficiency of an engine, and they reveal possibilities we could never have imagined, such as the energy inside an atom.
Occasionally over the past few centuries, a new equation endowed the next generation with a new magical tool, and so changed the course of history. Here are some of the most pivotal.
1. Newton's second law of motion (1687)
What does it say?
Force equals mass times acceleration.
In other words ...
It's easier to push an empty shopping cart than a full one.
What did it teach us?
Together with Isaac Newton’s other two laws of motion (the first says you need a force to move something, the third says every action has an equal and opposite reaction), this equation forms the foundation of classical mechanics.
F=ma allowed physicists and engineers to calculate the value of a force. For instance, your weight (measured in newtons) is your mass (in kilograms) multiplied by acceleration due to gravity (on Earth, about 10 metres per second squared).
Saying you “weigh” 60 kilograms is incorrect in physics terms – your actual weight is about 600 newtons. This is the force pushing down on your bathroom scales.
But was it practical?
This equation was crucial to the arrival of the mechanical age. It’s used in almost every calculation which involves using force to cause movement.
It tells you how powerful an engine needs to be to power a car, how much lift an aircraft needs to take-off, how much thrust to lift a rocket, how far a cannonball flies.
2. Newton's law of universal gravitation (1687)
What does it say?
Any two massive objects pull on one another across space. But the force decreases rapidly the further apart they are.
In other words ...
We’re stuck to the Earth’s surface because our planet is comparatively big with lots more mass.
What did it teach us?
For centuries, the Universe had been divided into two realms – the earthly and the celestial. But Newton’s law of gravitation applied to everything. The same tug that causes an apple to fall from a tree keeps the Moon orbiting the Earth. Newton gave us the first direct connection between everyday life and the movement of the heavens.
But was it practical?
For a long time, the equation's main use was to calculate the orbits of planets. The space-age of the 1950s and 60s saw it used in practice – to send satellites into orbit and astronauts to the Moon.
One failing, which Newton himself admitted, was that he did not know "why" gravity operated. It took nearly 230 years for Albert Einstein to come along and explain gravity as arising from the warping of spacetime by massive objects in his theory of general relativity.
Even so, general relativity is only used in extreme situations, such as when gravity is very strong, or when great precision is required, such as for GPS satellites. In most cases Newton’s 330-year-old equation is still good enough.
3. Second law of thermodynamics (1824)
What does it say?
Entropy (a measure of disorder) always increases.
In other words ...
It’s no good crying over spilt milk. Disorder and mess are inevitable in the Universe.
What did it teach us?
While trying to analyse steam engine efficiency in the 19th century, French physicist Sadi Carnot stumbled upon one of the most profound equations in all of science.
It tells us some processes are irreversible, and may even be responsible for the arrow of time. In one of its simplest forms, it says heat always travels from a warm object to a cold one.
It can also be applied to the grandest scales. Some have applied it to describe the ultimate fate of the Universe in the form of “heat death” where all the stars are burnt out and nothing’s left but waste heat.
Others have used it to wind back through time and describe the origin of the Universe in a moment of zero entropy (or perfect order) at the instant of the Big Bang.
But was it practical?
This law was important for developing technologies of the industrial revolution, from steam to internal combustion engines, to refrigerators and chemical engineering.
In real engines, some energy is always wasted – so the law also showed any efforts at perpetual motion were ultimately futile.
4. The Maxwell-Faraday equation (1831 and 1865)
What does it say?
You can create a changing electric field (left side of the equation) from a changing magnetic field (on the right) and vice versa.
In other words ...
Electricity and magnetism are related!
What did it teach us?
In 1831, Michael Faraday discovered the connection between two natural forces, electricity and magnetism, when he found a changing magnetic field induced a current in a nearby wire.
Later, James Clark Maxwell generalised Faraday’s observation as one of his four fundamental equations of electromagnetism.
But was it practical?
This is the equation that powers the world. Most electric generators (whether in a wind turbine, coal-fired plant or a hydroelectric dam) work by converting mechanical energy (from steam or water) to rotate a magnet. By running this process in reverse, you get the electric motor.
More generally, Maxwell’s equations are still used in almost every application of electrical engineering, communications technology and optics.
5. Einstein's mass-energy equivalence (1905)
What does it say?
Energy equals mass multiplied by the speed of light squared.
In other words ...
Mass is really just a super-condensed form of energy.
What did it teach us?
Because of the size of the constant in the equation (the speed of light squared, an unimaginably huge number) a colossal amount of energy can be released through converting a tiny amount of mass.
But was it practical?
Einstein's most famous equation hinted at the potential for the huge amounts of energy released in nuclear fission, when a large unstable nucleus breaks into two smaller ones. This is because the mass of the two smaller nuclei together is always less than the mass of the original big nucleus – and the missing mass is converted into energy.
The "Fat Man" atomic bomb dropped over Nagasaki in Japan on 9 August 1945 converted just one gram of mass to energy, but produced an explosion the equivalent around 20,000 tonnes of TNT.
Einstein himself had signed a letter to US president at the time Franklin Roosevelt recommending the atom bomb be developed – a decision he later regarded as the “one great mistake” of his life.
6. The Schrödinger wavefunction (1925)
What does it say?
It describes how the change of a particle’s wavefunction (represented by psi, the candlestick shaped symbol) can be calculated from its kinetic energy (movement) and its potential energy (the interactions on it).
In other words ...
It’s the quantum version of F=ma.
What did it teach us?
When Erwin Schrödinger formulated his equation in 1925, it placed the new theory of quantum mechanics on firm footing by allowing physicists to calculate how quantum particles move and interact.
The equation looks a bit weird because it uses the mathematics of waves. (Subatomic particles are "wavy", so their interaction is described as interference of waves, rather than like billiard balls.)
But was it practical?
In one of its simplest forms, it describes the structure of the atom, such as the arrangement of electrons around the nucleus, and all chemical bonding.
More generally it’s used for many calculations in quantum mechanics and is fundamental to much of modern technology from lasers to transistors, and the future development of quantum computers.https://cosmosmagazine.com/physics/six-physics-equations-changed-course-history
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
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\begin{document}
\title{\bf Announcement 326: The division by zero z/0=0 - its impact to human beings through education and research\\
(2016.10.17)}
\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 to education and research on our wrong world history.
\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.
We should define $F(b,0)= b/0 =0$, in general.
\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 impact 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 and by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki - repeated subtraction method,
\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, not the point at infinity.
\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.
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 impact on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and physical problems.
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 impact to our basic ideas on the universe.
For our ideas on the division by zero, see the survey style announcements.
\section{Basic Materials of Mathematics}
(1): First, we should declare that the divison by zero is possible in the natural and uniquley determined sense and its importance.
(2): In the elementary school, we should introduce the concept of division by the idea of repeated subtraction method by H. Michiwaki whoes method is applied in computer algorithmu and in old days for calculation of division. This method will give a simple and clear method for calculation of division and students will be happy to apply this simple method at the first stage. At this time, they will be able to understand that the division by zero is clear and trivial as $a/0=0$ for any $a$. Note that Michiwaki knows how to apply his method to the complex number field.
(3): For the introduction of the elemetary function $y= 1/x$, we should give the definition of the function at the origin $x=0$ as $y = 0$ by the division by zero idea and we should apply this definition for the occasions of its appearences, step by step, following the curriculum and the results of the division by zero.
(4): For the idea of the Euclidean space (plane), we should introduce, at the first stage, the concept of steleographic projection and the concept of the point at infinity -
one point compactification. Then, we will be able to see the whole Euclidean plane, however, by the division by zero, the point at infinity is represented by zero. We can teach the very important fact with many geometric and analytic geometry methods. These topics will give great pleasant feelings to many students.
Interesting topics are: parallel lines, what is a line? - a line contains the origin as an isolated
point for the case that the native line does not through the origin. All the lines pass the origin, our space is not the Eulcildean space and is not Aristoteles for the strong discontinuity at the point at infinity (at the origin). - Here note that an orthogonal coordinates should be fixed first for our all arguments.
(5): The inversion of the origin with respect to a circle with center the origin is the origin itself, not the point at infinity - the very classical result is wrong. We can also prove this elementary result by many elementary ways.
(6): We should change the concept of gradients; on the usual orthogonal coordinates $(x,y)$,
the gradient of the $y$ axis is zero; this is given and proved by the fundamental result
$\tan (\pi/2) =0$. The result is trivial in the definition of the Yamada field. This result is derived also from the {\bf division by zero calculus}:
\medskip
For any formal Laurent expansion around $z=a$,
\begin{equation}
f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n,
\end{equation}
we obtain the identity, by the division by zero
\begin{equation}
f(a) = C_0.
\end{equation}
\medskip
This fundamental result leads to the important new definition:
From the viewpoint of the division by zero, when there exists the limit, at $ x$
\begin{equation}
f^\prime(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h} =\infty
\end{equation}
or
\begin{equation}
f^\prime(x) = -\infty,
\end{equation}
both cases, we can write them as follows:
\begin{equation}
f^\prime(x) = 0.
\end{equation}
\medskip
For the elementary ordinary differential equation
\begin{equation}
y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0,
\end{equation}
how will be the case at the point $x = 0$? From its general solution, with a general constant $C$
\begin{equation}
y = \log x + C,
\end{equation}
we see that, by the division by zero,
\begin{equation}
y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0,
\end{equation}
that will mean that the division by zero (1.2) is very natural.
In addition, note that the function $y = \log x$ has infinite order derivatives and all the values are zero at the origin, in the sense of the division by zero.
However, for the derivative of the function $y = \log x$, we have to fix the sense at the origin, clearly, because the function is not differentiable, but it has a singularity at the origin. For $x >0$, there is no problem for (2.6) and (2.7). At $x = 0$, we see that we can not consider the limit in the sense (2.3). However, for $x >0$ we have (2.6) and
\begin{equation}
\lim_{x \to +0} \left(\log x \right)^\prime = +\infty.
\end{equation}
In the usual sense, the limit is $+\infty$, but in the present case, in the sense of the division by zero, we have:
\begin{equation}
\left[ \left(\log x \right)^\prime \right]_{x=0}= 0
\end{equation}
and we will be able to understand its sense graphycally.
By the new interpretation for the derivative, we can arrange many formulas for derivatives, by the division by zero. We can modify many formulas and statements in calculus and we can apply our concept to the differential equation theory and the universe in connetion with derivatives.
(7): We shall introduce the typical division by zero calculus.
For the integral
\begin{equation}
\int x(x^{2}+1)^{a}dx=\frac{(x^{2}+1)^{a+1}}{2(a+1)}\quad(a\ne-1),
\end{equation}
we obtain, by the division by zero,
\begin{equation}
\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}.
\end{equation}
We will consider the fundamental ordinary differential equations
\begin{equation}
x^{\prime \prime}(t) =g -kx^{\prime}(t)
\end{equation}
with the initial conditions
\begin{equation}
x(0) = -h, x^{\prime}(0) =0.
\end{equation}
Then we have the solution
\begin{equation}
x(t) = \frac{g}{k}t + \frac{g(e^{-kt}- 1)}{k^2} - h.
\end{equation}
Then, for $k=0$, we obtain, immediately, by the division by zero
\begin{equation}
x(t) = \frac{1}{2}g t^2 -h.
\end{equation}
In those examples, we were able to give valuable functions for denominator zero cases. The division by zero calculus may be applied to many cases as a new fundamental calculus over l'Hôpital's rule.
(8): When we apply the division by zero to functions, we can consider, in general, many ways. For example,
for the function $z/(z-1)$, when we insert $z=1$ in numerator and denominator, we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0.
\end{equation}
However,
from the identity --
the Laurent expansion around $z=1$,
\begin{equation}
\frac{z}{z-1} = \frac{1}{z-1} + 1,
\end{equation}
we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = 1.
\end{equation}
For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as the division by zero calculus, however, the values by means of the Laurent expansions are not always reasonable. We will need to consider many interpretations for reasonable values. In many formulas in mathematics and physics, however, we can see that the division by zero calculus is reasonably valid. See \cite{kmsy,msy}.
\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.
Note that the Big Bang also may be related to the division by zero like the blackholes.
\section{Computer systems}
The above Professors listed are wishing the contributions in order to avoid the division by zero trouble in computers. Now, we should arrange new computer systems in order not to meet the division by zero trouble in computer systems.
By the division by zero calculus, we will be able to overcome troubles in Maple for specialization problems.
\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 introduced.
\bigskip
We are standing on a new generation and in front of the new world, as in the discovery of the Americas. Should we push the research and education on the division by zero?
\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 $z/0=0$, Advances in Linear Algebra
\& Matrix Theory, 6, 51-58. http://dx.doi.org/10.4236/alamt.2016.62007 http://www.scirp.org/journal/alamt
\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.
\bibitem{ann300}
Announcement 300 (2016.05.22): New challenges on the division by zero z/0=0.
\end{thebibliography}
\end{document}
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