How to Deceive with Statistics: Distortions Due to Diminutive Denominators
We all learned in elementary school that "you can't divide by zero." But what happens when you divide by a number very close to zero, a small fraction? The quotient shoots way up to a very large value.
Pick any number. If you divide 27 by 1, you get 27. If you divide 27 by 0.1, you get 270. Divide 27 by 0.001, and you get 27,000. And so on. Any such division exercise blows up to a huge result as the denominator gets closer and closer to zero.
There are several indices being cited these days that get people's attention because of the big numbers displayed. But the reality is that those particular big numbers come entirely from having very small denominators when calculating a ratio. Three prominent examples of this mathematical artifact are the feedback effect in global warming models, the "Global Warming Potential," and the "Happy Planet Index." Each of these is afflicted by the enormous distortion that results when a denominator is small.
The Happy Planet Index
The "Happy Planet Index" is the easiest to explain. It is used to compare different countries and is formed by the combination of
(a x b x c) / d.
In this equation,
a = well-being – "how satisfied the residents of each country feel with life overall" (based on a Gallup poll)
b = life expectancy
c = inequalities of outcomes ("the inequalities between people within a country in terms of how long they live, and how happy they feel, based on the distribution in each country's life expectancy and well-being data")
d = ecological footprint ("the average impact that each resident of a country places on the environment, based on data prepared by the Global Footprint Network")
How do the assorted countries come out? Using this index, Costa Rica with a score of 44.7 is number 1; Mexico with a score of 40.7 is number 2; Bangladesh with a score of 38.4 is number 8; Venezuela with a score of 33.6 is 29; and the USA with a score of 20.7 is number 108 – out of 140 countries considered.
Beyond such obvious questions as "Why are so many people from Mexico coming to the USA while almost none are going the other way?," it is instructive to look at the role of the denominator (factor d) in arriving at those numerical index values.
Any country with a very low level of economic activity will have a low value of "ecological footprint." Uninhabited jungle or barren desert scores very low in that category. With a very small number for factor (d), it doesn't make a whole lot of difference what the numbers for (a), (b), and (c) are; the tiny denominator guarantees that the quotient will be large. Hence the large index reported for some truly squalid places.
The underlying reason why the "Happy Planet Index" is so misleading is because it includes division by a number that for some countries gets pretty close to zero.
Global Warming Potential
The second example of this effect is the parameter "Global Warming Potential," which is used to compare the relative strength of assorted greenhouse gases. The misuse of numbers here has led to all sorts of dreadful predictions about the need to do away with trace gases like methane (CH4), N2O, and others.
"Global Warming Potential" was first introduced in the IPCC's second assessment report and later formalized by the IPCC in its Fourth Assessment Report of 2007 (AR-4). It is described in section 2.10.2 of the text by Working Group 1. To grasp what it means, it is first necessary to understand how molecules absorb and re-emit radiation.
Every gas absorbs radiation in certain spectral bands. The more of a gas is present, the more it absorbs. Nitrogen (N2), 77% of the atmosphere, absorbs in the near-UV part of the spectrum, but not in the visible or infrared range. Water vapor (H2O) is a sufficiently strong absorber in the infrared that it causes the greenhouse effect and warms the Earth by over 30˚C, making our planet much more habitable. In places where little water vapor is present, there is less absorption, less greenhouse effect, and it soon gets cold (think of nighttime in the desert).
Once a molecule absorbs a photon, it gains energy and goes into an excited state. Until that energy is lost (via re-radiation or collisions), that molecule won't absorb another photon. A consequence of this is that the total absorption by any gas gradually saturates as the amount of that gas increases. A tiny amount of a gas absorbs very effectively, but if the amount is doubled, the total absorption will be less than twice as much as at first and similarly if doubled again and again. We say the absorption has logarithmic dependence on the concentration of the particular gas. The curve of how total absorption falls off varies according to the exponential function, exp (-X/A), where X is the amount of a gas present (typically expressed in parts per million, ppm), and A is a constant related to the physics of the molecule. Each gas will have a different value, denoted B, C, D, etc. Getting these numbers within ±15% is considered pretty good.
There is so much water vapor in the atmosphere (variable, above 10,000 ppm, or 1% in concentration) that its absorption is completely saturated, so there's not much to discuss. By contrast, the gas CO2 is a steady value of about 400 ppm, and its absorption is about 98% saturated. That coincides with the coefficient A being roughly equivalent to 100 ppm.
This excursion into the physics of absorption pays off when we look at the mathematics that goes into calculating the "Global Warming Potential" (GWP) of a trace gas. GWP is defined in terms of the ratio of the slopes of the absorption curves for two gases: specifically, the slope for the gas of interest divided by the slope for carbon dioxide. The slope of any curve is the first derivative of that curve. Economists speak of the "marginal" change in a function. For a change of 1 ppm in the concentration, what is the change in the radiative efficiency?
At this point, it is crucial to observe that every other gas is compared to CO2 to determine its GWP value. In other words, whatever GWP value is determined for CO2, that value is reset equal to 1 so that the calculation of GWP for a gas produces a number compared to CO2. The slope of the absorption curve for CO2 becomes the denominator of the calculation to find the GWP of every other gas.
Now let's calculate that denominator. When the absorption function is exp (-X/A), it is a mathematical fact that the first derivative = [-1/A][exp(-X/A)]. In the case of CO2 concentration being 400 ppm, when A = 100 ppm, that slope is [-1/100][exp (-4)] = - 0.000183. That is one mighty flat curve, with an extremely gentle slope that is slightly negative.
Next, examine the gas that's to be compared with CO2, and calculate the numerator.
It bears mentioning that the calculation of GWP also contains a factor related to the atmospheric lifetime of each gas. Here we'll concentrate on the change in absorption due to a small change in concentration. The slope of the absorption curve will be comparatively steep, because that molecule is at low concentration, able to catch all the photons that come its way.
To be numerically specific, consider methane (CH4), with an atmospheric concentration of about Y = 1.7 ppm, or N2O, at concentration Z = 0.3 ppm. Perhaps their numerical coefficients are B ~ 50 or C ~ 150; they won't be terribly far from the value of A for CO2. Taking the first derivative gives [-1/B][exp{-Y/B)]. Look at this closely: with Y or Z so close to zero, the exponential factor will be approximately 1, so the derivative is just 1/B (or 1/C, etc.). Maybe that number is 1/50 or 1/150 – but it won't be as small as 0.000183, the CO2 slope that appears in the denominator.
In fact, the denominator (the slope of the CO2 curve as it nears saturation) is guaranteed to be a factor of about [exp (-4)] smaller than the numerator 7 – for the very simple reason that there is ~ 400 times as much CO2 present, and its job of absorbing photons is nearly all done.
When a normal-sized numerator is divided by a tiny denominator, the quotient blows up. The GWP for assorted gases come out to very large numbers, like 25 for CH4 and 300 for N2O. The atmospheric lifetime factor swings some of these numbers around still farther: some of the hydrofluorocarbons (trade name Freon) have gigantic GWPs. HFC-134a, used in most auto air conditioners, winds up with GWP above 1,300. The IPCC suggests an error bracket of ±35% on these estimates. However, the reality is that every one of the GWPs calculated is enormously inflated due to division by the extremely small denominator associated with the slope of the CO2 absorption curve.
The calculation of GWP is not so much a warning about other gases, but rather an indictment of CO2, which (at 400 ppm) would not change its absorption perceptibly if CO2 concentration increased or decreased by 1 ppm.
The Feedback Effect of Global Warming
The third example comes from the estimates of the “feedback effect” in computational models of global warming. The term “Climate Sensitivity” expresses how much the temperature will rise if the greenhouse gas CO2 doubles in concentration. A relevant parameter in the calculation is “radiative forcing,” which can be treated either with or without feedback effects associated with water vapor in the atmosphere. Setting aside a lot of details, the “no feedback” case involves a factor L that characterizes the strength of the warming effect of CO2. But with feedback, that factor changes to [ L /(1 – KL)], where K is the sum of assorted feedback terms, such as reflection of radiation from clouds and other physical mechanisms; each of those is assigned a numerical quantity. The value of L tends to be around 0.3. The collected sum of the feedback terms is widely variable and hotly debated, but in the computational models used by the IPCC in prior years, the value of K tended to be about K = 2.8.
Notice that as L tends toward 1/3 and K goes to 3, the denominator goes to zero. For the particular case of L = 0.3 and K = 2.8, the denominator is 0.16 and the “feedback factor” becomes 6.25. It was that small denominator and consequent exaggerated feedback factor that increased the estimate of “Climate Sensitivity” from under 1 oC in the no-feedback case to alarmingly large estimates of temperature change. Some newspapers spoke of “11 oF increases in global temperatures.” Nobody paid attention to the numerical details.
In more recent years, the study of various positive and negative contributions to feedback improved, and the value of the sum K dropped to about 1, reducing the feedback factor to about 1.4. The value of the "Climate Sensitivity" estimated 30 years ago in the "Charney Report" was 3˚C ± 1.5˚C. Today, the IPCC gingerly speaks of projected Climate Sensitivity being "near the lower end of the range." That sobering revision can be traced to the change from a tiny denominator to a normal denominator.
The take-home lesson in all of this is to beware of tiny denominators. Any numerical factor that is cranked out is increasingly meaningless as the denominator shrinks.
When some parameter (such as "Climate Sensitivity" or "Global Warming Potential" or "Happy Planet Index") has built into it a small denominator, don't believe it. Such parameters have no meaning or purpose other than generating alarm and headlines.
とても興味深く読みました：ゼロ除算の発見
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\begin{document}
\title{\bf Announcement 388: Information and ideas on zero and division by zero\\
(a project)\\
(2017.10.29)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
}
\date{\today}
\maketitle
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotelēs (BC384 - BC322) and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta (598 - 668 ?).
In particular, Brahmagupta defined as $0/0=0$ in Brāhmasphuṭasiddhānta (628), however, our world history stated that his definition $0/0=0$ is wrong over 1300 years, but, we showed that his definition is suitable.
For the details, see the references and the site: http://okmr.yamatoblog.net/
We would like to write some story on zero and division by zero. For this purpose, we would like to gather some wide ideas and feelings on the zero and division by zero. For some precise facts and some wide viewpoints on these topics, please kindly send your ideas and feelings. For some valuable ones, we would like to immediately distribute them as in examples on the division by zero (now over 670 items).
For your kind comments, several lines will be well-comed
and or in A4 one page in word.
Please kindly send your ideas to the e-mail address:
\medskip
kbdmm360@yahoo.co.jp
\medskip
We would like to hear your valuable and interesting ideas on these topics.
\bibliographystyle{plain}
\begin{thebibliography}{10}
\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{ms16}
T. Matsuura and S. Saitoh,
Matrices and division by zero z/0=0,
Advances in Linear Algebra \& Matrix Theory, 2016, 6, 51-58
Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt
\\ http://dx.doi.org/10.4236/alamt.2016.62007.
\bibitem{ms18}
T. Matsuura and S. Saitoh,
Division by zero calculus and singular integrals. (Submitted for publication)
\bibitem{mms18}
T. Matsuura, H. Michiwaki and S. Saitoh,
$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.
\bibitem{msy15}
H. Michiwaki, S. Saitoh and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mos17}
H. Michiwaki, H. Okumura and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces,
International Journal of Mathematics and Computation, 28(2017); Issue 1, 2017), 1-16.
\bibitem{osm17}
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), 1(2017), 70-77.
\bibitem{ps18}
S. Pinelas and S. Saitoh,
Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.
\bibitem{s14}
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{s16}
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .
\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.
\bibitem{ann326}
Announcement 326 (2016.10.17): The division by zero z/0=0 - its impact to human beings through education and research.
\bibitem{ann352}
Announcement 352(2017.2.2): On the third birthday of the division by zero z/0=0.
\bibitem{ann354}
Announcement 354(2017.2.8): What are $n = 2,1,0$ regular polygons inscribed in a disc? -- relations of $0$ and infinity.
\bibitem{362}
Announcement 362(2017.5.5): Discovery of the division by zero as $0/0=1/0=z/0=0$
\bibitem{380}
Announcement 380 (2017.8.21): What is the zero?
\end{thebibliography}
\end{document}
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12276045402.html
1/0=0、0/0=0、z/0=0
ソクラテス・プラトン・アリストテレス その他
ゼロ除算の論文リスト：
List of division by zero:
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
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.
T. Matsuura and S. Saitoh,
Matrices and division by zero z/0=0,
Advances in Linear Algebra \& Matrix Theory, 2016, 6, 51-58
Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt
\\ http://dx.doi.org/10.4236/alamt.201....
T. Matsuura and S. Saitoh,
Division by zero calculus and singular integrals. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.)
T. Matsuura, H. Michiwaki and S. Saitoh,
$\log 0= \log \infty =0$ and applications. (Submitted for publication).
H. Michiwaki, S. Saitoh and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1....
H. Michiwaki, H. Okumura and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces,
International Journal of Mathematics and Computation, 28(2017); Issue 1, 2017), 1-16.
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), 1(2017), 70-77.
S. Pinelas and S. Saitoh,
Division by zero calculus and differential equations. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics).
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/
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .
List of division by zero:
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
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.
T. Matsuura and S. Saitoh,
Matrices and division by zero z/0=0,
Advances in Linear Algebra \& Matrix Theory, 2016, 6, 51-58
Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt
\\ http://dx.doi.org/10.4236/alamt.201....
T. Matsuura and S. Saitoh,
Division by zero calculus and singular integrals. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.)
T. Matsuura, H. Michiwaki and S. Saitoh,
$\log 0= \log \infty =0$ and applications. (Submitted for publication).
H. Michiwaki, S. Saitoh and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1....
H. Michiwaki, H. Okumura and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces,
International Journal of Mathematics and Computation, 28(2017); Issue 1, 2017), 1-16.
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), 1(2017), 70-77.
S. Pinelas and S. Saitoh,
Division by zero calculus and differential equations. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics).
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/
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
ドキュメンタリー 2017: 神の数式 第２回 宇宙はなぜ生まれたのか
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ゼロ除算の論文
Mysterious Properties of the Point at Infinity
Mysterious Properties of the Point at Infinity
Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces
Author: Jakub Czajko, 92(2) (2018) 171-197
WSN 92(2) (2018) 171-197
Author: Jakub Czajko, 92(2) (2018) 171-197
WSN 92(2) (2018) 171-197 
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