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.

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.

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 CO 2 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 CO 2 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 CO 2 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 CO 2 . 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.